diff --git a/previews/PR328/.documenter-siteinfo.json b/previews/PR328/.documenter-siteinfo.json index 2ca8f8da2..9c4df3cc0 100644 --- a/previews/PR328/.documenter-siteinfo.json +++ b/previews/PR328/.documenter-siteinfo.json @@ -1 +1 @@ -{"documenter":{"julia_version":"1.10.5","generation_timestamp":"2024-09-24T06:02:43","documenter_version":"1.7.0"}} \ No newline at end of file +{"documenter":{"julia_version":"1.10.5","generation_timestamp":"2024-09-24T13:53:40","documenter_version":"1.7.0"}} \ No newline at end of file diff --git a/previews/PR328/benchmarks/hodgkinhuxley/index.html b/previews/PR328/benchmarks/hodgkinhuxley/index.html index 1893e5571..73753cd69 100644 --- a/previews/PR328/benchmarks/hodgkinhuxley/index.html +++ b/previews/PR328/benchmarks/hodgkinhuxley/index.html @@ -653,4 +653,4 @@ [8e850b90] libblastrampoline_jll v5.8.0+1 [8e850ede] nghttp2_jll v1.52.0+1 [3f19e933] p7zip_jll v17.4.0+2 -Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m` +Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m` diff --git a/previews/PR328/benchmarks/lotkavolterra/index.html b/previews/PR328/benchmarks/lotkavolterra/index.html index 691d9341d..e1dbb9a5b 100644 --- a/previews/PR328/benchmarks/lotkavolterra/index.html +++ b/previews/PR328/benchmarks/lotkavolterra/index.html @@ -748,4 +748,4 @@ [8e850b90] libblastrampoline_jll v5.8.0+1 [8e850ede] nghttp2_jll v1.52.0+1 [3f19e933] p7zip_jll v17.4.0+2 -Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m` +Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m` diff --git a/previews/PR328/benchmarks/multi-language-wrappers/index.html b/previews/PR328/benchmarks/multi-language-wrappers/index.html index 2240b944b..8309b5d7f 100644 --- a/previews/PR328/benchmarks/multi-language-wrappers/index.html +++ b/previews/PR328/benchmarks/multi-language-wrappers/index.html @@ -692,4 +692,4 @@ [8e850b90] libblastrampoline_jll v5.8.0+1 [8e850ede] nghttp2_jll v1.52.0+1 [3f19e933] p7zip_jll v17.4.0+2 -Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m` +Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m` diff --git a/previews/PR328/benchmarks/orego/index.html b/previews/PR328/benchmarks/orego/index.html index 83c1dd2c9..9947be920 100644 --- a/previews/PR328/benchmarks/orego/index.html +++ b/previews/PR328/benchmarks/orego/index.html @@ -521,4 +521,4 @@ [8e850b90] libblastrampoline_jll v5.8.0+1 [8e850ede] nghttp2_jll v1.52.0+1 [3f19e933] p7zip_jll v17.4.0+2 -Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m` +Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m` diff --git a/previews/PR328/benchmarks/pleiades/index.html b/previews/PR328/benchmarks/pleiades/index.html index ff179e353..c10fe55a4 100644 --- a/previews/PR328/benchmarks/pleiades/index.html +++ b/previews/PR328/benchmarks/pleiades/index.html @@ -564,4 +564,4 @@ [8e850b90] libblastrampoline_jll v5.8.0+1 [8e850ede] nghttp2_jll v1.52.0+1 [3f19e933] p7zip_jll v17.4.0+2 -Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m` +Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m` diff --git a/previews/PR328/benchmarks/rober/index.html b/previews/PR328/benchmarks/rober/index.html index 5508026ea..835ca3989 100644 --- a/previews/PR328/benchmarks/rober/index.html +++ b/previews/PR328/benchmarks/rober/index.html @@ -517,4 +517,4 @@ [8e850b90] libblastrampoline_jll v5.8.0+1 [8e850ede] nghttp2_jll v1.52.0+1 [3f19e933] p7zip_jll v17.4.0+2 -Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m` +Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m` diff --git a/previews/PR328/benchmarks/vanderpol/index.html b/previews/PR328/benchmarks/vanderpol/index.html index 97bcac0d2..31de0e812 100644 --- a/previews/PR328/benchmarks/vanderpol/index.html +++ b/previews/PR328/benchmarks/vanderpol/index.html @@ -658,4 +658,4 @@ [8e850b90] libblastrampoline_jll v5.8.0+1 [8e850ede] nghttp2_jll v1.52.0+1 [3f19e933] p7zip_jll v17.4.0+2 -Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m` +Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m` diff --git a/previews/PR328/diffusions/index.html b/previews/PR328/diffusions/index.html index 9e2545e6e..bbbf98d1e 100644 --- a/previews/PR328/diffusions/index.html +++ b/previews/PR328/diffusions/index.html @@ -5,4 +5,4 @@ \text{d} Y^{(i)}(t) &= Y^{(i+1)}(t) \ \text{d}t, \qquad i = 0, \dots, q-1, \\ \text{d} Y^{(q)}(t) &= \textcolor{#389826}{A} Y(t) \ \text{d}t + \textcolor{#4063D8}{\Gamma} \ \text{d}W(t), \\ Y(0) &\sim \textcolor{purple}{ \mathcal{N} \left( \mu_0, \Sigma_0 \right) }. -\end{aligned}\]

Then $Y^{(i)}(t)$ models the $i$-th derivative of $y(t)$. In this section, we consider choices relating to the "diffusion" $\textcolor{#4063D8}{\Gamma}$. If you're more interested in the drift matrix $\textcolor{#389826}{A}$ check out the Priors section, and for info on the initial distribution $\textcolor{purple}{ \mathcal{N} \left( \mu_0, \Sigma_0 \right) }$ check out the Initialization section.

Diffusion and calibration

We call $\textcolor{#4063D8}{\Gamma}$ the "diffusion" parameter. Since it is typically not known we need to estimate it; this is called "calibration".

ProbNumDiffEq.jl provides a few different choices for how to model and estimate $\textcolor{#4063D8}{\Gamma}$:

Or more compactly:

Isotropic:Diagonal (only for the EK0)
Time-varyingDynamicDiffusionDynamicMVDiffusion
Time-fixedFixedDiffusionFixedMVDiffusion

For more details on diffusions and calibration, check out this paper [9].

API

ProbNumDiffEq.DynamicDiffusion — Type
DynamicDiffusion()

Time-varying, isotropic diffusion, which is quasi-maximum-likelihood-estimated at each step.

This is the recommended diffusion when using adaptive step-size selection, and in particular also when solving stiff systems.

source
ProbNumDiffEq.FixedDiffusion — Type
FixedDiffusion(; initial_diffusion=1.0, calibrate=true)

Time-fixed, isotropic diffusion, which is (optionally) quasi-maximum-likelihood-estimated.

This is the recommended diffusion when using fixed steps.

By default with calibrate=true, all covariances are re-scaled at the end of the solve with the MLE diffusion. Set calibrate=false to skip this step, e.g. when setting the initial_diffusion and then estimating the diffusion outside of the solver (e.g. with Fenrir.jl).

source
ProbNumDiffEq.DynamicMVDiffusion — Type
DynamicMVDiffusion()

Time-varying, diagonal diffusion, which is quasi-maximum-likelihood-estimated at each step.

Only works with the EK0!

A multi-variate version of DynamicDiffusion, where instead of an isotropic matrix, a diagonal matrix is estimated. This can be helpful to get more expressive posterior covariances when using the EK0, since the individual dimensions can be adjusted separately.

References

  • [9] Bosch et al, "Calibrated Adaptive Probabilistic ODE Solvers", AISTATS (2021)
source
ProbNumDiffEq.FixedMVDiffusion — Type
FixedMVDiffusion(; initial_diffusion=1.0, calibrate=true)

Time-fixed, diagonal diffusion, which is quasi-maximum-likelihood-estimated at each step.

Only works with the EK0!

A multi-variate version of FixedDiffusion, where instead of an isotropic matrix, a diagonal matrix is estimated. This can be helpful to get more expressive posterior covariances when using the EK0, since the individual dimensions can be adjusted separately.

References

  • [9] Bosch et al, "Calibrated Adaptive Probabilistic ODE Solvers", AISTATS (2021)
source

References

[9]
N. Bosch, P. Hennig and F. Tronarp. Calibrated Adaptive Probabilistic ODE Solvers. In: Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, Vol. 130 of Proceedings of Machine Learning Research, edited by A. Banerjee and K. Fukumizu (PMLR, 13–15 Apr 2021); pp. 3466–3474.
+\end{aligned}\]

Then $Y^{(i)}(t)$ models the $i$-th derivative of $y(t)$. In this section, we consider choices relating to the "diffusion" $\textcolor{#4063D8}{\Gamma}$. If you're more interested in the drift matrix $\textcolor{#389826}{A}$ check out the Priors section, and for info on the initial distribution $\textcolor{purple}{ \mathcal{N} \left( \mu_0, \Sigma_0 \right) }$ check out the Initialization section.

Diffusion and calibration

We call $\textcolor{#4063D8}{\Gamma}$ the "diffusion" parameter. Since it is typically not known we need to estimate it; this is called "calibration".

ProbNumDiffEq.jl provides a few different choices for how to model and estimate $\textcolor{#4063D8}{\Gamma}$:

Or more compactly:

Isotropic:Diagonal (only for the EK0)
Time-varyingDynamicDiffusionDynamicMVDiffusion
Time-fixedFixedDiffusionFixedMVDiffusion

For more details on diffusions and calibration, check out this paper [9].

API

ProbNumDiffEq.DynamicDiffusion — Type
DynamicDiffusion()

Time-varying, isotropic diffusion, which is quasi-maximum-likelihood-estimated at each step.

This is the recommended diffusion when using adaptive step-size selection, and in particular also when solving stiff systems.

source
ProbNumDiffEq.FixedDiffusion — Type
FixedDiffusion(; initial_diffusion=1.0, calibrate=true)

Time-fixed, isotropic diffusion, which is (optionally) quasi-maximum-likelihood-estimated.

This is the recommended diffusion when using fixed steps.

By default with calibrate=true, all covariances are re-scaled at the end of the solve with the MLE diffusion. Set calibrate=false to skip this step, e.g. when setting the initial_diffusion and then estimating the diffusion outside of the solver (e.g. with Fenrir.jl).

source
ProbNumDiffEq.DynamicMVDiffusion — Type
DynamicMVDiffusion()

Time-varying, diagonal diffusion, which is quasi-maximum-likelihood-estimated at each step.

Only works with the EK0!

A multi-variate version of DynamicDiffusion, where instead of an isotropic matrix, a diagonal matrix is estimated. This can be helpful to get more expressive posterior covariances when using the EK0, since the individual dimensions can be adjusted separately.

References

  • [9] Bosch et al, "Calibrated Adaptive Probabilistic ODE Solvers", AISTATS (2021)
source
ProbNumDiffEq.FixedMVDiffusion — Type
FixedMVDiffusion(; initial_diffusion=1.0, calibrate=true)

Time-fixed, diagonal diffusion, which is quasi-maximum-likelihood-estimated at each step.

Only works with the EK0!

A multi-variate version of FixedDiffusion, where instead of an isotropic matrix, a diagonal matrix is estimated. This can be helpful to get more expressive posterior covariances when using the EK0, since the individual dimensions can be adjusted separately.

References

  • [9] Bosch et al, "Calibrated Adaptive Probabilistic ODE Solvers", AISTATS (2021)
source

References

[9]
N. Bosch, P. Hennig and F. Tronarp. Calibrated Adaptive Probabilistic ODE Solvers. In: Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, Vol. 130 of Proceedings of Machine Learning Research, edited by A. Banerjee and K. Fukumizu (PMLR, 13–15 Apr 2021); pp. 3466–3474.
diff --git a/previews/PR328/filtering/index.html b/previews/PR328/filtering/index.html index cc158592d..51d33b7da 100644 --- a/previews/PR328/filtering/index.html +++ b/previews/PR328/filtering/index.html @@ -1,26 +1,26 @@ -Filtering and Smoothing · ProbNumDiffEq.jl
GitHub

Gaussian Filtering and Smoothing

Predict

ProbNumDiffEq.predict — Function
predict(x::Gaussian, A::AbstractMatrix, Q::AbstractMatrix)

Prediction step in Kalman filtering for linear dynamics models.

Given a Gaussian $x = \mathcal{N}(μ, Σ)$, compute and return $\mathcal{N}(A μ, A Σ A^T + Q)$.

See also the non-allocating square-root version predict!.

source
ProbNumDiffEq.predict! — Function
predict!(x_out, x_curr, Ah, Qh, cachemat)

In-place and square-root implementation of predict which saves the result into x_out.

Only works with PSDMatrices.PSDMatrix types as Ah, Qh, and in the covariances of x_curr and x_out (both of type Gaussian). To prevent allocations, a cache matrix cachemat of size $D \times 2D$ (where $D \times D$ is the size of Ah and Qh) needs to be passed.

See also: predict.

source

Update

ProbNumDiffEq.update — Function
update(x, measurement, H)

Update step in Kalman filtering for linear dynamics models.

Given a Gaussian $x = \mathcal{N}(μ, Σ)$ and a measurement $z = \mathcal{N}(\hat{z}, S)$, with $S = H Σ H^T$, compute

\[\begin{aligned} +Filtering and Smoothing · ProbNumDiffEq.jl

GitHub

Gaussian Filtering and Smoothing

Predict

ProbNumDiffEq.predict — Function
predict(x::Gaussian, A::AbstractMatrix, Q::AbstractMatrix)

Prediction step in Kalman filtering for linear dynamics models.

Given a Gaussian $x = \mathcal{N}(μ, Σ)$, compute and return $\mathcal{N}(A μ, A Σ A^T + Q)$.

See also the non-allocating square-root version predict!.

source
ProbNumDiffEq.predict! — Function
predict!(x_out, x_curr, Ah, Qh, cachemat)

In-place and square-root implementation of predict which saves the result into x_out.

Only works with PSDMatrices.PSDMatrix types as Ah, Qh, and in the covariances of x_curr and x_out (both of type Gaussian). To prevent allocations, a cache matrix cachemat of size $D \times 2D$ (where $D \times D$ is the size of Ah and Qh) needs to be passed.

See also: predict.

source

Update

ProbNumDiffEq.update — Function
update(x, measurement, H)

Update step in Kalman filtering for linear dynamics models.

Given a Gaussian $x = \mathcal{N}(μ, Σ)$ and a measurement $z = \mathcal{N}(\hat{z}, S)$, with $S = H Σ H^T$, compute

\[\begin{aligned} K &= Σ^P H^T S^{-1}, \\ μ^F &= μ + K (0 - \hat{z}), \\ Σ^F &= Σ - K S K^T, -\end{aligned}\]

and return an updated state \mathcal{N}(μ^F, Σ^F). Note that this assumes zero-measurements. When called with ProbNumDiffEq.SquarerootMatrix type arguments it performs the update in Joseph / square-root form.

For better performance, we recommend to use the non-allocating update!.

source
ProbNumDiffEq.update! — Function
update!(x_out, x_pred, measurement, H, K_cache, M_cache, S_cache)

In-place and square-root implementation of update which saves the result into x_out.

Implemented in Joseph Form to retain the PSDMatrix covariances:

\[\begin{aligned} +\end{aligned}\]

and return an updated state \mathcal{N}(μ^F, Σ^F). Note that this assumes zero-measurements. When called with ProbNumDiffEq.SquarerootMatrix type arguments it performs the update in Joseph / square-root form.

For better performance, we recommend to use the non-allocating update!.

source
ProbNumDiffEq.update! — Function
update!(x_out, x_pred, measurement, H, K_cache, M_cache, S_cache)

In-place and square-root implementation of update which saves the result into x_out.

Implemented in Joseph Form to retain the PSDMatrix covariances:

\[\begin{aligned} K &= Σ^P H^T S^{-1}, \\ μ^F &= μ + K (0 - \hat{z}), \\ \sqrt{Σ}^F &= (I - KH) \sqrt(Σ), -\end{aligned}\]

where $\sqrt{M}$ denotes the left square-root of a matrix M, i.e. $M = \sqrt{M} \sqrt{M}^T$.

To prevent allocations, write into caches K_cache and M_cache, both of size D × D, and S_cache of same type as measurement.Σ.

See also: update.

source

Smooth

ProbNumDiffEq.smooth — Function
smooth(x_curr, x_next_smoothed, A, Q)

Update step of the Kalman smoother, aka. Rauch-Tung-Striebel smoother, for linear dynamics models.

Given Gaussians $x_n = \mathcal{N}(μ_{n}, Σ_{n})$ and $x_{n+1} = \mathcal{N}(μ_{n+1}^S, Σ_{n+1}^S)$, compute

\[\begin{aligned} +\end{aligned}\]

where $\sqrt{M}$ denotes the left square-root of a matrix M, i.e. $M = \sqrt{M} \sqrt{M}^T$.

To prevent allocations, write into caches K_cache and M_cache, both of size D × D, and S_cache of same type as measurement.Σ.

See also: update.

source

Smooth

ProbNumDiffEq.smooth — Function
smooth(x_curr, x_next_smoothed, A, Q)

Update step of the Kalman smoother, aka. Rauch-Tung-Striebel smoother, for linear dynamics models.

Given Gaussians $x_n = \mathcal{N}(μ_{n}, Σ_{n})$ and $x_{n+1} = \mathcal{N}(μ_{n+1}^S, Σ_{n+1}^S)$, compute

\[\begin{aligned} μ_{n+1}^P &= A μ_n^F, \\ P_{n+1}^P &= A Σ_n^F A + Q, \\ G &= Σ_n^S A^T (Σ_{n+1}^P)^{-1}, \\ μ_n^S &= μ_n^F + G (μ_{n+1}^S - μ_{n+1}^P), \\ Σ_n^S &= (I - G A) Σ_n^F (I - G A)^T + G Q G^T + G Σ_{n+1}^S G^T, -\end{aligned}\]

and return a smoothed state \mathcal{N}(μ_n^S, Σ_n^S). When called with ProbNumDiffEq.SquarerootMatrix type arguments it performs the update in Joseph / square-root form.

source

Markov Kernels

ProbNumDiffEq.AffineNormalKernel — Type
AffineNormalKernel(A[, b], C)

Structure to represent affine Normal Markov kernels, i.e. conditional distributions of the form

\[\begin{aligned} +\end{aligned}\]

and return a smoothed state \mathcal{N}(μ_n^S, Σ_n^S). When called with ProbNumDiffEq.SquarerootMatrix type arguments it performs the update in Joseph / square-root form.

source

Markov Kernels

ProbNumDiffEq.AffineNormalKernel — Type
AffineNormalKernel(A[, b], C)

Structure to represent affine Normal Markov kernels, i.e. conditional distributions of the form

\[\begin{aligned} y \mid x \sim \mathcal{N} \left( y; A x + b, C \right). -\end{aligned}\]

At the point of writing, AffineNormalKernels are only used to precompute and store the backward representation of the posterior (via compute_backward_kernel!) and for smoothing (via marginalize!).

source
ProbNumDiffEq.marginalize! — Function
marginalize!(
     xout::Gaussian{Vector{T},PSDMatrix{T,S}}
     x::Gaussian{Vector{T},PSDMatrix{T,S}},
     K::AffineNormalKernel{<:AbstractMatrix,Union{<:Number,<:AbstractVector,Missing},<:PSDMatrix};
     C_DxD, C_3DxD
-)

Basically the same as predict!), but in kernel language and with support for affine transitions. At the time of writing, this is only used to smooth the posterior using it's backward representation, where the kernels are precomputed with compute_backward_kernel!.

Note that this function assumes certain shapes:

  • size(x.μ) == (D, D)
  • size(x.Σ) == (D, D)
  • size(K.A) == (D, D)
  • size(K.b) == (D,), or missing
  • size(K.C) == (D, D), _but with a tall square-root size(K.C.R) == (3D, D)

xout is assumes to have the same shapes as x.

source
ProbNumDiffEq.compute_backward_kernel! — Function
compute_backward_kernel!(Kout, xpred, x, K; C_DxD[, diffusion=1])

Compute the backward representation of the posterior, i.e. the conditional distribution of the current state given the next state and the transition kernel.

More precisely, given a distribution (x)

\[\begin{aligned} +)

Basically the same as predict!), but in kernel language and with support for affine transitions. At the time of writing, this is only used to smooth the posterior using it's backward representation, where the kernels are precomputed with compute_backward_kernel!.

Note that this function assumes certain shapes:

  • size(x.μ) == (D, D)
  • size(x.Σ) == (D, D)
  • size(K.A) == (D, D)
  • size(K.b) == (D,), or missing
  • size(K.C) == (D, D), _but with a tall square-root size(K.C.R) == (3D, D)

xout is assumes to have the same shapes as x.

source
ProbNumDiffEq.compute_backward_kernel! — Function
compute_backward_kernel!(Kout, xpred, x, K; C_DxD[, diffusion=1])

Compute the backward representation of the posterior, i.e. the conditional distribution of the current state given the next state and the transition kernel.

More precisely, given a distribution (x)

\[\begin{aligned} x \sim \mathcal{N} \left( x; μ, Σ \right), \end{aligned}\]

a kernel (K)

\[\begin{aligned} y \mid x \sim \mathcal{N} \left( y; A x + b, C \right), @@ -34,4 +34,4 @@ G &= Σ A^\top (Σ^P)^{-1}, \\ d &= μ - G μ^P, \\ Λ &= Σ - G Σ^P G^\top. -\end{aligned}\]

Everything is computed in square-root form and with minimal allocations (thus the cache C_DxD), so the actual formulas implemented here differ a bit.

The resulting backward kernels are used to smooth the posterior, via marginalize!.

source
+\end{aligned}\]

Everything is computed in square-root form and with minimal allocations (thus the cache C_DxD), so the actual formulas implemented here differ a bit.

The resulting backward kernels are used to smooth the posterior, via marginalize!.

source
diff --git a/previews/PR328/implementation/index.html b/previews/PR328/implementation/index.html index d9bd658c3..31720a8e0 100644 --- a/previews/PR328/implementation/index.html +++ b/previews/PR328/implementation/index.html @@ -1,2 +1,2 @@ -Implementation via OrdinaryDiffEq.jl · ProbNumDiffEq.jl
GitHub

Solver Implementation via OrdinaryDiffEq.jl

ProbNumDiffEq.jl builds directly on OrdinaryDiffEq.jl to benefit from its iterator interface, flexible step-size control, and efficient Jacobian calculations. But, this requires extending non-public APIs. This page is meant to provide an overview on which parts exactly ProbNumDiffEq.jl builds on.

For more discussion on the pros and cons of building on OrdinaryDiffEq.jl, see this thread on discourse.

Building on OrdinaryDiffEq.jl

ProbNumDiffEq.jl shares most of OrdinaryDiffEq.jl's implementation. In particular:

  1. OrdinaryDiffEq.__init builds the cache and the integrator, and calls OrdinaryDiffEq.initialize!
  2. OrdinaryDiffEq.solve! implements the actual iterator structure, with
    • OrdinaryDiffEq.loopheader!
    • OrdinaryDiffEq.perform_step!
    • OrdinaryDiffEq.loopfooter!
    • OrdinaryDiffEq.postamble!

ProbNumDiffEq.jl builds around this structure and overloads some of the parts:

  • Algorithms: EK0/EK1 <: AbstractEK <: OrdinaryDiffEq.OrdinaryDiffEqAdaptiveAlgorithm
    • ./src/algorithms.jl provides the algorithms themselves
    • ./src/alg_utils.jl implements many traits (relating to automatic differentiation, implicitness, step-size control, etc)
  • Cache: EKCache <: AbstractODEFilterCache <: OrdinaryDiffEq.OrdinaryDiffEqCache
    • ./src/caches.jl implements the cache and its main constructor: OrdinaryDiffEq.alg_cache
  • Initialization and perform_step!: via OrdinaryDiffEq.initialize! and OrdinaryDiffEq.perform_step!. Implemented in ./src/perform_step.jl.
  • Custom postamble by overloading OrdinaryDiffEq.postamble! (which should always call OrdinaryDiffEq._postamble!). This is where we do the "smoothing" of the solution. Implemented in ./src/integrator_utils.jl.
  • Custom saving by overloading OrdinaryDiffEq.savevalues! (which should always call OrdinaryDiffEq._savevalues!). Implemented in ./src/integrator_utils.jl.

Building on DiffEqBase.jl

  • DiffEqBase.__init is currently overloaded to transform OOP problems into IIP problems (in ./src/solve.jl).
  • The solution object: ProbODESolution <: AbstractProbODESolution <: DiffEqBase.AbstractODESolution
    • ./src/solution.jl implements the main parts. Note that the main constructor DiffEqBase.build_solution is called by OrdinaryDiffEq.__init, so OrdinaryDiffEq.jl has control over its inputs.
    • MeanProbODESolution <: DiffEqBase.AbstractODESolution is a wrapper that allows handling the mean of a probabilistic ODE solution the same way one would handle any "standard" ODE solution, by just ignoring the covariances.
    • AbstractODEFilterPosterior <: DiffEqBase.AbstractDiffEqInterpolation handles the interpolation.
    • Plot recipe in ./ext/RecipesBaseExt.jl
    • Sampling in ./src/solution_sampling.jl
  • DiffEqBase.prepare_alg(::EK1{0}); closely follows a similar function implemented in OrdinaryDiffEq.jl ./src/alg_utils.jl
    • this also required DiffEqBase.remake(::EK1)

Other packages

  • DiffEqDevTools.appxtrue: We extend this function to work with ProbODESolution. This also enables DiffEqDevTools.WorkPrecision to work out of the box.
+Implementation via OrdinaryDiffEq.jl · ProbNumDiffEq.jl
GitHub

Solver Implementation via OrdinaryDiffEq.jl

ProbNumDiffEq.jl builds directly on OrdinaryDiffEq.jl to benefit from its iterator interface, flexible step-size control, and efficient Jacobian calculations. But, this requires extending non-public APIs. This page is meant to provide an overview on which parts exactly ProbNumDiffEq.jl builds on.

For more discussion on the pros and cons of building on OrdinaryDiffEq.jl, see this thread on discourse.

Building on OrdinaryDiffEq.jl

ProbNumDiffEq.jl shares most of OrdinaryDiffEq.jl's implementation. In particular:

  1. OrdinaryDiffEq.__init builds the cache and the integrator, and calls OrdinaryDiffEq.initialize!
  2. OrdinaryDiffEq.solve! implements the actual iterator structure, with
    • OrdinaryDiffEq.loopheader!
    • OrdinaryDiffEq.perform_step!
    • OrdinaryDiffEq.loopfooter!
    • OrdinaryDiffEq.postamble!

ProbNumDiffEq.jl builds around this structure and overloads some of the parts:

  • Algorithms: EK0/EK1 <: AbstractEK <: OrdinaryDiffEq.OrdinaryDiffEqAdaptiveAlgorithm
    • ./src/algorithms.jl provides the algorithms themselves
    • ./src/alg_utils.jl implements many traits (relating to automatic differentiation, implicitness, step-size control, etc)
  • Cache: EKCache <: AbstractODEFilterCache <: OrdinaryDiffEq.OrdinaryDiffEqCache
    • ./src/caches.jl implements the cache and its main constructor: OrdinaryDiffEq.alg_cache
  • Initialization and perform_step!: via OrdinaryDiffEq.initialize! and OrdinaryDiffEq.perform_step!. Implemented in ./src/perform_step.jl.
  • Custom postamble by overloading OrdinaryDiffEq.postamble! (which should always call OrdinaryDiffEq._postamble!). This is where we do the "smoothing" of the solution. Implemented in ./src/integrator_utils.jl.
  • Custom saving by overloading OrdinaryDiffEq.savevalues! (which should always call OrdinaryDiffEq._savevalues!). Implemented in ./src/integrator_utils.jl.

Building on DiffEqBase.jl

  • DiffEqBase.__init is currently overloaded to transform OOP problems into IIP problems (in ./src/solve.jl).
  • The solution object: ProbODESolution <: AbstractProbODESolution <: DiffEqBase.AbstractODESolution
    • ./src/solution.jl implements the main parts. Note that the main constructor DiffEqBase.build_solution is called by OrdinaryDiffEq.__init, so OrdinaryDiffEq.jl has control over its inputs.
    • MeanProbODESolution <: DiffEqBase.AbstractODESolution is a wrapper that allows handling the mean of a probabilistic ODE solution the same way one would handle any "standard" ODE solution, by just ignoring the covariances.
    • AbstractODEFilterPosterior <: DiffEqBase.AbstractDiffEqInterpolation handles the interpolation.
    • Plot recipe in ./ext/RecipesBaseExt.jl
    • Sampling in ./src/solution_sampling.jl
  • DiffEqBase.prepare_alg(::EK1{0}); closely follows a similar function implemented in OrdinaryDiffEq.jl ./src/alg_utils.jl
    • this also required DiffEqBase.remake(::EK1)

Other packages

  • DiffEqDevTools.appxtrue: We extend this function to work with ProbODESolution. This also enables DiffEqDevTools.WorkPrecision to work out of the box.
diff --git a/previews/PR328/index.html b/previews/PR328/index.html index cff6716cd..38fb2dbec 100644 --- a/previews/PR328/index.html +++ b/previews/PR328/index.html @@ -1,3 +1,3 @@ Home · ProbNumDiffEq.jl
GitHub

Probabilistic Numerical Differential Equation Solvers

Banner

ProbNumDiffEq.jl provides probabilistic numerical solvers to the DifferentialEquations.jl ecosystem. The implemented ODE filters solve differential equations via Bayesian filtering and smoothing and compute not just a single point estimate of the true solution, but a posterior distribution that contains an estimate of its numerical approximation error.

For a short intro video, check out our poster presentation at JuliaCon2021.

Installation

Run Julia, enter ] to bring up Julia's package manager, and add the ProbNumDiffEq.jl package:

julia> ]
-(v1.10) pkg> add ProbNumDiffEq

Getting Started

For a quick introduction check out the "Solving ODEs with Probabilistic Numerics" tutorial.

Features

  • probdiffeq: Fast and feature-rich filtering-based probabilistic ODE solvers in JAX.
  • ProbNum: Probabilistic numerics in Python. It has not only probabilistic ODE solvers, but also probabilistic linear solvers, Bayesian quadrature, and many filtering and smoothing implementations.
+(v1.10) pkg> add ProbNumDiffEq

Getting Started

For a quick introduction check out the "Solving ODEs with Probabilistic Numerics" tutorial.

Features

diff --git a/previews/PR328/initialization/index.html b/previews/PR328/initialization/index.html index 3e92ba2b0..b4c094992 100644 --- a/previews/PR328/initialization/index.html +++ b/previews/PR328/initialization/index.html @@ -11,4 +11,4 @@ \end{aligned}\]

It is clear that this contains quite some information for $Y(0)$: The initial value $y_0$ and the vector field $f$ imply

\[\begin{aligned} Y^{(0)}(0) &= y_0, \\ Y^{(1)}(0) &= f(y_0, 0). -\end{aligned}\]

It turns out that we can also compute higher-order derivatives of $y$ with the chain rule, and then use these to better initialize $Y^{(i)}(0)$. This, done efficiently with Taylor-mode automatic differentiation by using TaylorIntegration.jl, is what TaylorModeInit does. See also [1].

In the vast majority of cases, just stick to the exact Taylor-mode initialization TaylorModeInit!

API

ProbNumDiffEq.TaylorModeInit — Type
TaylorModeInit(order)

Exact initialization via Taylor-mode automatic differentiation up to order order.

This is the recommended initialization method!

It uses TaylorIntegration.jl to efficiently compute the higher-order derivatives of the solution at the initial value, via Taylor-mode automatic differentiation.

In some special cases it can happen that TaylorIntegration.jl is incompatible with the given problem (typically because the problem definition does not allow for elements of type Taylor). If this happens, try one of SimpleInit, ForwardDiffInit (for low enough orders), ClassicSolverInit.

References

  • [7] Krämer et al, "Stable Implementation of Probabilistic ODE Solvers" (2020)
source
ProbNumDiffEq.ForwardDiffInit — Type
ForwardDiffInit(order)

Exact initialization via ForwardDiff.jl up to order order.

Warning: This does not scale well to high orders! For orders > 3, TaylorModeInit most likely performs better.

source
ProbNumDiffEq.SimpleInit — Type
SimpleInit()

Simple initialization, only with the given initial value and derivative.

The remaining derivatives are set to zero with unit covariance (unless specified otherwise by setting a custom FixedDiffusion).

source
ProbNumDiffEq.ClassicSolverInit — Type
ClassicSolverInit(; alg=OrdinaryDiffEqCore.Tsit5(), init_on_ddu=false)

Initialization via regression on a few steps of a classic ODE solver.

In a nutshell, instead of specifying $\mu_0$ exactly and setting $\Sigma_0=0$ (which is what TaylorModeInit does), use a classic ODE solver to compute a few steps of the solution, and then regress on the computed values (by running a smoother) to compute $\mu_0$ and $\Sigma_0$ as the mean and covariance of the smoothing posterior at time 0. See also [2].

The initial value and derivative are set directly from the given initial value problem; optionally the second derivative can also be set via automatic differentiation by setting init_on_ddu=true.

Arguments

  • alg: The solver to be used. Can be any solver from OrdinaryDiffEqCore.jl.
  • init_on_ddu: If true, the second derivative is also initialized exactly via automatic differentiation with ForwardDiff.jl.

References

  • [7] Krämer et al, "Stable Implementation of Probabilistic ODE Solvers" (2020)
  • [8] Schober et al, "A probabilistic model for the numerical solution of initial value problems", Statistics and Computing (2019)
source

References

[7]
N. Krämer and P. Hennig. Stable Implementation of Probabilistic ODE Solvers. CoRR (2020), arXiv:2012.10106 [stat.ML].
[8]
M. Schober, S. Särkkä and P. Hennig. A probabilistic model for the numerical solution of initial value problems. Statistics and Computing 29, 99–122 (2019).
+\end{aligned}\]

It turns out that we can also compute higher-order derivatives of $y$ with the chain rule, and then use these to better initialize $Y^{(i)}(0)$. This, done efficiently with Taylor-mode automatic differentiation by using TaylorIntegration.jl, is what TaylorModeInit does. See also [1].

In the vast majority of cases, just stick to the exact Taylor-mode initialization TaylorModeInit!

API

ProbNumDiffEq.TaylorModeInit — Type
TaylorModeInit(order)

Exact initialization via Taylor-mode automatic differentiation up to order order.

This is the recommended initialization method!

It uses TaylorIntegration.jl to efficiently compute the higher-order derivatives of the solution at the initial value, via Taylor-mode automatic differentiation.

In some special cases it can happen that TaylorIntegration.jl is incompatible with the given problem (typically because the problem definition does not allow for elements of type Taylor). If this happens, try one of SimpleInit, ForwardDiffInit (for low enough orders), ClassicSolverInit.

References

  • [7] Krämer et al, "Stable Implementation of Probabilistic ODE Solvers" (2020)
source
ProbNumDiffEq.ForwardDiffInit — Type
ForwardDiffInit(order)

Exact initialization via ForwardDiff.jl up to order order.

Warning: This does not scale well to high orders! For orders > 3, TaylorModeInit most likely performs better.

source
ProbNumDiffEq.SimpleInit — Type
SimpleInit()

Simple initialization, only with the given initial value and derivative.

The remaining derivatives are set to zero with unit covariance (unless specified otherwise by setting a custom FixedDiffusion).

source
ProbNumDiffEq.ClassicSolverInit — Type
ClassicSolverInit(; alg=OrdinaryDiffEqCore.Tsit5(), init_on_ddu=false)

Initialization via regression on a few steps of a classic ODE solver.

In a nutshell, instead of specifying $\mu_0$ exactly and setting $\Sigma_0=0$ (which is what TaylorModeInit does), use a classic ODE solver to compute a few steps of the solution, and then regress on the computed values (by running a smoother) to compute $\mu_0$ and $\Sigma_0$ as the mean and covariance of the smoothing posterior at time 0. See also [2].

The initial value and derivative are set directly from the given initial value problem; optionally the second derivative can also be set via automatic differentiation by setting init_on_ddu=true.

Arguments

  • alg: The solver to be used. Can be any solver from OrdinaryDiffEqCore.jl.
  • init_on_ddu: If true, the second derivative is also initialized exactly via automatic differentiation with ForwardDiff.jl.

References

  • [7] Krämer et al, "Stable Implementation of Probabilistic ODE Solvers" (2020)
  • [8] Schober et al, "A probabilistic model for the numerical solution of initial value problems", Statistics and Computing (2019)
source

References

[7]
N. Krämer and P. Hennig. Stable Implementation of Probabilistic ODE Solvers. CoRR (2020), arXiv:2012.10106 [stat.ML].
[8]
M. Schober, S. Särkkä and P. Hennig. A probabilistic model for the numerical solution of initial value problems. Statistics and Computing 29, 99–122 (2019).
diff --git a/previews/PR328/likelihoods/index.html b/previews/PR328/likelihoods/index.html index 9924a2a2a..ff14f9611 100644 --- a/previews/PR328/likelihoods/index.html +++ b/previews/PR328/likelihoods/index.html @@ -8,7 +8,7 @@ data, kwargs... ) -

Compute the Fenrir [3] approximate negative log-likelihood (NLL) of the data.

This is a convenience function that

  1. Solves the ODE with a ProbNumDiffEq.EK1 of the specified order and with a diffusion as provided by the diffusion_var argument, and
  2. Fits the ODE posterior to the data via Kalman filtering and thereby computes the log-likelihood of the data on the way.

You can control the step-size behaviour of the solver as you would for a standard ODE solve, but additionally the solver always steps through the data.t locations by adding them to tstops.

You can also choose steps adaptively by setting adaptive=true, but this is not well-tested so use at your own risk!

Arguments

Reference

source
ProbNumDiffEq.dalton_data_loglik — Function
dalton_data_loglik(
+

Compute the Fenrir [3] approximate negative log-likelihood (NLL) of the data.

This is a convenience function that

  1. Solves the ODE with a ProbNumDiffEq.EK1 of the specified order and with a diffusion as provided by the diffusion_var argument, and
  2. Fits the ODE posterior to the data via Kalman filtering and thereby computes the log-likelihood of the data on the way.

You can control the step-size behaviour of the solver as you would for a standard ODE solve, but additionally the solver always steps through the data.t locations by adding them to tstops.

You can also choose steps adaptively by setting adaptive=true, but this is not well-tested so use at your own risk!

Arguments

  • prob::SciMLBase.AbstractODEProblem: the initial value problem of interest
  • alg::AbstractEK: the probabilistic ODE solver to be used; use EK1 for best results.
  • data::NamedTuple{(:t, :u)}: the data to be fitted
  • observation_matrix::Union{AbstractMatrix,UniformScaling}: the matrix which maps the ODE state to the measurements; typically a projection matrix
  • observation_noise_cov::Union{Number,AbstractMatrix}: the scalar observation noise variance

Reference

  • [3] Tronarp et al, "Fenrir: Physics-Enhanced Regression for Initial Value Problems", ICML (2022)
source
ProbNumDiffEq.dalton_data_loglik — Function
dalton_data_loglik(
     prob::SciMLBase.AbstractODEProblem,
     alg::ProbNumDiffEq.AbstractEK,
     args...;
@@ -17,4 +17,4 @@
     data,
     kwargs...
 )
-

Compute the DALTON [10] approximate negative log-likelihood (NLL) of the data.

You can control the step-size behaviour of the solver as you would for a standard ODE solve, but additionally the solver always steps through the data.t locations by adding them to tstops. You can also choose steps adaptively by setting adaptive=true, but this is not well-tested so use at your own risk!

Arguments

  • prob::SciMLBase.AbstractODEProblem: the initial value problem of interest
  • alg::AbstractEK: the probabilistic ODE solver to be used; use EK1 for best results.
  • data::NamedTuple{(:t, :u)}: the data to be fitted
  • observation_matrix::Union{AbstractMatrix,UniformScaling}: the matrix which maps the ODE state to the measurements; typically a projection matrix
  • observation_noise_cov::Union{Number,AbstractMatrix}: the scalar observation noise variance

Reference

  • [10] Wu et al, "Data-Adaptive Probabilistic Likelihood Approximation for Ordinary Differential Equations", arXiv (2023)
source
+

Compute the DALTON [10] approximate negative log-likelihood (NLL) of the data.

You can control the step-size behaviour of the solver as you would for a standard ODE solve, but additionally the solver always steps through the data.t locations by adding them to tstops. You can also choose steps adaptively by setting adaptive=true, but this is not well-tested so use at your own risk!

Arguments

Reference

source diff --git a/previews/PR328/priors/131fdfb5.svg b/previews/PR328/priors/131fdfb5.svg deleted file mode 100644 index 01c998984..000000000 --- a/previews/PR328/priors/131fdfb5.svg +++ /dev/null @@ -1,176 +0,0 @@ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - diff --git a/previews/PR328/priors/2953cdf6.svg b/previews/PR328/priors/2953cdf6.svg deleted file mode 100644 index a2bb5c4d8..000000000 --- a/previews/PR328/priors/2953cdf6.svg +++ /dev/null @@ -1,102 +0,0 @@ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - diff --git a/previews/PR328/priors/488fd57b.svg b/previews/PR328/priors/488fd57b.svg new file mode 100644 index 000000000..78cc1d029 --- /dev/null +++ b/previews/PR328/priors/488fd57b.svg @@ -0,0 +1,174 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR328/priors/5ba424c2.svg b/previews/PR328/priors/5ba424c2.svg new file mode 100644 index 000000000..bd66f7f08 --- /dev/null +++ b/previews/PR328/priors/5ba424c2.svg @@ -0,0 +1,174 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR328/priors/61ffd76b.svg b/previews/PR328/priors/61ffd76b.svg deleted file mode 100644 index 0576a1566..000000000 --- a/previews/PR328/priors/61ffd76b.svg +++ /dev/null @@ -1,178 +0,0 @@ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - diff --git a/previews/PR328/priors/7dfa769b.svg b/previews/PR328/priors/7dfa769b.svg new file mode 100644 index 000000000..49421d64e --- /dev/null +++ b/previews/PR328/priors/7dfa769b.svg @@ -0,0 +1,178 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR328/priors/9c39903f.svg b/previews/PR328/priors/9c39903f.svg deleted file mode 100644 index a9c402eb5..000000000 --- a/previews/PR328/priors/9c39903f.svg +++ /dev/null @@ -1,174 +0,0 @@ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - diff --git a/previews/PR328/priors/c0d9b015.svg b/previews/PR328/priors/c0d9b015.svg new file mode 100644 index 000000000..de6f45e5d --- /dev/null +++ b/previews/PR328/priors/c0d9b015.svg @@ -0,0 +1,100 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR328/priors/cefb3836.svg b/previews/PR328/priors/cefb3836.svg deleted file mode 100644 index 60eb7d099..000000000 --- a/previews/PR328/priors/cefb3836.svg +++ /dev/null @@ -1,174 +0,0 @@ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - diff --git a/previews/PR328/priors/df90853f.svg b/previews/PR328/priors/df90853f.svg new file mode 100644 index 000000000..907e81441 --- /dev/null +++ b/previews/PR328/priors/df90853f.svg @@ -0,0 +1,174 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR328/priors/index.html b/previews/PR328/priors/index.html index 01a8d579e..006e1a2e1 100644 --- a/previews/PR328/priors/index.html +++ b/previews/PR328/priors/index.html @@ -10,7 +10,7 @@ \end{aligned}\]

defined as the solution of the stochastic differential equation

\[\begin{aligned} \text{d} Y^{(i)}(t) &= Y^{(i+1)}(t) \ \text{d}t, \qquad i = 0, \dots, q-1 \\ \text{d} Y^{(q)}(t) &= \Gamma \ \text{d}W(t). -\end{aligned}\]

Then, $Y^{(0)}(t)$ is the $q$-times integrated Wiener process (IWP) and $Y^{(i)}(t)$ is the $i$-th derivative of the IWP, for $i = 1, \dots, q$.

Examples

julia> solve(prob, EK1(prior=IWP(2)))
source

Here is how the IWP looks for varying smoothness parameters $q$:

using ProbNumDiffEq, Plots
+\end{aligned}\]
Then, $Y^{(0)}(t)$ is the $q$-times integrated Wiener process (IWP) and $Y^{(i)}(t)$ is the $i$-th derivative of the IWP, for $i = 1, \dots, q$.
Examples
julia> solve(prob, EK1(prior=IWP(2)))
source
Here is how the IWP looks for varying smoothness parameters $q$:
using ProbNumDiffEq, Plots
 plotrange = range(0, 10, length=250)
 plot(
     plot(IWP(1), plotrange; title="q=1"),
@@ -18,22 +18,22 @@
     plot(IWP(3), plotrange; title="q=3"),
     plot(IWP(4), plotrange; title="q=4");
     ylims=(-20,20),
-)
Example block output
In the context of ODE solvers, the smoothness parameter $q$ influences the convergence rate of the solver, and so it is typically chose similarly to the order of a Runge–Kutta method: lower order for low accuracy, higher order for high accuracy.
Integrated Ornstein–Uhlenbeck process (IOUP)
The $q$-times integrated Ornstein–Uhlenbeck process prior IOUP is a generalization of the IWP prior, where the drift matrix $\textcolor{#389826}{A}$ is not zero:
ProbNumDiffEq.IOUP — Type
IOUP([dim::Integer=1,]
+)
Example block output
In the context of ODE solvers, the smoothness parameter $q$ influences the convergence rate of the solver, and so it is typically chose similarly to the order of a Runge–Kutta method: lower order for low accuracy, higher order for high accuracy.
Integrated Ornstein–Uhlenbeck process (IOUP)
The $q$-times integrated Ornstein–Uhlenbeck process prior IOUP is a generalization of the IWP prior, where the drift matrix $\textcolor{#389826}{A}$ is not zero:
ProbNumDiffEq.IOUP — Type
IOUP([dim::Integer=1,]
      num_derivatives::Integer,
      rate_parameter::Union{Number,AbstractVector,AbstractMatrix})
Integrated Ornstein–Uhlenbeck process.
This prior is mostly used in the context of Probabilistic Exponential Integrators to include the linear part of a semi-linear ODE in the prior, so it is used in the ExpEK and the RosenbrockExpEK.
In math
The IOUP is a Gauss–Markov process, which we model with a state representation
\[\begin{aligned}
 Y(t) = \left[ Y^{(0)}(t), Y^{(1)}(t), \dots Y^{(q)}(t) \right],
 \end{aligned}\]
defined as the solution of the stochastic differential equation
\[\begin{aligned}
 \text{d} Y^{(i)}(t) &= Y^{(i+1)}(t) \ \text{d}t, \qquad i = 0, \dots, q-1 \\
 \text{d} Y^{(q)}(t) &= L Y^{(q)}(t) \ \text{d}t + \Gamma \ \text{d}W(t),
-\end{aligned}\]
where $L$ is called the rate_parameter. Then, $Y^{(0)}(t)$ is the $q$-times integrated Ornstein–Uhlenbeck process (IOUP) and $Y^{(i)}(t)$ is the $i$-th derivative of the IOUP, for $i = 1, \dots, q$.
Examples
julia> solve(prob, EK1(prior=IOUP(2, -1)))
source
Here is how the IOUP looks for varying rate parameters:
using ProbNumDiffEq, Plots
+\end{aligned}\]
where $L$ is called the rate_parameter. Then, $Y^{(0)}(t)$ is the $q$-times integrated Ornstein–Uhlenbeck process (IOUP) and $Y^{(i)}(t)$ is the $i$-th derivative of the IOUP, for $i = 1, \dots, q$.
Examples
julia> solve(prob, EK1(prior=IOUP(2, -1)))
source
Here is how the IOUP looks for varying rate parameters:
using ProbNumDiffEq, Plots
 plotrange = range(0, 10, length=250)
 plot(
     plot(IOUP(1, -1), plotrange; title="q=1,L=-1", ylims=(-20,20)),
     plot(IOUP(1, 1), plotrange; title="q=1,L=1", ylims=(-20,20)),
     plot(IOUP(4, -1), plotrange; title="q=4,L=-1", ylims=(-50,50)),
     plot(IOUP(4, 1), plotrange; title="q=4,L=1", ylims=(-50,50));
-)
Example block output
In the context of Probabilistic Exponential Integrators, the rate parameter is often chosen according to the given ODE. Here is an example for a damped oscillator:
plot(IOUP(dim=2, num_derivatives=1, rate_parameter=[-0.2 -2π; 2π -0.2]),
-     plotrange; plot_title="damped oscillator prior")
Example block output
Matérn process (Matern)
ProbNumDiffEq.Matern — Type
Matern([dim::Integer=1,]
+)
Example block output
In the context of Probabilistic Exponential Integrators, the rate parameter is often chosen according to the given ODE. Here is an example for a damped oscillator:
plot(IOUP(dim=2, num_derivatives=1, rate_parameter=[-0.2 -2π; 2π -0.2]),
+     plotrange; plot_title="damped oscillator prior")
Example block output
Matérn process (Matern)
ProbNumDiffEq.Matern — Type
Matern([dim::Integer=1,]
        num_derivatives::Integer,
        lengthscale::Number)
Matern process.
The class of Matern processes is well-known in the Gaussian process literature, and they also have a corresponding SDE representation similarly to the IWP and the IOUP. See also [6] for more details.
In math
A Matern process is a Gauss–Markov process, which we model with a state representation
\[\begin{aligned}
 Y(t) = \left[ Y^{(0)}(t), Y^{(1)}(t), \dots Y^{(q)}(t) \right],
@@ -43,22 +43,22 @@
   \begin{pmatrix} q+1 \\ j \end{pmatrix}
   \left( \frac{\sqrt{2q - 1}}{l} \right)^{q-j}
   Y^{(j)}(t) \right) \ \text{d}t + \Gamma \ \text{d}W(t).
-\end{aligned}\]
where $l$ is called the lengthscale parameter. Then, $Y^{(0)}(t)$ is a Matern process and $Y^{(i)}(t)$ is the $i$-th derivative of this process, for $i = 1, \dots, q$.
Examples
julia> solve(prob, EK1(prior=Matern(2, 1)))
source
Here is how the Matern looks for varying smoothness parameters $q$:
using ProbNumDiffEq, Plots
+\end{aligned}\]
where $l$ is called the lengthscale parameter. Then, $Y^{(0)}(t)$ is a Matern process and $Y^{(i)}(t)$ is the $i$-th derivative of this process, for $i = 1, \dots, q$.
Examples
julia> solve(prob, EK1(prior=Matern(2, 1)))
source
Here is how the Matern looks for varying smoothness parameters $q$:
using ProbNumDiffEq, Plots
 plotrange = range(0, 10, length=250)
 plot(
     plot(Matern(1, 1), plotrange; title="q=1"),
     plot(Matern(2, 1), plotrange; title="q=2"),
     plot(Matern(3, 1), plotrange; title="q=3"),
     plot(Matern(4, 1), plotrange; title="q=4");
-)
Example block output
and for varying length scales $\ell$:
plot(
+)
Example block output
and for varying length scales $\ell$:
plot(
     plot(Matern(2, 5), plotrange; title="l=5"),
     plot(Matern(2, 2), plotrange; title="l=2"),
     plot(Matern(2, 1), plotrange; title="l=1"),
     plot(Matern(2, 0.5), plotrange; title="l=0.5"),
-)
Example block output
API
ProbNumDiffEq.AbstractGaussMarkovProcess — Type
AbstractGaussMarkovProcess{elType}
Abstract type for Gauss-Markov processes.
Gauss-Markov processes are solutions to linear time-invariant stochastic differential equations (SDEs). Here we assume SDEs of the form
\[\begin{aligned}
+)
Example block output

API

ProbNumDiffEq.AbstractGaussMarkovProcess — Type
AbstractGaussMarkovProcess{elType}

Abstract type for Gauss-Markov processes.

Gauss-Markov processes are solutions to linear time-invariant stochastic differential equations (SDEs). Here we assume SDEs of the form

\[\begin{aligned} dX_t &= F X_t dt + L dW_t \\ X_0 &= \mathcal{N} \left( X_0; \mu_0, \Sigma_0 \right) -\end{aligned}\]

where $X_t$ is the state, $W_t$ is a Wiener process, and $F$ and $L$ are matrices.

Currently, ProbNumDiffEq.jl makes many assumptions about the structure of the SDEs that it can solve. In particular, it assumes that the state vector $X_t$ contains a range of dervatives, and that the Wiener process only enters the highest one. It also assumes a certain ordering of dimensions and derivatives. This is not a limitation of the underlying mathematics, but rather a limitation of the current implementation. In the future, we hope to remove these limitations.

Warning

We currently strongly recommended to not implement your own Gauss-Markov process by subtyping this type! The interface is not yet stable, and the implementation is not yet sufficiently documented. Proceed at your own risk.

source
ProbNumDiffEq.LTISDE — Type
LTISDE(F::AbstractMatrix, L::AbstractMatrix)

Linear time-invariant stochastic differential equation.

A LTI-SDE is a stochastic differential equation of the form

\[dX_t = F X_t dt + L dW_t\]

where $X_t$ is the state, $W_t$ is a Wiener process, and $F$ and $L$ are matrices. This LTISDE object holds the matrices $F$ and $L$. It also provides some functionality to discretize the SDE via a matrix-fraction decomposition. See: discretize(::LTISDE, ::Real).

source
ProbNumDiffEq.dim — Function
dim(p::AbstractGaussMarkovProcess)

Return the dimension of the process.

This is not the dimension of the "state" that is used to efficiently model the prior process as a state-space model, but it is the dimension of the process itself that we aim to model.

See AbstractGaussMarkovProcess for more details on Gauss-Markov processes in ProbNumDiffEq.

source
ProbNumDiffEq.num_derivatives — Function
num_derivatives(p::AbstractGaussMarkovProcess)

Return the number of derivatives that are represented by the processes state.

See AbstractGaussMarkovProcess for more details on Gauss-Markov processes in ProbNumDiffEq.

source
ProbNumDiffEq.to_sde — Function
to_sde(p::AbstractGaussMarkovProcess)

Convert the prior to the corresponding SDE.

Gauss-Markov processes are solutions to linear time-invariant stochastic differential equations (SDEs) of the form

\[\begin{aligned} +\end{aligned}\]

where $X_t$ is the state, $W_t$ is a Wiener process, and $F$ and $L$ are matrices.

Currently, ProbNumDiffEq.jl makes many assumptions about the structure of the SDEs that it can solve. In particular, it assumes that the state vector $X_t$ contains a range of dervatives, and that the Wiener process only enters the highest one. It also assumes a certain ordering of dimensions and derivatives. This is not a limitation of the underlying mathematics, but rather a limitation of the current implementation. In the future, we hope to remove these limitations.

Warning

We currently strongly recommended to not implement your own Gauss-Markov process by subtyping this type! The interface is not yet stable, and the implementation is not yet sufficiently documented. Proceed at your own risk.

source
ProbNumDiffEq.LTISDE — Type
LTISDE(F::AbstractMatrix, L::AbstractMatrix)

Linear time-invariant stochastic differential equation.

A LTI-SDE is a stochastic differential equation of the form

\[dX_t = F X_t dt + L dW_t\]

where $X_t$ is the state, $W_t$ is a Wiener process, and $F$ and $L$ are matrices. This LTISDE object holds the matrices $F$ and $L$. It also provides some functionality to discretize the SDE via a matrix-fraction decomposition. See: discretize(::LTISDE, ::Real).

source
ProbNumDiffEq.dim — Function
dim(p::AbstractGaussMarkovProcess)

Return the dimension of the process.

This is not the dimension of the "state" that is used to efficiently model the prior process as a state-space model, but it is the dimension of the process itself that we aim to model.

See AbstractGaussMarkovProcess for more details on Gauss-Markov processes in ProbNumDiffEq.

source
ProbNumDiffEq.num_derivatives — Function
num_derivatives(p::AbstractGaussMarkovProcess)

Return the number of derivatives that are represented by the processes state.

See AbstractGaussMarkovProcess for more details on Gauss-Markov processes in ProbNumDiffEq.

source
ProbNumDiffEq.to_sde — Function
to_sde(p::AbstractGaussMarkovProcess)

Convert the prior to the corresponding SDE.

Gauss-Markov processes are solutions to linear time-invariant stochastic differential equations (SDEs) of the form

\[\begin{aligned} dX_t &= F X_t dt + L dW_t \\ X_0 &= \mathcal{N} \left( X_0; \mu_0, \Sigma_0 \right) -\end{aligned}\]

where $X_t$ is the state, $W_t$ is a Wiener process, and $F$ and $L$ are matrices. This function returns the corresponding SDE, i.e. the matrices $F$ and $L$, as a LTISDE.

source
ProbNumDiffEq.discretize — Function
discretize(p::AbstractGaussMarkovProcess, step_size::Real)

Compute the transition matrices of the process for a given step size.

source
discretize(p::LTISDE, step_size::Real)

Compute the transition matrices of the SDE solution for a given step size.

source
ProbNumDiffEq.initial_distribution — Function
initial_distribution(p::AbstractGaussMarkovProcess)

Return the initial distribution of the process.

Currently this is always a Gaussian distribution with zero mean and unit variance, unless explicitly overwitten (e.g. for Matern processes to have the stationary distribution). This implementation is likely to change in the future to allow for more flexibility.

source

Convenience functions to analyze and visualize priors

ProbNumDiffEq.marginalize — Function
marginalize(process::AbstractGaussMarkovProcess, times)

Compute the marginal distributions of the process at the given time points.

This function computes the marginal distributions of the process at the given times. It does so by discretizing the process with the given step sizes (using ProbNumDiffEq.discretize), and then computing the marginal distributions of the resulting Gaussian distributions.

See also: sample.

source
ProbNumDiffEq.sample — Function
sample(process::AbstractGaussMarkovProcess, times, N=1)

Samples from the Gauss-Markov process on the given time grid.

See also: marginalize.

source
+\end{aligned}\]

where $X_t$ is the state, $W_t$ is a Wiener process, and $F$ and $L$ are matrices. This function returns the corresponding SDE, i.e. the matrices $F$ and $L$, as a LTISDE.

source
ProbNumDiffEq.discretize — Function
discretize(p::AbstractGaussMarkovProcess, step_size::Real)

Compute the transition matrices of the process for a given step size.

source
discretize(p::LTISDE, step_size::Real)

Compute the transition matrices of the SDE solution for a given step size.

source
ProbNumDiffEq.initial_distribution — Function
initial_distribution(p::AbstractGaussMarkovProcess)

Return the initial distribution of the process.

Currently this is always a Gaussian distribution with zero mean and unit variance, unless explicitly overwitten (e.g. for Matern processes to have the stationary distribution). This implementation is likely to change in the future to allow for more flexibility.

source

Convenience functions to analyze and visualize priors

ProbNumDiffEq.marginalize — Function
marginalize(process::AbstractGaussMarkovProcess, times)

Compute the marginal distributions of the process at the given time points.

This function computes the marginal distributions of the process at the given times. It does so by discretizing the process with the given step sizes (using ProbNumDiffEq.discretize), and then computing the marginal distributions of the resulting Gaussian distributions.

See also: sample.

source
ProbNumDiffEq.sample — Function
sample(process::AbstractGaussMarkovProcess, times, N=1)

Samples from the Gauss-Markov process on the given time grid.

See also: marginalize.

source
diff --git a/previews/PR328/references/index.html b/previews/PR328/references/index.html index 6db4d0cf7..d3529feb5 100644 --- a/previews/PR328/references/index.html +++ b/previews/PR328/references/index.html @@ -1,2 +1,2 @@ -References · ProbNumDiffEq.jl
GitHub

References

[1]
N. Bosch, F. Tronarp and P. Hennig. Pick-and-Mix Information Operators for Probabilistic ODE Solvers. In: Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, Vol. 151 of Proceedings of Machine Learning Research, edited by G. Camps-Valls, F. J. Ruiz and I. Valera (PMLR, 28–30 Mar 2022); pp. 10015–10027.
[2]
N. Bosch, P. Hennig and F. Tronarp. Probabilistic Exponential Integrators. In: Thirty-seventh Conference on Neural Information Processing Systems (2023).
[3]
F. Tronarp, N. Bosch and P. Hennig. Fenrir: Physics-Enhanced Regression for Initial Value Problems. In: Proceedings of the 39th International Conference on Machine Learning, Vol. 162 of Proceedings of Machine Learning Research, edited by K. Chaudhuri, S. Jegelka, L. Song, C. Szepesvari, G. Niu and S. Sabato (PMLR, 17–23 Jul 2022); pp. 21776–21794.
[4]
[5]
N. Krämer, N. Bosch, J. Schmidt and P. Hennig. Probabilistic ODE Solutions in Millions of Dimensions. In: Proceedings of the 39th International Conference on Machine Learning, Vol. 162 of Proceedings of Machine Learning Research, edited by K. Chaudhuri, S. Jegelka, L. Song, C. Szepesvari, G. Niu and S. Sabato (PMLR, 17–23 Jul 2022); pp. 11634–11649.
[6]
S. Särkkä and A. Solin. Applied Stochastic Differential Equations. Institute of Mathematical Statistics Textbooks (Cambridge University Press, 2019).
[7]
[8]
[9]
N. Bosch, P. Hennig and F. Tronarp. Calibrated Adaptive Probabilistic ODE Solvers. In: Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, Vol. 130 of Proceedings of Machine Learning Research, edited by A. Banerjee and K. Fukumizu (PMLR, 13–15 Apr 2021); pp. 3466–3474.
[10]
[11]
P. Hennig, M. A. Osborne and H. P. Kersting. Probabilistic Numerics: Computation as Machine Learning (Cambridge University Press, 2022).
[12]
H. Kersting and P. Hennig. Active Uncertainty Calibration in Bayesian ODE Solvers. In: Proceedings of the Thirty-Second Conference on Uncertainty in Artificial Intelligence, UAI'16 (AUAI Press, 2016); pp. 309–318.
[13]
H. Kersting, T. J. Sullivan and P. Hennig. Convergence rates of Gaussian ODE filters. Statistics and Computing 30, 1791–1816 (2020).
[14]
+References · ProbNumDiffEq.jl
GitHub

References

[1]
N. Bosch, F. Tronarp and P. Hennig. Pick-and-Mix Information Operators for Probabilistic ODE Solvers. In: Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, Vol. 151 of Proceedings of Machine Learning Research, edited by G. Camps-Valls, F. J. Ruiz and I. Valera (PMLR, 28–30 Mar 2022); pp. 10015–10027.
[2]
N. Bosch, P. Hennig and F. Tronarp. Probabilistic Exponential Integrators. In: Thirty-seventh Conference on Neural Information Processing Systems (2023).
[3]
F. Tronarp, N. Bosch and P. Hennig. Fenrir: Physics-Enhanced Regression for Initial Value Problems. In: Proceedings of the 39th International Conference on Machine Learning, Vol. 162 of Proceedings of Machine Learning Research, edited by K. Chaudhuri, S. Jegelka, L. Song, C. Szepesvari, G. Niu and S. Sabato (PMLR, 17–23 Jul 2022); pp. 21776–21794.
[4]
[5]
N. Krämer, N. Bosch, J. Schmidt and P. Hennig. Probabilistic ODE Solutions in Millions of Dimensions. In: Proceedings of the 39th International Conference on Machine Learning, Vol. 162 of Proceedings of Machine Learning Research, edited by K. Chaudhuri, S. Jegelka, L. Song, C. Szepesvari, G. Niu and S. Sabato (PMLR, 17–23 Jul 2022); pp. 11634–11649.
[6]
S. Särkkä and A. Solin. Applied Stochastic Differential Equations. Institute of Mathematical Statistics Textbooks (Cambridge University Press, 2019).
[7]
[8]
[9]
N. Bosch, P. Hennig and F. Tronarp. Calibrated Adaptive Probabilistic ODE Solvers. In: Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, Vol. 130 of Proceedings of Machine Learning Research, edited by A. Banerjee and K. Fukumizu (PMLR, 13–15 Apr 2021); pp. 3466–3474.
[10]
[11]
P. Hennig, M. A. Osborne and H. P. Kersting. Probabilistic Numerics: Computation as Machine Learning (Cambridge University Press, 2022).
[12]
H. Kersting and P. Hennig. Active Uncertainty Calibration in Bayesian ODE Solvers. In: Proceedings of the Thirty-Second Conference on Uncertainty in Artificial Intelligence, UAI'16 (AUAI Press, 2016); pp. 309–318.
[13]
H. Kersting, T. J. Sullivan and P. Hennig. Convergence rates of Gaussian ODE filters. Statistics and Computing 30, 1791–1816 (2020).
[14]
diff --git a/previews/PR328/search_index.js b/previews/PR328/search_index.js index 2978e43cb..94021fac0 100644 --- a/previews/PR328/search_index.js +++ b/previews/PR328/search_index.js @@ -1,3 +1,3 @@ var documenterSearchIndex = {"docs": -[{"location":"references/#references","page":"References","title":"References","text":"","category":"section"},{"location":"references/","page":"References","title":"References","text":"N. Bosch, F. Tronarp and P. Hennig. Pick-and-Mix Information Operators for Probabilistic ODE Solvers. In: Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, Vol. 151 of Proceedings of Machine Learning Research, edited by G. Camps-Valls, F. J. Ruiz and I. Valera (PMLR, 28–30 Mar 2022); pp. 10015–10027.\n\n\n\nN. Bosch, P. Hennig and F. Tronarp. Probabilistic Exponential Integrators. In: Thirty-seventh Conference on Neural Information Processing Systems (2023).\n\n\n\nF. Tronarp, N. Bosch and P. Hennig. Fenrir: Physics-Enhanced Regression for Initial Value Problems. In: Proceedings of the 39th International Conference on Machine Learning, Vol. 162 of Proceedings of Machine Learning Research, edited by K. Chaudhuri, S. Jegelka, L. Song, C. Szepesvari, G. Niu and S. Sabato (PMLR, 17–23 Jul 2022); pp. 21776–21794.\n\n\n\nF. Tronarp, H. Kersting, S. Särkkä and P. Hennig. Probabilistic solutions to ordinary differential equations as nonlinear Bayesian filtering: a new perspective. Statistics and Computing 29, 1297–1315 (2019).\n\n\n\nN. Krämer, N. Bosch, J. Schmidt and P. Hennig. Probabilistic ODE Solutions in Millions of Dimensions. In: Proceedings of the 39th International Conference on Machine Learning, Vol. 162 of Proceedings of Machine Learning Research, edited by K. Chaudhuri, S. Jegelka, L. Song, C. Szepesvari, G. Niu and S. Sabato (PMLR, 17–23 Jul 2022); pp. 11634–11649.\n\n\n\nS. Särkkä and A. Solin. Applied Stochastic Differential Equations. Institute of Mathematical Statistics Textbooks (Cambridge University Press, 2019).\n\n\n\nN. Krämer and P. Hennig. Stable Implementation of Probabilistic ODE Solvers. CoRR (2020), arXiv:2012.10106 [stat.ML].\n\n\n\nM. Schober, S. Särkkä and P. Hennig. A probabilistic model for the numerical solution of initial value problems. Statistics and Computing 29, 99–122 (2019).\n\n\n\nN. Bosch, P. Hennig and F. Tronarp. Calibrated Adaptive Probabilistic ODE Solvers. In: Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, Vol. 130 of Proceedings of Machine Learning Research, edited by A. Banerjee and K. Fukumizu (PMLR, 13–15 Apr 2021); pp. 3466–3474.\n\n\n\nM. Wu and M. Lysy. Data-Adaptive Probabilistic Likelihood Approximation for Ordinary Differential Equations. CoRR (2023), arXiv:2306.05566 [stat.ML].\n\n\n\nP. Hennig, M. A. Osborne and H. P. Kersting. Probabilistic Numerics: Computation as Machine Learning (Cambridge University Press, 2022).\n\n\n\nH. Kersting and P. Hennig. Active Uncertainty Calibration in Bayesian ODE Solvers. In: Proceedings of the Thirty-Second Conference on Uncertainty in Artificial Intelligence, UAI'16 (AUAI Press, 2016); pp. 309–318.\n\n\n\nH. Kersting, T. J. Sullivan and P. Hennig. Convergence rates of Gaussian ODE filters. Statistics and Computing 30, 1791–1816 (2020).\n\n\n\nF. Tronarp, S. Särkkä and P. Hennig. Bayesian ODE solvers: the maximum a posteriori estimate. Statistics and Computing 31, 23 (2021).\n\n\n\n","category":"page"},{"location":"benchmarks/vanderpol/#Van-der-Pol-benchmark","page":"Van der Pol","title":"Van der Pol benchmark","text":"","category":"section"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"note: Summary\nVan der Pol is a low-dimensional, stiff, second-order ODE. We see that:The EK1 is very well able to solve stiff problems.\nSince Van der Pol is actually a second-order ODE, do solve it as a second-order ODE.\nUse the TaylorInit or ForwardDiffInit initialization. While SimpleInit works well for lower orders, it fails for higher orders. And since Taylor-mode initialization is fast and works well, there is no reason not to use it.","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Code:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"using LinearAlgebra, Statistics, Distributions\nusing DiffEqDevTools, ParameterizedFunctions, SciMLBase, OrdinaryDiffEq, Plots\nusing ProbNumDiffEq\n\nPlots.theme(\n :dao;\n markerstrokewidth=0.5,\n legend=:outertopright,\n margin=5Plots.mm,\n xticks = 10.0 .^ (-16:1:16)\n)\n\nfunction plot_chisq_interval!(df, q=0.01)\n dist = Chisq(df)\n low, high, mid = quantile(dist, [q, 1-q])..., mean(dist)\n hline!([low, high], linestyle=:dash, color=:black, label=\"\",\n fill_between=[high nothing], fillcolor=:green, fillalpha=0.15)\n hline!([mid], linestyle=:solid, color=:black, label=\"\")\nend","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Code:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"function vanderpol!(du, u, p, t)\n du[1] = u[2]\n du[2] = p[1] * ((1 - u[1]^2) * u[2] - u[1])\nend\np = [1e5]\ntspan = (0.0, 2.0)\nu0 = [2.0, 0.0]\nprob = ODEProblem(vanderpol!, u0, tspan, p)\n\ntest_sol = solve(prob, RadauIIA5(), abstol=1/10^14, reltol=1/10^14)\nplot(test_sol, title=\"Van der Pol Solution\", legend=false, ylims=(-5, 5), xticks=:auto)","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"(Image: )","category":"page"},{"location":"benchmarks/vanderpol/#vdp_main_results","page":"Van der Pol","title":"EK1 across orders","text":"","category":"section"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Code:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"DENSE = false;\nSAVE_EVERYSTEP = false;\n\n_setups = [\n \"EK1($order)\" => Dict(:alg => EK1(order=order, smooth=DENSE))\n for order in 3:7\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nabstols = 1.0 ./ 10.0 .^ (6:11)\nreltols = 1.0 ./ 10.0 .^ (3:8)\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n numruns = 5,\n)\n\nplot(wp, palette=Plots.palette([:blue, :red], length(_setups)))\n\n_ref_setups = [\n \"Rosenbrock23\" => Dict(:alg => Rosenbrock23())\n \"Rodas4P\" => Dict(:alg => Rodas4P())\n \"RadauIIA5\" => Dict(:alg => RadauIIA5())\n]\nref_labels = first.(_ref_setups)\nref_setups = last.(_ref_setups)\nref_wp_final = WorkPrecisionSet(\n prob, abstols, reltols, ref_setups;\n names = ref_labels,\n appxsol = test_sol,\n dense = false,\n save_everystep = false,\n maxiters = Int(1e7),\n)\nref_wp_dense = WorkPrecisionSet(\n prob, abstols, reltols, ref_setups;\n names = ref_labels,\n appxsol = test_sol,\n dense = true,\n save_everystep = true,\n maxiters = Int(1e7),\n)\n\nplot!(ref_wp_final, x=:final, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"(Image: )","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Discrete time-series errors (l2):","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Code:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"DENSE = true;\nSAVE_EVERYSTEP = true;\n\n_setups = [\n \"EK1($order)\" => Dict(:alg => EK1(order=order, smooth=DENSE))\n for order in 3:7\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nabstols = 1.0 ./ 10.0 .^ (6:11)\nreltols = 1.0 ./ 10.0 .^ (3:8)\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n numruns = 5,\n)\n\nplot(wp, x=:l2, palette=Plots.palette([:blue, :red], length(_setups)))\nplot!(ref_wp_dense, x=:l2, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"(Image: )","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Interpolation errors (L2):","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Code:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"plot(wp, x=:L2, palette=Plots.palette([:blue, :red], length(_setups)))\nplot!(ref_wp_dense, x=:L2, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"(Image: )","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/#Calibration","page":"Van der Pol","title":"Calibration","text":"","category":"section"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Code:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"plot(wp, x=:final, y=:chi2_final, yguide=\"Chi-squared (final)\",\n palette=Plots.palette([:blue, :red], length(_setups)))\nplot_chisq_interval!(2)","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"(Image: )","category":"page"},{"location":"benchmarks/vanderpol/#vdp_initialization","page":"Van der Pol","title":"Comparison of the different initialization schemes","text":"","category":"section"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Code:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"DENSE = false;\nSAVE_EVERYSTEP = false;\n\nabstols = 1.0 ./ 10.0 .^ (6:11)\nreltols = 1.0 ./ 10.0 .^ (3:8)\n\norders = (3, 5, 8)\nps = []\nfor o in orders\n _setups = [\n \"EK1($o) TaylorInit\" => Dict(:alg => EK1(order=o, smooth=DENSE, initialization=TaylorModeInit(o)))\n \"EK1($o) ForwardDiffInit\" => Dict(:alg => EK1(order=o, smooth=DENSE, initialization=ForwardDiffInit(o)))\n \"EK1($o) SimpleInit\" => Dict(:alg => EK1(order=o, smooth=DENSE, initialization=SimpleInit()))\n # \"EK1($o) ClassicSolverInit\" => Dict(:alg => EK1(order=o, smooth=DENSE, initialization=ClassicSolverInit())) # unstable\n ]\n\n labels = first.(_setups)\n setups = last.(_setups)\n\n wp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n numruns = 5,\n )\n\n p = plot(wp, color=[2 4 5 6], xticks = 10.0 .^ (-16:1:5))\n push!(ps, p)\nend\nplot(\n ps...,\n layout=(length(orders), 1),\n size = (1000, length(orders)*300),\n xlabel=[\"\" \"\" \"\" \"Error\"],\n)","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"(Image: )","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Code:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"DENSE = false;\nSAVE_EVERYSTEP = false;\n\nabstols = 1.0 ./ 10.0 .^ (5:8)\nreltols = 1.0 ./ 10.0 .^ (2:5)\n\n_setups = [\n \"EK1(8) TaylorInit(8)\" => Dict(:alg => EK1(order=8, smooth=DENSE, initialization=TaylorModeInit(8)))\n \"EK1(8) TaylorInit(7)\" => Dict(:alg => EK1(order=8, smooth=DENSE, initialization=TaylorModeInit(7)))\n \"EK1(8) TaylorInit(6)\" => Dict(:alg => EK1(order=8, smooth=DENSE, initialization=TaylorModeInit(6)))\n \"EK1(8) TaylorInit(5)\" => Dict(:alg => EK1(order=8, smooth=DENSE, initialization=TaylorModeInit(5)))\n \"EK1(8) TaylorInit(4)\" => Dict(:alg => EK1(order=8, smooth=DENSE, initialization=TaylorModeInit(4)))\n \"EK1(8) TaylorInit(3)\" => Dict(:alg => EK1(order=8, smooth=DENSE, initialization=TaylorModeInit(3)))\n \"EK1(8) TaylorInit(2)\" => Dict(:alg => EK1(order=8, smooth=DENSE, initialization=TaylorModeInit(2)))\n # \"EK1(8) TaylorInit(1)\" => Dict(:alg => EK1(order=8, smooth=DENSE, initialization=TaylorModeInit(1))) # fails, see above\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n numruns = 5,\n)\n\nplot(wp, palette=Plots.palette([:blue, :red], length(_setups)), xticks = 10.0 .^ (-16:1:5))","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"(Image: )","category":"page"},{"location":"benchmarks/vanderpol/#vdp_second_order","page":"Van der Pol","title":"Solving the first- vs second-order ODE","text":"","category":"section"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Code:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"function vanderpol2!(ddu, du, u, p, t)\n ddu[1] = p[1] * ((1 - u[1]^2) * du[1] - u[1])\nend\np = [1e5]\ntspan = (0.0, 2.0)\nu0 = [2.0]\ndu0 = [0.0]\nprob2 = SecondOrderODEProblem(vanderpol2!, du0, u0, tspan, p)\n\ntest_sol2 = solve(prob2, RadauIIA5(), abstol=1/10^14, reltol=1/10^14)\n# plot(test_sol2, title=\"Van der Pol Solution (2nd order)\", legend=false, ylims=(-5, 5), xticks=:auto)\nnothing","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Code:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"DENSE = true;\nSAVE_EVERYSTEP = true;\n\n_setups = [\n \"EK1(3) 1st order\" => Dict(:alg => EK1(order=3, smooth=DENSE))\n \"EK1(4) 1st order\" => Dict(:alg => EK1(order=4, smooth=DENSE))\n \"EK1(5) 1st order\" => Dict(:alg => EK1(order=5, smooth=DENSE))\n \"EK1(6) 1st order\" => Dict(:alg => EK1(order=6, smooth=DENSE))\n \"EK1(4) 2nd order\" => Dict(:prob_choice => 2, :alg => EK1(order=4, smooth=DENSE))\n \"EK1(5) 2nd order\" => Dict(:prob_choice => 2, :alg => EK1(order=5, smooth=DENSE))\n \"EK1(6) 2nd order\" => Dict(:prob_choice => 2, :alg => EK1(order=6, smooth=DENSE))\n \"EK1(7) 2nd order\" => Dict(:prob_choice => 2, :alg => EK1(order=7, smooth=DENSE))\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nabstols = 1.0 ./ 10.0 .^ (6:11)\nreltols = 1.0 ./ 10.0 .^ (3:8)\n\nwp = WorkPrecisionSet(\n [prob, prob2], abstols, reltols, setups;\n names = labels,\n appxsol = [test_sol, test_sol2],\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n numruns = 5,\n)\n\ncolor = [1 1 1 1 2 2 2 2]\nplot(wp; x=:final, color)\nplot!(ref_wp_dense, x=:final, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"(Image: )","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Interpolation errors (L2):","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Code:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"plot(wp; x=:L2, color)\nplot!(ref_wp_dense, x=:L2, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"(Image: )","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/#Calibration-2","page":"Van der Pol","title":"Calibration","text":"","category":"section"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Code:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"plot(wp; x=:final, y=:chi2_final, yguide=\"Chi-squared (final)\", color)\nplot_chisq_interval!(2)","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"(Image: )","category":"page"},{"location":"benchmarks/vanderpol/#Appendix","page":"Van der Pol","title":"Appendix","text":"","category":"section"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Computer information:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"using InteractiveUtils\nInteractiveUtils.versioninfo()","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"Julia Version 1.10.0\nCommit 3120989f39b (2023-12-25 18:01 UTC)\nBuild Info:\n Official https://julialang.org/ release\nPlatform Info:\n OS: Linux (x86_64-linux-gnu)\n CPU: 12 × Intel(R) Core(TM) i7-6800K CPU @ 3.40GHz\n WORD_SIZE: 64\n LIBM: libopenlibm\n LLVM: libLLVM-15.0.7 (ORCJIT, broadwell)\n Threads: 17 on 12 virtual cores\nEnvironment:\n JULIA_NUM_THREADS = auto","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Package information:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"using Pkg\nPkg.status()","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Project.toml`\n [f3b72e0c] DiffEqDevTools v2.44.1\n [31c24e10] Distributions v0.25.107\n [7073ff75] IJulia v1.24.2\n [7f56f5a3] LSODA v0.7.5\n [e6f89c97] LoggingExtras v1.0.3\n [e2752cbe] MATLABDiffEq v1.2.0\n⌃ [961ee093] ModelingToolkit v8.73.2\n [54ca160b] ODEInterface v0.5.0\n [09606e27] ODEInterfaceDiffEq v3.13.3\n⌃ [1dea7af3] OrdinaryDiffEq v6.66.0\n [65888b18] ParameterizedFunctions v5.16.0\n [91a5bcdd] Plots v1.39.0\n [bf3e78b0] ProbNumDiffEq v0.14.0 `~/.julia/dev/ProbNumDiffEq`\n⌃ [0bca4576] SciMLBase v2.10.0\n [505e40e9] SciPyDiffEq v0.2.1\n [ce78b400] SimpleUnPack v1.1.0\n [90137ffa] StaticArrays v1.9.1\n [c3572dad] Sundials v4.23.1\n [44d3d7a6] Weave v0.10.12\n [0518478a] deSolveDiffEq v0.1.1\nInfo Packages marked with ⌃ have new versions available and may be upgradable.","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
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Full manifest:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"Pkg.status(mode=Pkg.PKGMODE_MANIFEST)","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Manifest.toml`\n [47edcb42] ADTypes v0.2.6\n⌅ [c3fe647b] AbstractAlgebra v0.34.7\n [621f4979] AbstractFFTs v1.5.0\n [1520ce14] AbstractTrees v0.4.4\n [7d9f7c33] Accessors v0.1.35\n⌅ [79e6a3ab] Adapt v3.7.2\n [ec485272] ArnoldiMethod v0.2.0\n [c9d4266f] ArrayAllocators v0.3.0\n [4fba245c] ArrayInterface v7.7.0\n [4c555306] ArrayLayouts v1.5.2\n [6e4b80f9] BenchmarkTools v1.4.0\n [e2ed5e7c] Bijections v0.1.6\n [d1d4a3ce] BitFlags v0.1.8\n [62783981] BitTwiddlingConvenienceFunctions v0.1.5\n [fa961155] CEnum v0.5.0\n [2a0fbf3d] CPUSummary v0.2.4\n [00ebfdb7] CSTParser v3.4.0\n [49dc2e85] Calculus v0.5.1\n [324d7699] CategoricalArrays v0.10.8\n [d360d2e6] ChainRulesCore v1.19.1\n 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Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/rober/#ROBER-benchmark","page":"ROBER","title":"ROBER benchmark","text":"","category":"section"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"note: Summary\nThe EK1 can solve mass-matrix DAEs. But for this problem, it only works well for low tolerances.\nFor this problem it only works well for low tolerances, but the order-to-error-tolerance heuristic should in principle still hold: lower tolerance level rightarrow higher order.","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"Adapted from SciMLBenchmarks.jl.","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"
Code:","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"using LinearAlgebra, Statistics, Distributions\nusing DiffEqDevTools, ParameterizedFunctions, SciMLBase, OrdinaryDiffEq, Sundials, Plots\nusing ModelingToolkit\nusing ProbNumDiffEq\n\nPlots.theme(\n :dao;\n markerstrokewidth=0.5,\n legend=:outertopright,\n margin=5Plots.mm,\n xticks=10.0 .^ (-16:1:16),\n)\n\nfunction plot_chisq_interval!(df, q=0.01)\n dist = Chisq(df)\n low, high, mid = quantile(dist, [q, 1-q])..., mean(dist)\n hline!([low, high], linestyle=:dash, color=:black, label=\"\",\n fill_between=[high nothing], fillcolor=:green, fillalpha=0.15)\n hline!([mid], linestyle=:solid, color=:black, label=\"\")\nend","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"
","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"
Code:","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"@variables t y₁(t)=1.0 y₂(t)=0.0 y₃(t)=0.0\n@parameters k₁=0.04 k₂=3e7 k₃=1e4\nD = Differential(t)\neqs = [\n D(y₁) ~ -k₁*y₁ + k₃*y₂*y₃\n D(y₂) ~ k₁*y₁ - k₃*y₂*y₃ - k₂*y₂^2\n 0 ~ y₁ + y₂ + y₃ - 1\n]\n@named sys = ODESystem(eqs)\nmmprob = ODEProblem(sys,[],(0.0,1e5))\ndaeprob = DAEProblem(sys,[D(y₁)=>-0.04, D(y₂)=>0.04, D(y₃)=>0.0],[],(0.0,1e5)) # can't handle this yet\nodaeprob = ODAEProblem(structural_simplify(sys),[],(0.0,1e5)) # can't handle this yet\n\nref_sol = solve(daeprob,IDA(),abstol=1/10^14,reltol=1/10^14)\nplot(ref_sol, idxs=[y₁,y₂,y₃], title=\"ROBER Solution\", legend=false, ylims=(0, 1), xticks=:auto)","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"
","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"(Image: )","category":"page"},{"location":"benchmarks/rober/#rober_results","page":"ROBER","title":"EK1 across orders","text":"","category":"section"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"
Code:","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"DENSE = false;\nSAVE_EVERYSTEP = false;\n\n_setups = [\n \"EK1($order)\" => Dict(:alg => EK1(order=order, smooth=DENSE))\n for order in 2:4\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nabstols = 1.0 ./ 10.0 .^ (4:7)\nreltols = 1.0 ./ 10.0 .^ (1:4)\n\nwp = WorkPrecisionSet(\n mmprob, abstols, reltols, setups;\n names = labels,\n appxsol = ref_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n numruns = 10,\n maxiters = Int(1e7),\n)\n\nplot(wp, palette=Plots.palette([:blue, :red], length(_setups)))\n\n\n_ref_setups = [\n \"Rosenbrock23\" => Dict(:alg => Rosenbrock23())\n \"Rodas4P\" => Dict(:alg => Rodas4P())\n \"RadauIIA\" => Dict(:alg => RadauIIA5())\n]\nref_labels = first.(_ref_setups)\nref_setups = last.(_ref_setups)\nref_wp = WorkPrecisionSet(\n mmprob, abstols ./ 10000, reltols ./ 10000, ref_setups;\n names = ref_labels,\n appxsol = ref_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n)\nplot!(ref_wp, x=:final, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"
","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"(Image: )","category":"page"},{"location":"benchmarks/rober/#Calibration","page":"ROBER","title":"Calibration","text":"","category":"section"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"
Code:","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"plot(wp; x=:final, y=:chi2_final, yguide=\"Chi-squared (final)\",\n palette=Plots.palette([:blue, :red], length(_setups)))\nplot_chisq_interval!(3)","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"
","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"(Image: )","category":"page"},{"location":"benchmarks/rober/#Appendix","page":"ROBER","title":"Appendix","text":"","category":"section"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"
Computer information:","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"using InteractiveUtils\nInteractiveUtils.versioninfo()","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"Julia Version 1.10.0\nCommit 3120989f39b (2023-12-25 18:01 UTC)\nBuild Info:\n Official https://julialang.org/ release\nPlatform Info:\n OS: Linux (x86_64-linux-gnu)\n CPU: 12 × Intel(R) Core(TM) i7-6800K CPU @ 3.40GHz\n WORD_SIZE: 64\n LIBM: libopenlibm\n LLVM: libLLVM-15.0.7 (ORCJIT, broadwell)\n Threads: 17 on 12 virtual cores\nEnvironment:\n JULIA_NUM_THREADS = auto","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"
","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"
Package information:","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"using Pkg\nPkg.status()","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Project.toml`\n [f3b72e0c] DiffEqDevTools v2.44.1\n [31c24e10] Distributions v0.25.107\n [7073ff75] IJulia v1.24.2\n [7f56f5a3] LSODA v0.7.5\n [e6f89c97] LoggingExtras v1.0.3\n [e2752cbe] MATLABDiffEq v1.2.0\n⌃ [961ee093] ModelingToolkit v8.73.2\n [54ca160b] ODEInterface v0.5.0\n [09606e27] ODEInterfaceDiffEq v3.13.3\n⌃ [1dea7af3] OrdinaryDiffEq v6.66.0\n [65888b18] ParameterizedFunctions v5.16.0\n [91a5bcdd] Plots v1.39.0\n [bf3e78b0] ProbNumDiffEq v0.14.0 `~/.julia/dev/ProbNumDiffEq`\n⌃ [0bca4576] SciMLBase v2.10.0\n [505e40e9] SciPyDiffEq v0.2.1\n [ce78b400] SimpleUnPack v1.1.0\n [90137ffa] StaticArrays v1.9.1\n [c3572dad] Sundials v4.23.1\n [44d3d7a6] Weave v0.10.12\n [0518478a] deSolveDiffEq v0.1.1\nInfo Packages marked with ⌃ have new versions available and may be upgradable.","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"
","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"
Full manifest:","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"Pkg.status(mode=Pkg.PKGMODE_MANIFEST)","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Manifest.toml`\n [47edcb42] ADTypes v0.2.6\n⌅ [c3fe647b] AbstractAlgebra v0.34.7\n [621f4979] AbstractFFTs v1.5.0\n [1520ce14] AbstractTrees v0.4.4\n [7d9f7c33] Accessors v0.1.35\n⌅ [79e6a3ab] Adapt v3.7.2\n [ec485272] ArnoldiMethod v0.2.0\n [c9d4266f] ArrayAllocators v0.3.0\n [4fba245c] ArrayInterface v7.7.0\n [4c555306] ArrayLayouts v1.5.2\n [6e4b80f9] BenchmarkTools v1.4.0\n [e2ed5e7c] Bijections v0.1.6\n [d1d4a3ce] BitFlags v0.1.8\n [62783981] BitTwiddlingConvenienceFunctions v0.1.5\n [fa961155] CEnum v0.5.0\n [2a0fbf3d] CPUSummary v0.2.4\n [00ebfdb7] CSTParser v3.4.0\n [49dc2e85] Calculus v0.5.1\n [324d7699] CategoricalArrays v0.10.8\n [d360d2e6] ChainRulesCore v1.19.1\n [fb6a15b2] CloseOpenIntervals 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Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"
","category":"page"},{"location":"likelihoods/#Data-Likelihoods","page":"Data Likelihoods","title":"Data Likelihoods","text":"","category":"section"},{"location":"likelihoods/","page":"Data Likelihoods","title":"Data Likelihoods","text":"ProbNumDiffEq.DataLikelihoods.fenrir_data_loglik\nProbNumDiffEq.DataLikelihoods.dalton_data_loglik","category":"page"},{"location":"likelihoods/#ProbNumDiffEq.fenrir_data_loglik","page":"Data Likelihoods","title":"ProbNumDiffEq.fenrir_data_loglik","text":"fenrir_data_loglik(\n prob::SciMLBase.AbstractODEProblem,\n alg::ProbNumDiffEq.AbstractEK,\n args...;\n observation_matrix,\n observation_noise_cov,\n data,\n kwargs...\n)\n\n\nCompute the Fenrir [3] approximate negative log-likelihood (NLL) of the data.\n\nThis is a convenience function that\n\nSolves the ODE with a ProbNumDiffEq.EK1 of the specified order and with a diffusion as provided by the diffusion_var argument, and\nFits the ODE posterior to the data via Kalman filtering and thereby computes the log-likelihood of the data on the way.\n\nYou can control the step-size behaviour of the solver as you would for a standard ODE solve, but additionally the solver always steps through the data.t locations by adding them to tstops.\n\nYou can also choose steps adaptively by setting adaptive=true, but this is not well-tested so use at your own risk!\n\nArguments\n\nprob::SciMLBase.AbstractODEProblem: the initial value problem of interest\nalg::AbstractEK: the probabilistic ODE solver to be used; use EK1 for best results.\ndata::NamedTuple{(:t, :u)}: the data to be fitted\nobservation_matrix::Union{AbstractMatrix,UniformScaling}: the matrix which maps the ODE state to the measurements; typically a projection matrix\nobservation_noise_cov::Union{Number,AbstractMatrix}: the scalar observation noise variance\n\nReference\n\n[3] Tronarp et al, \"Fenrir: Physics-Enhanced Regression for Initial Value Problems\", ICML (2022)\n\n\n\n\n\n","category":"function"},{"location":"likelihoods/#ProbNumDiffEq.dalton_data_loglik","page":"Data Likelihoods","title":"ProbNumDiffEq.dalton_data_loglik","text":"dalton_data_loglik(\n prob::SciMLBase.AbstractODEProblem,\n alg::ProbNumDiffEq.AbstractEK,\n args...;\n observation_matrix,\n observation_noise_cov,\n data,\n kwargs...\n)\n\n\nCompute the DALTON [10] approximate negative log-likelihood (NLL) of the data.\n\nYou can control the step-size behaviour of the solver as you would for a standard ODE solve, but additionally the solver always steps through the data.t locations by adding them to tstops. You can also choose steps adaptively by setting adaptive=true, but this is not well-tested so use at your own risk!\n\nArguments\n\nprob::SciMLBase.AbstractODEProblem: the initial value problem of interest\nalg::AbstractEK: the probabilistic ODE solver to be used; use EK1 for best results.\ndata::NamedTuple{(:t, :u)}: the data to be fitted\nobservation_matrix::Union{AbstractMatrix,UniformScaling}: the matrix which maps the ODE state to the measurements; typically a projection matrix\nobservation_noise_cov::Union{Number,AbstractMatrix}: the scalar observation noise variance\n\nReference\n\n[10] Wu et al, \"Data-Adaptive Probabilistic Likelihood Approximation for Ordinary Differential Equations\", arXiv (2023)\n\n\n\n\n\n","category":"function"},{"location":"filtering/#Gaussian-Filtering-and-Smoothing","page":"Filtering and Smoothing","title":"Gaussian Filtering and Smoothing","text":"","category":"section"},{"location":"filtering/#Predict","page":"Filtering and Smoothing","title":"Predict","text":"","category":"section"},{"location":"filtering/","page":"Filtering and Smoothing","title":"Filtering and Smoothing","text":"ProbNumDiffEq.predict\nProbNumDiffEq.predict!","category":"page"},{"location":"filtering/#ProbNumDiffEq.predict","page":"Filtering and Smoothing","title":"ProbNumDiffEq.predict","text":"predict(x::Gaussian, A::AbstractMatrix, Q::AbstractMatrix)\n\nPrediction step in Kalman filtering for linear dynamics models.\n\nGiven a Gaussian x = mathcalN(μ Σ), compute and return mathcalN(A μ A Σ A^T + Q).\n\nSee also the non-allocating square-root version predict!.\n\n\n\n\n\n","category":"function"},{"location":"filtering/#ProbNumDiffEq.predict!","page":"Filtering and Smoothing","title":"ProbNumDiffEq.predict!","text":"predict!(x_out, x_curr, Ah, Qh, cachemat)\n\nIn-place and square-root implementation of predict which saves the result into x_out.\n\nOnly works with PSDMatrices.PSDMatrix types as Ah, Qh, and in the covariances of x_curr and x_out (both of type Gaussian). To prevent allocations, a cache matrix cachemat of size D times 2D (where D times D is the size of Ah and Qh) needs to be passed.\n\nSee also: predict.\n\n\n\n\n\n","category":"function"},{"location":"filtering/#Update","page":"Filtering and Smoothing","title":"Update","text":"","category":"section"},{"location":"filtering/","page":"Filtering and Smoothing","title":"Filtering and Smoothing","text":"ProbNumDiffEq.update\nProbNumDiffEq.update!","category":"page"},{"location":"filtering/#ProbNumDiffEq.update","page":"Filtering and Smoothing","title":"ProbNumDiffEq.update","text":"update(x, measurement, H)\n\nUpdate step in Kalman filtering for linear dynamics models.\n\nGiven a Gaussian x = mathcalN(μ Σ) and a measurement z = mathcalN(hatz S), with S = H Σ H^T, compute\n\nbeginaligned\nK = Σ^P H^T S^-1 \nμ^F = μ + K (0 - hatz) \nΣ^F = Σ - K S K^T\nendaligned\n\nand return an updated state \\mathcal{N}(μ^F, Σ^F). Note that this assumes zero-measurements. When called with ProbNumDiffEq.SquarerootMatrix type arguments it performs the update in Joseph / square-root form.\n\nFor better performance, we recommend to use the non-allocating update!.\n\n\n\n\n\n","category":"function"},{"location":"filtering/#ProbNumDiffEq.update!","page":"Filtering and Smoothing","title":"ProbNumDiffEq.update!","text":"update!(x_out, x_pred, measurement, H, K_cache, M_cache, S_cache)\n\nIn-place and square-root implementation of update which saves the result into x_out.\n\nImplemented in Joseph Form to retain the PSDMatrix covariances:\n\nbeginaligned\nK = Σ^P H^T S^-1 \nμ^F = μ + K (0 - hatz) \nsqrtΣ^F = (I - KH) sqrt(Σ)\nendaligned\n\nwhere sqrtM denotes the left square-root of a matrix M, i.e. M = sqrtM sqrtM^T.\n\nTo prevent allocations, write into caches K_cache and M_cache, both of size D × D, and S_cache of same type as measurement.Σ.\n\nSee also: update.\n\n\n\n\n\n","category":"function"},{"location":"filtering/#Smooth","page":"Filtering and Smoothing","title":"Smooth","text":"","category":"section"},{"location":"filtering/","page":"Filtering and Smoothing","title":"Filtering and Smoothing","text":"ProbNumDiffEq.smooth","category":"page"},{"location":"filtering/#ProbNumDiffEq.smooth","page":"Filtering and Smoothing","title":"ProbNumDiffEq.smooth","text":"smooth(x_curr, x_next_smoothed, A, Q)\n\nUpdate step of the Kalman smoother, aka. Rauch-Tung-Striebel smoother, for linear dynamics models.\n\nGiven Gaussians x_n = mathcalN(μ_n Σ_n) and x_n+1 = mathcalN(μ_n+1^S Σ_n+1^S), compute\n\nbeginaligned\nμ_n+1^P = A μ_n^F \nP_n+1^P = A Σ_n^F A + Q \nG = Σ_n^S A^T (Σ_n+1^P)^-1 \nμ_n^S = μ_n^F + G (μ_n+1^S - μ_n+1^P) \nΣ_n^S = (I - G A) Σ_n^F (I - G A)^T + G Q G^T + G Σ_n+1^S G^T\nendaligned\n\nand return a smoothed state \\mathcal{N}(μ_n^S, Σ_n^S). When called with ProbNumDiffEq.SquarerootMatrix type arguments it performs the update in Joseph / square-root form.\n\n\n\n\n\n","category":"function"},{"location":"filtering/#Markov-Kernels","page":"Filtering and Smoothing","title":"Markov Kernels","text":"","category":"section"},{"location":"filtering/","page":"Filtering and Smoothing","title":"Filtering and Smoothing","text":"ProbNumDiffEq.AffineNormalKernel\nProbNumDiffEq.marginalize!\nProbNumDiffEq.compute_backward_kernel!","category":"page"},{"location":"filtering/#ProbNumDiffEq.AffineNormalKernel","page":"Filtering and Smoothing","title":"ProbNumDiffEq.AffineNormalKernel","text":"AffineNormalKernel(A[, b], C)\n\nStructure to represent affine Normal Markov kernels, i.e. conditional distributions of the form\n\nbeginaligned\ny mid x sim mathcalN left( y A x + b C right)\nendaligned\n\nAt the point of writing, AffineNormalKernels are only used to precompute and store the backward representation of the posterior (via compute_backward_kernel!) and for smoothing (via marginalize!).\n\n\n\n\n\n","category":"type"},{"location":"filtering/#ProbNumDiffEq.marginalize!","page":"Filtering and Smoothing","title":"ProbNumDiffEq.marginalize!","text":"marginalize!(\n xout::Gaussian{Vector{T},PSDMatrix{T,S}}\n x::Gaussian{Vector{T},PSDMatrix{T,S}},\n K::AffineNormalKernel{<:AbstractMatrix,Union{<:Number,<:AbstractVector,Missing},<:PSDMatrix};\n C_DxD, C_3DxD\n)\n\nBasically the same as predict!), but in kernel language and with support for affine transitions. At the time of writing, this is only used to smooth the posterior using it's backward representation, where the kernels are precomputed with compute_backward_kernel!.\n\nNote that this function assumes certain shapes:\n\nsize(x.μ) == (D, D)\nsize(x.Σ) == (D, D)\nsize(K.A) == (D, D)\nsize(K.b) == (D,), or missing\nsize(K.C) == (D, D), _but with a tall square-root size(K.C.R) == (3D, D)\n\nxout is assumes to have the same shapes as x.\n\n\n\n\n\n","category":"function"},{"location":"filtering/#ProbNumDiffEq.compute_backward_kernel!","page":"Filtering and Smoothing","title":"ProbNumDiffEq.compute_backward_kernel!","text":"compute_backward_kernel!(Kout, xpred, x, K; C_DxD[, diffusion=1])\n\nCompute the backward representation of the posterior, i.e. the conditional distribution of the current state given the next state and the transition kernel.\n\nMore precisely, given a distribution (x)\n\nbeginaligned\nx sim mathcalN left( x μ Σ right)\nendaligned\n\na kernel (K)\n\nbeginaligned\ny mid x sim mathcalN left( y A x + b C right)\nendaligned\n\nand a distribution (xpred) obtained via marginalization\n\nbeginaligned\ny sim mathcalN left( y μ^P Σ^P right) \nμ^P = A μ + b \nΣ^P = A Σ A^top + C\nendaligned\n\nthis function computes the conditional distribution\n\nbeginaligned\nx mid y sim mathcalN left( x G y + d Λ right)\nendaligned\n\nwhere\n\nbeginaligned\nG = Σ A^top (Σ^P)^-1 \nd = μ - G μ^P \nΛ = Σ - G Σ^P G^top\nendaligned\n\nEverything is computed in square-root form and with minimal allocations (thus the cache C_DxD), so the actual formulas implemented here differ a bit.\n\nThe resulting backward kernels are used to smooth the posterior, via marginalize!.\n\n\n\n\n\n","category":"function"},{"location":"benchmarks/orego/#OREGO-benchmark","page":"OREGO","title":"OREGO benchmark","text":"","category":"section"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"note: Summary\nThe EK1 is able to solve mass-matrix DAEs. To achieve low error, use order 4 or higher.\nThe order-to-error-tolerance heuristic holds: lower tolerance level rightarrow higher order.","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"Adapted from SciMLBenchmarks.jl.","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"
Code:","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"using LinearAlgebra, Statistics, Distributions\nusing DiffEqDevTools, ParameterizedFunctions, SciMLBase, OrdinaryDiffEq, Sundials, Plots\nusing ModelingToolkit\nusing ProbNumDiffEq\n\nPlots.theme(\n :dao;\n markerstrokewidth=0.5,\n legend=:outertopright,\n margin=5Plots.mm,\n xticks=10.0 .^ (-16:1:16),\n)\n\nfunction plot_chisq_interval!(df, q=0.01)\n dist = Chisq(df)\n low, high, mid = quantile(dist, [q, 1-q])..., mean(dist)\n hline!([low, high], linestyle=:dash, color=:black, label=\"\",\n fill_between=[high nothing], fillcolor=:green, fillalpha=0.15)\n hline!([mid], linestyle=:solid, color=:black, label=\"\")\nend","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"
","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"
Code:","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"@variables t y1(t)=1.0 y2(t)=2.0 y3(t)=3.0\n@parameters p1=77.27 p2=8.375e-6 p3=0.161\nD = Differential(t)\neqs = [\n D(y1) ~ p1*(y2+y1*(1-p2*y1-y2))\n D(y2) ~ (y3-(1+y1)*y2)/p1\n D(y3) ~ p3*(y1-y3)\n]\n@named sys = ODESystem(eqs)\nsimpsys = structural_simplify(sys)\nmmprob = ODEProblem(sys,[],(0.0,30.0))\ndaeprob = DAEProblem(sys,[D(y1)=>77.26935286375,\n D(y2)=>-0.012941633234114146,\n D(y3)=>-0.322],[],(0.0,30.0))\nodaeprob = ODAEProblem(simpsys,[],(0.0,30.0))\n\nref_sol = solve(daeprob,IDA(),abstol=1/10^14,reltol=1/10^14)\n\nplot(ref_sol, title=\"OREGO Solution\", legend=false, xticks=:auto)","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"
","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"(Image: )","category":"page"},{"location":"benchmarks/orego/#orego_results","page":"OREGO","title":"EK1 across orders","text":"","category":"section"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"
Code:","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"DENSE = false;\nSAVE_EVERYSTEP = false;\n\n_setups = [\n \"EK1($order)\" => Dict(:alg => EK1(order=order, smooth=DENSE))\n for order in 2:6\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nabstols = 1.0 ./ 10.0 .^ (6:10)\nreltols = 1.0 ./ 10.0 .^ (3:7)\n\nwp = WorkPrecisionSet(\n mmprob, abstols, reltols, setups;\n names = labels,\n appxsol = ref_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n numruns = 10,\n maxiters = Int(1e7),\n)\n\nplot(wp, palette=Plots.palette([:blue, :red], length(_setups)))\n\n\n_ref_setups = [\n \"Rosenbrock23\" => Dict(:alg => Rosenbrock23())\n \"Rodas4P\" => Dict(:alg => Rodas4P())\n \"RadauIIA\" => Dict(:alg => RadauIIA5())\n]\nref_labels = first.(_ref_setups)\nref_setups = last.(_ref_setups)\nref_wp = WorkPrecisionSet(\n mmprob, abstols, reltols, ref_setups;\n names = ref_labels,\n appxsol = ref_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n)\nplot!(ref_wp, x=:final, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"
","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"(Image: )","category":"page"},{"location":"benchmarks/orego/#Calibration","page":"OREGO","title":"Calibration","text":"","category":"section"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"
Code:","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"plot(wp; x=:final, y=:chi2_final, yguide=\"Chi-squared (final)\",\n palette=Plots.palette([:blue, :red], length(_setups)))\nplot_chisq_interval!(3)","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"
","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"(Image: )","category":"page"},{"location":"benchmarks/orego/#Appendix","page":"OREGO","title":"Appendix","text":"","category":"section"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"
Computer information:","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"using InteractiveUtils\nInteractiveUtils.versioninfo()","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"Julia Version 1.10.0\nCommit 3120989f39b (2023-12-25 18:01 UTC)\nBuild Info:\n Official https://julialang.org/ release\nPlatform Info:\n OS: Linux (x86_64-linux-gnu)\n CPU: 12 × Intel(R) Core(TM) i7-6800K CPU @ 3.40GHz\n WORD_SIZE: 64\n LIBM: libopenlibm\n LLVM: libLLVM-15.0.7 (ORCJIT, broadwell)\n Threads: 17 on 12 virtual cores\nEnvironment:\n JULIA_NUM_THREADS = auto","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"
","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"
Package information:","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"using Pkg\nPkg.status()","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Project.toml`\n [f3b72e0c] DiffEqDevTools v2.44.1\n [31c24e10] Distributions v0.25.107\n [7073ff75] IJulia v1.24.2\n [7f56f5a3] LSODA v0.7.5\n [e6f89c97] LoggingExtras v1.0.3\n [e2752cbe] MATLABDiffEq v1.2.0\n⌃ [961ee093] ModelingToolkit v8.73.2\n [54ca160b] ODEInterface v0.5.0\n [09606e27] ODEInterfaceDiffEq v3.13.3\n⌃ [1dea7af3] OrdinaryDiffEq v6.66.0\n [65888b18] ParameterizedFunctions v5.16.0\n [91a5bcdd] Plots v1.39.0\n [bf3e78b0] ProbNumDiffEq v0.14.0 `~/.julia/dev/ProbNumDiffEq`\n⌃ [0bca4576] SciMLBase v2.10.0\n [505e40e9] SciPyDiffEq v0.2.1\n [ce78b400] SimpleUnPack v1.1.0\n [90137ffa] StaticArrays v1.9.1\n [c3572dad] Sundials v4.23.1\n [44d3d7a6] Weave v0.10.12\n [0518478a] deSolveDiffEq v0.1.1\nInfo Packages marked with ⌃ have new versions available and may be upgradable.","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"
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Full manifest:","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"Pkg.status(mode=Pkg.PKGMODE_MANIFEST)","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Manifest.toml`\n [47edcb42] ADTypes v0.2.6\n⌅ [c3fe647b] AbstractAlgebra v0.34.7\n [621f4979] AbstractFFTs v1.5.0\n [1520ce14] AbstractTrees v0.4.4\n [7d9f7c33] Accessors v0.1.35\n⌅ [79e6a3ab] Adapt v3.7.2\n [ec485272] ArnoldiMethod v0.2.0\n [c9d4266f] ArrayAllocators v0.3.0\n [4fba245c] ArrayInterface v7.7.0\n [4c555306] ArrayLayouts v1.5.2\n [6e4b80f9] BenchmarkTools v1.4.0\n [e2ed5e7c] Bijections v0.1.6\n [d1d4a3ce] BitFlags v0.1.8\n [62783981] BitTwiddlingConvenienceFunctions v0.1.5\n [fa961155] CEnum v0.5.0\n [2a0fbf3d] CPUSummary v0.2.4\n [00ebfdb7] CSTParser v3.4.0\n [49dc2e85] Calculus v0.5.1\n [324d7699] CategoricalArrays v0.10.8\n [d360d2e6] ChainRulesCore v1.19.1\n [fb6a15b2] CloseOpenIntervals 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Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"
","category":"page"},{"location":"solvers/#Solvers","page":"Solvers","title":"Solvers","text":"","category":"section"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"ProbNumDiffEq.jl provides two solvers: the EK1 and the EK0. Both based on extended Kalman filtering and smoothing, but the latter relies on evaluating the Jacobian of the vector field.","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"Which solver should I use?","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"Use the EK1 to get the best uncertainty quantification and to solve stiff problems.\nUse the EK0 to get the fastest runtimes and to solve high-dimensional problems.","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"All solvers are compatible with DAEs in mass-matrix ODE form. They also specialize on second-order ODEs: If the problem is of type SecondOrderODEProblem, it solves the second-order problem directly; this is more efficient than solving the transformed first-order problem and provides more meaningful posteriors [1].","category":"page"},{"location":"solvers/#API","page":"Solvers","title":"API","text":"","category":"section"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"EK1\nEK0","category":"page"},{"location":"solvers/#ProbNumDiffEq.EK1","page":"Solvers","title":"ProbNumDiffEq.EK1","text":"EK1(; order=3,\n smooth=true,\n prior=IWP(order),\n diffusionmodel=DynamicDiffusion(),\n initialization=TaylorModeInit(num_derivatives(prior)),\n kwargs...)\n\nGaussian ODE filter with first-order vector field linearization.\n\nThis is a semi-implicit, L-stable ODE solver so it can handle stiffness quite well [4], and it generally produces more expressive posterior covariances than the EK0. However, as typical implicit ODE solvers it scales cubically with the ODE dimension [5], so if you're solving a high-dimensional non-stiff problem you might want to give the EK0 a try.\n\nArguments\n\norder::Integer: Order of the integrated Wiener process (IWP) prior.\nsmooth::Bool: Turn smoothing on/off; smoothing is required for dense output.\nprior::AbstractGaussMarkovProcess: Prior to be used by the ODE filter. By default, uses a 3-times integrated Wiener process prior IWP(3). See also: Priors.\ndiffusionmodel::ProbNumDiffEq.AbstractDiffusion: See Diffusion models and calibration.\ninitialization::ProbNumDiffEq.InitializationScheme: See Initialization.\n\nSome additional kwargs relating to implicit solvers are supported; check out DifferentialEquations.jl's Extra Options page. Right now, we support autodiff, chunk_size, and diff_type. In particular, autodiff=false can come in handy to use finite differences instead of ForwardDiff.jl to compute Jacobians.\n\nExamples\n\njulia> solve(prob, EK1())\n\nReferences\n\n\n\n\n\n","category":"type"},{"location":"solvers/#ProbNumDiffEq.EK0","page":"Solvers","title":"ProbNumDiffEq.EK0","text":"EK0(; order=3,\n smooth=true,\n prior=IWP(order),\n diffusionmodel=DynamicDiffusion(),\n initialization=TaylorModeInit(num_derivatives(prior)))\n\nGaussian ODE filter with zeroth-order vector field linearization.\n\nThis is an explicit ODE solver. It is fast and scales well to high-dimensional problems [5], but it is not L-stable [4]. So for stiff problems, use the EK1.\n\nWhenever possible this solver will use a Kronecker-factored implementation to achieve its linear scaling and to get the best runtimes. This can currently be done only with an IWP prior (default), with a scalar diffusion model (either DynamicDiffusion or FixedDiffusion). For other configurations the solver falls back to a dense implementation which scales cubically with the problem size.\n\nArguments\n\norder::Integer: Order of the integrated Wiener process (IWP) prior.\nsmooth::Bool: Turn smoothing on/off; smoothing is required for dense output.\nprior::AbstractGaussMarkovProcess: Prior to be used by the ODE filter. By default, uses a 3-times integrated Wiener process prior IWP(3). See also: Priors.\ndiffusionmodel::ProbNumDiffEq.AbstractDiffusion: See Diffusion models and calibration.\ninitialization::ProbNumDiffEq.InitializationScheme: See Initialization.\n\nExamples\n\njulia> solve(prob, EK0())\n\nReferences\n\n\n\n\n\n","category":"type"},{"location":"solvers/#Probabilistic-Exponential-Integrators","page":"Solvers","title":"Probabilistic Exponential Integrators","text":"","category":"section"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"ExpEK\nRosenbrockExpEK","category":"page"},{"location":"solvers/#ProbNumDiffEq.ExpEK","page":"Solvers","title":"ProbNumDiffEq.ExpEK","text":"ExpEK(; L, order=3, kwargs...)\n\nProbabilistic exponential integrator\n\nProbabilistic exponential integrators are a class of integrators for semi-linear stiff ODEs that provide improved stability by essentially solving the linear part of the ODE exactly. In probabilistic numerics, this amounts to including the linear part into the prior model of the solver.\n\nExpEK is therefore just a short-hand for EK0 with IOUP prior:\n\nExpEK(; order=3, L, kwargs...) = EK0(; prior=IOUP(order, L), kwargs...)\n\nSee also RosenbrockExpEK, EK0, EK1.\n\nArguments\n\nSee EK0 for available keyword arguments.\n\nExamples\n\njulia> prob = ODEProblem((du, u, p, t) -> (@. du = - u + sin(u)), [1.0], (0.0, 10.0))\njulia> solve(prob, ExpEK(L=-1))\n\nReference\n\n[2] Bosch et al, \"Probabilistic Exponential Integrators\", arXiv (2021)\n\n\n\n\n\n","category":"function"},{"location":"solvers/#ProbNumDiffEq.RosenbrockExpEK","page":"Solvers","title":"ProbNumDiffEq.RosenbrockExpEK","text":"RosenbrockExpEK(; order=3, kwargs...)\n\nProbabilistic Rosenbrock-type exponential integrator\n\nA probabilistic exponential integrator similar to ExpEK, but with automatic linearization along the mean numerical solution. This brings the advantage that the linearity does not need to be specified manually, and the more accurate local linearization can sometimes also improve stability; but since the \"prior\" is adjusted at each step the probabilistic interpretation becomes more complicated.\n\nRosenbrockExpEK is just a short-hand for EK1 with locally-updated IOUP prior:\n\nRosenbrockExpEK(; order=3, kwargs...) = EK1(; prior=IOUP(order, update_rate_parameter=true), kwargs...)\n\nSee also ExpEK, EK0, EK1.\n\nArguments\n\nSee EK1 for available keyword arguments.\n\nExamples\n\njulia> prob = ODEProblem((du, u, p, t) -> (@. du = - u + sin(u)), [1.0], (0.0, 10.0))\njulia> solve(prob, RosenbrockExpEK())\n\nReference\n\n[2] Bosch et al, \"Probabilistic Exponential Integrators\", NeurIPS (2023)\n\n\n\n\n\n","category":"function"},{"location":"solvers/#solversrefs","page":"Solvers","title":"References","text":"","category":"section"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"F. Tronarp, H. Kersting, S. Särkkä and P. Hennig. Probabilistic solutions to ordinary differential equations as nonlinear Bayesian filtering: a new perspective. Statistics and Computing 29, 1297–1315 (2019).\n\n\n\nN. Krämer, N. Bosch, J. Schmidt and P. Hennig. Probabilistic ODE Solutions in Millions of Dimensions. In: Proceedings of the 39th International Conference on Machine Learning, Vol. 162 of Proceedings of Machine Learning Research, edited by K. Chaudhuri, S. Jegelka, L. Song, C. Szepesvari, G. Niu and S. Sabato (PMLR, 17–23 Jul 2022); pp. 11634–11649.\n\n\n\nN. Bosch, P. Hennig and F. Tronarp. Probabilistic Exponential Integrators. In: Thirty-seventh Conference on Neural Information Processing Systems (2023).\n\n\n\n","category":"page"},{"location":"benchmarks/hodgkinhuxley/#Hodgkin-Huxley-benchmark","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley benchmark","text":"","category":"section"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"note: Summary\nHodgkin-Huxley is a four-dimensional ODE, which can be stiff or non-stiff depending on the parameters; here we consider a non-stiff version. We see that:EK0 is the fastest solver.\nRosenbrockExpEK is slowest; but suffers less from smoothing than EK0 and EK1.\nResults are similar for fixed time steps.","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
Code:","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"using LinearAlgebra, Statistics, Distributions\nusing DiffEqDevTools, SciMLBase, OrdinaryDiffEq, Plots, SimpleUnPack\nusing ProbNumDiffEq\n\nPlots.theme(\n :dao;\n markerstrokewidth=0.5,\n legend=:outertopright,\n margin=5Plots.mm,\n xticks=10.0 .^ (-16:1:16),\n yticks=10.0 .^ (-6:1:5),\n)","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
Code:","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"αm(V, VT) = -0.32 * (V - VT - 13) / (exp(-(V - VT - 13) / 4) - 1)\nβm(V, VT) = 0.28 * (V - VT - 40) / (exp((V - VT - 40) / 5) - 1)\n\nαn(V, VT) = -0.032 * (V - VT - 15) / (exp(-(V - VT - 15) / 5) - 1)\nβn(V, VT) = 0.5 * exp(-(V - VT - 10) / 40)\n\nαh(V, VT) = 0.128 * exp(-(V - VT - 17) / 18)\nβh(V, VT) = 4 / (1 + exp(-(V - VT - 40) / 5))\n\nconst current_tspan = (5, 40)\nInj(t) = (current_tspan[1] <= t <= current_tspan[2]) ? 500one(t) : zero(t)\n\nfunction f(du, u, p, t)\n @unpack gNa, gK, ENa, EK, area, C, Eleak, VT, gleak = p\n\n V, m, n, h = u\n\n I_inj = Inj(t) * 1e-6\n\n du[2] = dmdt = (αm(V, VT) * (1 - m) - βm(V, VT) * m)\n du[3] = dndt = (αn(V, VT) * (1 - n) - βn(V, VT) * n)\n du[4] = dhdt = (αh(V, VT) * (1 - h) - βh(V, VT) * h)\n\n INa = gNa * m^3 * h * (V - ENa) * area\n IK = gK * n^4 * (V - EK) * area\n Ileak = gleak * (V - Eleak) * area\n Cm = C * area\n du[1] = dVdt = -(Ileak + INa + IK - I_inj) / Cm\nend\n\np = (gNa=20.0, gK=15.0, ENa = 53, EK = -107, area = 15e-5, C = 1, Eleak = -70, VT = -60, gleak = 0.1, V0 = -70)\n\nm_inf(V, VT) = 1 / (1 + βm(V, VT) / αm(V, VT))\nn_inf(V, VT) = 1 / (1 + βn(V, VT) / αn(V, VT))\nh_inf(V, VT) = 1 / (1 + βh(V, VT) / αh(V, VT))\nu0 = [p.V0, m_inf(p.V0, p.VT), n_inf(p.V0, p.VT), h_inf(p.V0, p.VT)]\n\nprob = ODEProblem{true,SciMLBase.FullSpecialize()}(f, u0, (0.0, 50.0), p)\n\ntest_sol = solve(prob, Vern7(), abstol=1/10^14, reltol=1/10^14)\nplot(test_sol,\n legend=false,\n layout=(4,1),\n title=[\"Hodgkin-Huxley Solution\" \"\" \"\" \"\"],\n ylabel=[\"V(t)\" \"m(t)\" \"n(t)\" \"h(t)\"],\n xlabel=[\"\" \"\" \"\" \"t\"],\n size = (1000, 600),\n color=[1 2 3 4],\n xticks=:auto, yticks=:auto\n)","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"(Image: )","category":"page"},{"location":"benchmarks/hodgkinhuxley/#hh_solver_comparison","page":"Hodgkin-Huxley","title":"Solver comparison: EK0 vs. EK1 vs RosenbrockExpEK","text":"","category":"section"},{"location":"benchmarks/hodgkinhuxley/#Without-smoothing","page":"Hodgkin-Huxley","title":"Without smoothing","text":"","category":"section"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
Code:","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"DENSE = SAVE_EVERYSTEP = false\n\n_setups = [\n \"EK0(2)\" => Dict(:alg=>EK0(order=2, smooth=DENSE))\n \"EK0(3)\" => Dict(:alg=>EK0(order=3, smooth=DENSE))\n \"EK1(2)\" => Dict(:alg=>EK1(order=2, smooth=DENSE))\n \"EK1(3)\" => Dict(:alg=>EK1(order=3, smooth=DENSE))\n \"RosenbrockExpEK1(2)\" => Dict(:alg=>RosenbrockExpEK(order=2, smooth=DENSE))\n \"RosenbrockExpEK1(3)\" => Dict(:alg=>RosenbrockExpEK(order=3, smooth=DENSE))\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\ncolors = [1 1 2 2 3 3]\n\nabstols = 1.0 ./ 10.0 .^ (6:10)\nreltols = 1.0 ./ 10.0 .^ (3:7)\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n numruns = 5,\n tstops = current_tspan,\n)\n\nplot(wp, title=\"Adaptive steps - no smoothing\", color=colors)\n\n\n_ref_setups = [\n \"Tsit5\" => Dict(:alg => Tsit5())\n \"Vern7\" => Dict(:alg => Vern7())\n \"RadauIIA5\" => Dict(:alg => RadauIIA5())\n]\nref_labels = first.(_ref_setups)\nref_setups = last.(_ref_setups)\nref_wp_final = WorkPrecisionSet(\n prob, abstols ./ 1000, reltols ./ 1000, ref_setups;\n names = ref_labels,\n appxsol = test_sol,\n dense = false,\n save_everystep = false,\n maxiters = Int(1e7),\n tstops = current_tspan,\n)\nref_wp_dense = WorkPrecisionSet(\n prob, abstols ./ 1000, reltols ./ 1000, ref_setups;\n names = ref_labels,\n appxsol = test_sol,\n dense = true,\n save_everystep = true,\n maxiters = Int(1e7),\n tstops = current_tspan,\n)\n\nplot!(ref_wp_final, x=:final, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"(Image: )","category":"page"},{"location":"benchmarks/hodgkinhuxley/#With-smoothing","page":"Hodgkin-Huxley","title":"With smoothing","text":"","category":"section"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
Code:","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"DENSE = SAVE_EVERYSTEP = true\n\n_setups = [\n \"EK0(2)\" => Dict(:alg=>EK0(order=2, smooth=DENSE))\n \"EK0(3)\" => Dict(:alg=>EK0(order=3, smooth=DENSE))\n \"EK1(2)\" => Dict(:alg=>EK1(order=2, smooth=DENSE))\n \"EK1(3)\" => Dict(:alg=>EK1(order=3, smooth=DENSE))\n \"RosenbrockExpEK1(2)\" => Dict(:alg=>RosenbrockExpEK(order=2, smooth=DENSE))\n \"RosenbrockExpEK1(3)\" => Dict(:alg=>RosenbrockExpEK(order=3, smooth=DENSE))\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\ncolors = [1 1 2 2 3 3]\n\nabstols = 1.0 ./ 10.0 .^ (6:10)\nreltols = 1.0 ./ 10.0 .^ (3:7)\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n numruns = 5,\n tstops = current_tspan,\n)\n\nplot(wp, title=\"Adaptive steps - with smoothing\", color=colors)\nplot!(ref_wp_dense, x=:final, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"(Image: )","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
Interoplation errors (L2):","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
Code:","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"plot(wp, x=:L2, title=\"Adaptive steps - with smoothing\", color=colors)\nplot!(ref_wp_dense, x=:L2, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"(Image: )","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
","category":"page"},{"location":"benchmarks/hodgkinhuxley/#Calibration","page":"Hodgkin-Huxley","title":"Calibration","text":"","category":"section"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
Code:","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"plot(wp; x=:final, y=:chi2_final, yguide=\"Chi-squared (final)\", color=colors)\n\n# Should be distributed according to a Chi-squared distribution:\nfunction plot_chisq_interval!(df, q=0.01)\n dist = Chisq(df)\n low, high, mid = quantile(dist, [q, 1-q])..., mean(dist)\n hline!([low, high], linestyle=:dash, color=:black, label=\"\",\n fill_between=[high nothing], fillcolor=:green, fillalpha=0.15)\n hline!([mid], linestyle=:solid, color=:black, label=\"\")\nend\nplot_chisq_interval!(4)","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"(Image: )","category":"page"},{"location":"benchmarks/hodgkinhuxley/#hh_fixed_steps","page":"Hodgkin-Huxley","title":"Fixed-step solver comparison","text":"","category":"section"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"Without smothing:","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
Code:","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"DENSE = SAVE_EVERYSTEP = false\n\ndts = 10.0 .^ range(-2, -3, length=10)[begin:end-1]\nabstols = reltols = repeat([missing], length(dts))\n\nDM = FixedDiffusion()\n_setups = [\n \"EK0(3)\" => Dict(:alg=>EK0(order=3, diffusionmodel=DM, smooth=DENSE), :dts=>dts)\n \"EK1(3)\" => Dict(:alg=>EK1(order=3, diffusionmodel=DM, smooth=DENSE), :dts=>dts)\n \"RosenbrockExpEK1(3)\" => Dict(:alg=>RosenbrockExpEK(order=3, diffusionmodel=DM, smooth=DENSE), :dts=>dts)\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\ncolors = [1 2 3]\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n adaptive = false,\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n numruns = 5,\n)\n\nplot(wp, title=\"Fixed steps - no smoothing\", color=colors)","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"(Image: )","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
With smoothing:","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
Code:","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"DENSE = SAVE_EVERYSTEP = true\n\ndts = 10.0 .^ range(-2, -3, length=10)[begin:end-1]\nabstols = reltols = repeat([missing], length(dts))\n\nDM = FixedDiffusion()\n_setups = [\n \"EK0(3)\" => Dict(:alg=>EK0(order=3, diffusionmodel=DM, smooth=DENSE), :dts=>dts)\n \"EK1(3)\" => Dict(:alg=>EK1(order=3, diffusionmodel=DM, smooth=DENSE), :dts=>dts)\n \"RosenbrockExpEK1(3)\" => Dict(:alg=>RosenbrockExpEK(order=3, diffusionmodel=DM, smooth=DENSE), :dts=>dts)\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\ncolors = [1 2 3]\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n adaptive = false,\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n numruns = 5,\n)\n\nplot(wp, title=\"Fixed steps - with smoothing\", color=colors)","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"(Image: )","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
","category":"page"},{"location":"benchmarks/hodgkinhuxley/#Appendix","page":"Hodgkin-Huxley","title":"Appendix","text":"","category":"section"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
Computer information:","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"using InteractiveUtils\nInteractiveUtils.versioninfo()","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"Julia Version 1.10.0\nCommit 3120989f39b (2023-12-25 18:01 UTC)\nBuild Info:\n Official https://julialang.org/ release\nPlatform Info:\n OS: Linux (x86_64-linux-gnu)\n CPU: 12 × Intel(R) Core(TM) i7-6800K CPU @ 3.40GHz\n WORD_SIZE: 64\n LIBM: libopenlibm\n LLVM: libLLVM-15.0.7 (ORCJIT, broadwell)\n Threads: 17 on 12 virtual cores\nEnvironment:\n JULIA_NUM_THREADS = auto","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
Package information:","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"using Pkg\nPkg.status()","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Project.toml`\n [f3b72e0c] DiffEqDevTools v2.44.1\n [31c24e10] Distributions v0.25.107\n [7073ff75] IJulia v1.24.2\n [7f56f5a3] LSODA v0.7.5\n [e6f89c97] LoggingExtras v1.0.3\n [e2752cbe] MATLABDiffEq v1.2.0\n⌃ [961ee093] ModelingToolkit v8.73.2\n [54ca160b] ODEInterface v0.5.0\n [09606e27] ODEInterfaceDiffEq v3.13.3\n⌃ [1dea7af3] OrdinaryDiffEq v6.66.0\n [65888b18] ParameterizedFunctions v5.16.0\n [91a5bcdd] Plots v1.39.0\n [bf3e78b0] ProbNumDiffEq v0.14.0 `~/.julia/dev/ProbNumDiffEq`\n⌃ [0bca4576] SciMLBase v2.10.0\n [505e40e9] SciPyDiffEq v0.2.1\n [ce78b400] SimpleUnPack v1.1.0\n [90137ffa] StaticArrays v1.9.1\n [c3572dad] Sundials v4.23.1\n [44d3d7a6] Weave v0.10.12\n [0518478a] deSolveDiffEq v0.1.1\nInfo Packages marked with ⌃ have new versions available and may be upgradable.","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
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available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
","category":"page"},{"location":"tutorials/exponential_integrators/#probexpinttutorial","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"","category":"section"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"Exponential integrators are a class of numerical methods for solving semi-linear ordinary differential equations (ODEs) of the form","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"beginaligned\ndoty(t) = L y(t) + f(y(t) t) quad y(0) = y_0\nendaligned","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"where L is a linear operator and f is a nonlinear function. In a nutshell, exponential integrators solve the linear part of the ODE exactly, and only approximate the nonlinear part. Probabilistic exponential integrators [2] are the probabilistic numerics approach to exponential integrators.","category":"page"},{"location":"tutorials/exponential_integrators/#Example","page":"Probabilistic Exponential Integrators","title":"Example","text":"","category":"section"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"Let's consider a simple semi-linear ODE","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"beginaligned\ndoty(t) = - y(t) + sin(y(t)) quad y(0) = 10\nendaligned","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"We can solve this ODE reasonably well with the standard EK1 and adaptive steps (the default):","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"using ProbNumDiffEq, Plots, LinearAlgebra\ntheme(:default; palette=[\"#4063D8\", \"#389826\", \"#9558B2\", \"#CB3C33\"])\n\nf(du, u, p, t) = (@. du = -u + sin(u))\nu0 = [1.0]\ntspan = (0.0, 20.0)\nprob = ODEProblem(f, u0, tspan)\n\nref = solve(prob, EK1(), abstol=1e-10, reltol=1e-10)\nplot(ref, color=:black, linestyle=:dash, label=\"Reference\")","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"But for fixed (large) step sizes this ODE is more challenging: The explicit EK0 method oscillates and diverges due to the stiffness of the ODE, and the semi-implicit EK1 method is stable but the solution is not very accurate.","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"STEPSIZE = 4\nDM = FixedDiffusion() # recommended for fixed steps\n\n# we don't smooth the EK0 here to show the oscillations more clearly\nsol0 = solve(prob, EK0(smooth=false, diffusionmodel=DM), adaptive=false, dt=STEPSIZE, dense=false)\nsol1 = solve(prob, EK1(diffusionmodel=DM), adaptive=false, dt=STEPSIZE)\n\nplot(ylims=(0.3, 1.05))\nplot!(ref, color=:black, linestyle=:dash, label=\"Reference\")\nplot!(sol0, denseplot=false, marker=:o, markersize=2, label=\"EK0\", color=1)\nplot!(sol1, denseplot=false, marker=:o, markersize=2, label=\"EK1\", color=2)","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"Probabilistic exponential integrators leverage the semi-linearity of the ODE to compute more accurate solutions for the same fixed step size. You can use either the ExpEK method and provide the linear part (with the keyword argument L), or the RosenbrockExpEK to automatically linearize along the mean of the numerical solution:","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"sol_exp = solve(prob, ExpEK(L=-1, diffusionmodel=DM), adaptive=false, dt=STEPSIZE)\nsol_ros = solve(prob, RosenbrockExpEK(diffusionmodel=DM), adaptive=false, dt=STEPSIZE)\n\nplot(ylims=(0.3, 1.05))\nplot!(ref, color=:black, linestyle=:dash, label=\"Reference\")\nplot!(sol_exp, denseplot=false, marker=:o, markersize=2, label=\"ExpEK\", color=3)\nplot!(sol_ros, denseplot=false, marker=:o, markersize=2, label=\"RosenbrockExpEK\", color=4)","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"The solutions are indeed much more accurate than those of the standard EK1, for the same fixed step size!","category":"page"},{"location":"tutorials/exponential_integrators/#Background:-Integrated-Ornstein-Uhlenbeck-priors","page":"Probabilistic Exponential Integrators","title":"Background: Integrated Ornstein-Uhlenbeck priors","text":"","category":"section"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"Probabilistic exponential integrators \"solve the linear part exactly\" by including it into the prior model of the solver. Namely, the solver chooses a (q-times) integrated Ornstein-Uhlenbeck prior with rate parameter equal to the linearity. The ExpEK solver is just a short-hand for an EK0 with appropriate prior:","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"ExpEK(order=3, L=-1) == EK0(prior=IOUP(3, -1))","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"Similarly, the RosenbrockExpEK solver is also just a short-hand:","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"RosenbrockExpEK(order=3) == EK1(prior=IOUP(3, update_rate_parameter=true))","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"This means that you can also construct other probabilistic exponential integrators by hand! In this example the EK1 with IOUP prior with rate parameter -1 performs extremely well:","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"sol_expek1 = solve(prob, EK1(prior=IOUP(3, -1), diffusionmodel=DM), adaptive=false, dt=STEPSIZE)\n\nplot(ylims=(0.3, 1.05))\nplot!(ref, color=:black, linestyle=:dash, label=\"Reference\")\nplot!(sol_expek1, denseplot=false, marker=:o, markersize=2, label=\"EK1 + IOUP\")","category":"page"},{"location":"tutorials/exponential_integrators/#References","page":"Probabilistic Exponential Integrators","title":"References","text":"","category":"section"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"N. Bosch, P. Hennig and F. Tronarp. Probabilistic Exponential Integrators. In: Thirty-seventh Conference on Neural Information Processing Systems (2023).\n\n\n\n","category":"page"},{"location":"benchmarks/lotkavolterra/#Lotka-Volterra-benchmark","page":"Lotka-Volterra","title":"Lotka-Volterra benchmark","text":"","category":"section"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"note: Summary\nLotka-Volterra is a simple, low-dimensional, non-stiff ODE. We see that:EK0 and EK1 have a very similar runtime. But note that Lotka-Volterra is a non-stiff low-dimensional problem: if it were stiff, EK1 would be better; if it were high-dimensional, EK0 would be faster.\nOrders behave as in classic solvers: Use low order for low accuracy, medium order for medium accuracy, high order for high accuracy.\nDo not use EK0 with order > 5: The adaptive step size selection apparently does not work well for high orders right now.\nUse diffusionmodel=DynamicDiffusion: Error-wise, the performance of the diffusion models is similar, but the calibration of FixedDiffusion and FixedMVDiffusion with adaptive steps is currently broken.\nInitialization schemes are all similar, but initialization=TaylorModeInit performs best.\nIf you only need to solve for the last time point, set smooth=false, dense=false, and save_everystep=false. This greatly reduces the run time of the solver.","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"Benchmark adapted from SciMLBenchmarks.jl.","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"using LinearAlgebra, Statistics, Distributions\nusing DiffEqDevTools, ParameterizedFunctions, SciMLBase, OrdinaryDiffEq, Plots\nusing ProbNumDiffEq\n\nPlots.theme(\n :dao;\n markerstrokewidth=0.5,\n legend=:outertopright,\n margin=5Plots.mm,\n xticks=10.0 .^ (-16:1:16),\n)\n\nfunction plot_chisq_interval!(df, q=0.01)\n dist = Chisq(df)\n low, high, mid = quantile(dist, [q, 1-q])..., mean(dist)\n hline!([low, high], linestyle=:dash, color=:black, label=\"\",\n fill_between=[high nothing], fillcolor=:green, fillalpha=0.15)\n hline!([mid], linestyle=:solid, color=:black, label=\"\")\nend","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"f = @ode_def LotkaVolterra begin\n dx = a*x - b*x*y\n dy = -c*y + d*x*y\nend a b c d\np = [1.5, 1, 3, 1]\ntspan = (0.0, 10.0)\nu0 = [1.0, 1.0]\nprob = ODEProblem{true, SciMLBase.FullSpecialize}(f, u0, tspan, p)\n\ntest_sol = solve(prob, Vern7(), abstol=1/10^14, reltol=1/10^14)\nplot(test_sol, title=\"Lotka-Volterra Solution\", legend=false, xticks=:auto)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/#lv_ek0_comparison","page":"Lotka-Volterra","title":"EK0 Benchmark Across Orders","text":"","category":"section"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"DENSE = false;\nSAVE_EVERYSTEP = false;\n\n_setups = [\n \"EK0(order=$order)\" => Dict(:alg => EK0(order=order, smooth=DENSE))\n for order in 2:7\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nabstols = 1.0 ./ 10.0 .^ (4:12)\nreltols = 1.0 ./ 10.0 .^ (1:9)\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n)\n\nplot(wp, palette=Plots.palette([:blue, :red], length(_setups)))\n\n_ref_setups = [\n \"Tsit5\" => Dict(:alg => Tsit5())\n \"Vern7\" => Dict(:alg => Vern7())\n \"RadauIIA5\" => Dict(:alg => RadauIIA5())\n]\nref_labels = first.(_ref_setups)\nref_setups = last.(_ref_setups)\nref_wp_final = WorkPrecisionSet(\n prob, abstols, reltols, ref_setups;\n names = ref_labels,\n appxsol = test_sol,\n dense = false,\n save_everystep = false,\n maxiters = Int(1e7),\n)\nref_wp_dense = WorkPrecisionSet(\n prob, abstols, reltols, ref_setups;\n names = ref_labels,\n appxsol = test_sol,\n dense = true,\n save_everystep = true,\n maxiters = Int(1e7),\n)\n\nplot!(ref_wp_final, x=:final, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Discrete time-series errors (l2):","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"DENSE = true;\nSAVE_EVERYSTEP = true;\n\n_setups = [\n \"EK0(order=$order)\" => Dict(:alg => EK0(order=order, smooth=DENSE))\n for order in 2:7\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nabstols = 1.0 ./ 10.0 .^ (4:12)\nreltols = 1.0 ./ 10.0 .^ (1:9)\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n)\n\nplot(wp, x=:l2, palette=Plots.palette([:blue, :red], length(_setups)))\nplot!(ref_wp_dense, x=:l2, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Interoplation errors (L2):","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"plot(wp, x=:L2, palette=Plots.palette([:blue, :red], length(_setups)))\nplot!(ref_wp_dense, x=:L2, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/#lv_ek1_comparison","page":"Lotka-Volterra","title":"EK1 Benchmark Across Orders","text":"","category":"section"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"DENSE = false;\nSAVE_EVERYSTEP = false;\n\n_setups = [\n \"EK1(order=$order)\" => Dict(:alg => EK1(order=order, smooth=DENSE))\n for order in 2:7\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nabstols = 1.0 ./ 10.0 .^ (4:12)\nreltols = 1.0 ./ 10.0 .^ (1:9)\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n)\n\nplot(wp, palette=Plots.palette([:blue, :red], length(_setups)))\nplot!(ref_wp_final, x=:final, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Discrete time-series errors (l2):","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"DENSE = true;\nSAVE_EVERYSTEP = true;\n\n_setups = [\n \"EK1(order=$order)\" => Dict(:alg => EK1(order=order, smooth=DENSE))\n for order in 2:7\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nabstols = 1.0 ./ 10.0 .^ (4:12)\nreltols = 1.0 ./ 10.0 .^ (1:9)\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n)\n\nplot(wp, x=:l2, palette=Plots.palette([:blue, :red], length(_setups)))\nplot!(ref_wp_dense, x=:l2, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Interoplation errors (L2):","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"plot(wp, x=:L2, palette=Plots.palette([:blue, :red], length(_setups)))\nplot!(ref_wp_dense, x=:L2, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/#lv_ek0_vs_ek1","page":"Lotka-Volterra","title":"EK0 vs. EK1: Work-Precision","text":"","category":"section"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"DENSE = false;\nSAVE_EVERYSTEP = false;\n\n_setups = [\n \"EK0(order=2)\" => Dict(:alg => EK0(order=2, smooth=DENSE))\n \"EK0(order=3)\" => Dict(:alg => EK0(order=3, smooth=DENSE))\n \"EK0(order=4)\" => Dict(:alg => EK0(order=4, smooth=DENSE))\n \"EK0(order=5)\" => Dict(:alg => EK0(order=5, smooth=DENSE))\n \"EK1(order=2)\" => Dict(:alg => EK1(order=2, smooth=DENSE))\n \"EK1(order=3)\" => Dict(:alg => EK1(order=3, smooth=DENSE))\n \"EK1(order=4)\" => Dict(:alg => EK1(order=4, smooth=DENSE))\n \"EK1(order=5)\" => Dict(:alg => EK1(order=5, smooth=DENSE))\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nabstols = 1.0 ./ 10.0 .^ (4:12)\nreltols = 1.0 ./ 10.0 .^ (1:9)\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n)\n\nplot(wp, color=[1 1 1 1 2 2 2 2])\nplot!(ref_wp_final, x=:final, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Interoplation errors (L2):","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"DENSE = true;\nSAVE_EVERYSTEP = true;\n\n_setups = [\n \"EK0(order=2)\" => Dict(:alg => EK0(order=2, smooth=DENSE))\n \"EK0(order=3)\" => Dict(:alg => EK0(order=3, smooth=DENSE))\n \"EK0(order=4)\" => Dict(:alg => EK0(order=4, smooth=DENSE))\n \"EK0(order=5)\" => Dict(:alg => EK0(order=5, smooth=DENSE))\n \"EK1(order=2)\" => Dict(:alg => EK1(order=2, smooth=DENSE))\n \"EK1(order=3)\" => Dict(:alg => EK1(order=3, smooth=DENSE))\n \"EK1(order=4)\" => Dict(:alg => EK1(order=4, smooth=DENSE))\n \"EK1(order=5)\" => Dict(:alg => EK1(order=5, smooth=DENSE))\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nabstols = 1.0 ./ 10.0 .^ (4:12)\nreltols = 1.0 ./ 10.0 .^ (1:9)\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n)\n\nplot(wp, x=:L2, color=[1 1 1 1 2 2 2 2])\nplot!(ref_wp_dense, x=:L2, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/#[EK0](@ref)-vs.-[EK1](@ref):-Calibration","page":"Lotka-Volterra","title":"EK0 vs. EK1: Calibration","text":"","category":"section"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"plot(wp, x=:final, y=:chi2_final, color=[1 1 1 1 2 2 2 2], yguide=\"Chi-squared (final)\")\nplot_chisq_interval!(2)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/#lv_diffusion","page":"Lotka-Volterra","title":"Diffusion model comparison","text":"","category":"section"},{"location":"benchmarks/lotkavolterra/#[EK0](@ref)-with-different-diffusions","page":"Lotka-Volterra","title":"EK0 with different diffusions","text":"","category":"section"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"DENSE = false;\nSAVE_EVERYSTEP = false;\n\n_setups = [\n \"EK0(3) Dynamic\" => Dict(:alg => EK0(order=3, smooth=DENSE, diffusionmodel=DynamicDiffusion()))\n \"EK0(5) Dynamic\" => Dict(:alg => EK0(order=5, smooth=DENSE, diffusionmodel=DynamicDiffusion()))\n \"EK0(3) Fixed\" => Dict(:alg => EK0(order=3, smooth=DENSE, diffusionmodel=FixedDiffusion()))\n \"EK0(5) Fixed\" => Dict(:alg => EK0(order=5, smooth=DENSE, diffusionmodel=FixedDiffusion()))\n \"EK0(3) DynamicMV\" => Dict(:alg => EK0(order=3, smooth=DENSE, diffusionmodel=DynamicMVDiffusion()))\n \"EK0(5) DynamicMV\" => Dict(:alg => EK0(order=5, smooth=DENSE, diffusionmodel=DynamicMVDiffusion()))\n \"EK0(3) FixedMV\" => Dict(:alg => EK0(order=3, smooth=DENSE, diffusionmodel=FixedMVDiffusion()))\n \"EK0(5) FixedMV\" => Dict(:alg => EK0(order=5, smooth=DENSE, diffusionmodel=FixedMVDiffusion()))\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nabstols = 1.0 ./ 10.0 .^ (4:12)\nreltols = 1.0 ./ 10.0 .^ (1:9)\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n)\n\ncolor = [2 2 3 3 4 4 5 5]\nlinestyle = [:solid :dash :solid :dash :solid :dash :solid :dash]\nplot(wp; color, linestyle)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"plot(wp; x=:final, y=:chi2_final, color, linestyle, yguide=\"Chi-squared (final)\")\nplot_chisq_interval!(2)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/#[EK1](@ref)-with-different-diffusions","page":"Lotka-Volterra","title":"EK1 with different diffusions","text":"","category":"section"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"DENSE = false;\nSAVE_EVERYSTEP = false;\n\n_setups = [\n \"EK1(3) Dynamic\" => Dict(:alg => EK1(order=3, smooth=DENSE, diffusionmodel=DynamicDiffusion()))\n \"EK1(5) Dynamic\" => Dict(:alg => EK1(order=5, smooth=DENSE, diffusionmodel=DynamicDiffusion()))\n \"EK1(3) Fixed\" => Dict(:alg => EK1(order=3, smooth=DENSE, diffusionmodel=FixedDiffusion()))\n \"EK1(5) Fixed\" => Dict(:alg => EK1(order=5, smooth=DENSE, diffusionmodel=FixedDiffusion()))\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nabstols = 1.0 ./ 10.0 .^ (4:12)\nreltols = 1.0 ./ 10.0 .^ (1:9)\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n)\n\ncolor = [2 2 3 3]\nlinestyle = [:solid :dash :solid :dash]\nplot(wp; color, linestyle)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"plot(wp; x=:final, y=:chi2_final, color, linestyle, yguide=\"Chi-squared (final)\")\nplot_chisq_interval!(2)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/#lv_initialization","page":"Lotka-Volterra","title":"Initialization scheme comparison","text":"","category":"section"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"DENSE = false;\nSAVE_EVERYSTEP = false;\n\nabstols = 1.0 ./ 10.0 .^ (4:12)\nreltols = 1.0 ./ 10.0 .^ (1:9)\n\norders = (2, 3, 5, 8)\nps = []\nfor o in orders\n _setups = [\n \"EK1($o) TaylorInit\" => Dict(:alg => EK1(order=o, smooth=DENSE, initialization=TaylorModeInit(o)))\n \"EK1($o) ForwardDiffInit\" => Dict(:alg => EK1(order=o, smooth=DENSE, initialization=ForwardDiffInit(o)))\n \"EK1($o) SimpleInit\" => Dict(:alg => EK1(order=o, smooth=DENSE, initialization=SimpleInit()))\n \"EK1($o) ClassicSolverInit\" => Dict(:alg => EK1(order=o, smooth=DENSE, initialization=ClassicSolverInit()))\n ]\n\n labels = first.(_setups)\n setups = last.(_setups)\n\n wp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n )\n\n p = plot(wp, color=[2 4 5 6], xticks = 10.0 .^ (-16:1:5), title = \"Order $o\")\n push!(ps, p)\nend\nplot(\n ps...,\n layout=(length(orders), 1),\n size = (800, length(orders)*300),\n xlabel=[\"\" \"\" \"\" \"Error (final)\"],\n)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/#Appendix","page":"Lotka-Volterra","title":"Appendix","text":"","category":"section"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Computer information:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"using InteractiveUtils\nInteractiveUtils.versioninfo()","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"Julia Version 1.10.0\nCommit 3120989f39b (2023-12-25 18:01 UTC)\nBuild Info:\n Official https://julialang.org/ release\nPlatform Info:\n OS: Linux (x86_64-linux-gnu)\n CPU: 12 × Intel(R) Core(TM) i7-6800K CPU @ 3.40GHz\n WORD_SIZE: 64\n LIBM: libopenlibm\n LLVM: libLLVM-15.0.7 (ORCJIT, broadwell)\n Threads: 17 on 12 virtual cores\nEnvironment:\n JULIA_NUM_THREADS = auto","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Package information:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"using Pkg\nPkg.status()","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Project.toml`\n [f3b72e0c] DiffEqDevTools v2.44.1\n [31c24e10] Distributions v0.25.107\n [7073ff75] IJulia v1.24.2\n [7f56f5a3] LSODA v0.7.5\n [e6f89c97] LoggingExtras v1.0.3\n [e2752cbe] MATLABDiffEq v1.2.0\n⌃ [961ee093] ModelingToolkit v8.73.2\n [54ca160b] ODEInterface v0.5.0\n [09606e27] ODEInterfaceDiffEq v3.13.3\n⌃ [1dea7af3] OrdinaryDiffEq v6.66.0\n [65888b18] ParameterizedFunctions v5.16.0\n [91a5bcdd] Plots v1.39.0\n [bf3e78b0] ProbNumDiffEq v0.14.0 `~/.julia/dev/ProbNumDiffEq`\n⌃ [0bca4576] SciMLBase v2.10.0\n [505e40e9] SciPyDiffEq v0.2.1\n [ce78b400] SimpleUnPack v1.1.0\n [90137ffa] StaticArrays v1.9.1\n [c3572dad] Sundials v4.23.1\n [44d3d7a6] Weave v0.10.12\n [0518478a] deSolveDiffEq v0.1.1\nInfo Packages marked with ⌃ have new versions available and may be upgradable.","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Full manifest:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"Pkg.status(mode=Pkg.PKGMODE_MANIFEST)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Manifest.toml`\n [47edcb42] ADTypes v0.2.6\n⌅ [c3fe647b] AbstractAlgebra v0.34.7\n [621f4979] AbstractFFTs v1.5.0\n [1520ce14] AbstractTrees v0.4.4\n [7d9f7c33] Accessors v0.1.35\n⌅ [79e6a3ab] Adapt v3.7.2\n [ec485272] ArnoldiMethod v0.2.0\n [c9d4266f] ArrayAllocators v0.3.0\n [4fba245c] ArrayInterface v7.7.0\n [4c555306] ArrayLayouts v1.5.2\n [6e4b80f9] BenchmarkTools v1.4.0\n [e2ed5e7c] Bijections v0.1.6\n [d1d4a3ce] BitFlags v0.1.8\n [62783981] BitTwiddlingConvenienceFunctions v0.1.5\n [fa961155] CEnum v0.5.0\n [2a0fbf3d] CPUSummary v0.2.4\n [00ebfdb7] CSTParser v3.4.0\n [49dc2e85] Calculus v0.5.1\n [324d7699] CategoricalArrays v0.10.8\n [d360d2e6] 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available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"initialization/#Initialization","page":"Initialization","title":"Initialization","text":"","category":"section"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"The notion of \"initialization\" relates to the prior part of the model.","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"TL;DR: In the vast majority of cases, just stick to the exact Taylor-mode initialization TaylorModeInit!","category":"page"},{"location":"initialization/#Background:-The-prior","page":"Initialization","title":"Background: The prior","text":"","category":"section"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"We model the ODE solution y(t) with a Gauss–Markov prior. More precisely, let","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"beginaligned\nY(t) = left Y^(0)(t) Y^(1)(t) dots Y^(q)(t) right\nendaligned","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"be the solution to the SDE","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"beginaligned\ntextd Y^(i)(t) = Y^(i+1)(t) textdt qquad i = 0 dots q-1 \ntextd Y^(q)(t) = textcolor389826A Y(t) textdt + textcolor4063D8Gamma textdW(t) \nY(0) sim textcolor9558B2 mathcalN left( mu_0 Sigma_0 right) \nendaligned","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"Then Y^(i)(t) models the i-th derivative of y(t). In this section, we consider the initial distribution textcolorpurple mathcalN left( mu_0 Sigma_0 right) . If you're more interested in the drift matrix textcolor389826A check out the Priors section, and for more info on the diffusion textcolor4063D8Gamma check out the Diffusion models and calibration section.","category":"page"},{"location":"initialization/#Setting-the-initial-distribution","page":"Initialization","title":"Setting the initial distribution","text":"","category":"section"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"Let's assume an initial value problem of the form","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"beginaligned\ndoty(t) = f(y(t) t) qquad 0 T \ny(0) = y_0\nendaligned","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"It is clear that this contains quite some information for Y(0): The initial value y_0 and the vector field f imply","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"beginaligned\nY^(0)(0) = y_0 \nY^(1)(0) = f(y_0 0)\nendaligned","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"It turns out that we can also compute higher-order derivatives of y with the chain rule, and then use these to better initialize Y^(i)(0). This, done efficiently with Taylor-mode automatic differentiation by using TaylorIntegration.jl, is what TaylorModeInit does. See also [1].","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"In the vast majority of cases, just stick to the exact Taylor-mode initialization TaylorModeInit!","category":"page"},{"location":"initialization/#API","page":"Initialization","title":"API","text":"","category":"section"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"TaylorModeInit\nForwardDiffInit\nSimpleInit\nClassicSolverInit","category":"page"},{"location":"initialization/#ProbNumDiffEq.TaylorModeInit","page":"Initialization","title":"ProbNumDiffEq.TaylorModeInit","text":"TaylorModeInit(order)\n\nExact initialization via Taylor-mode automatic differentiation up to order order.\n\nThis is the recommended initialization method!\n\nIt uses TaylorIntegration.jl to efficiently compute the higher-order derivatives of the solution at the initial value, via Taylor-mode automatic differentiation.\n\nIn some special cases it can happen that TaylorIntegration.jl is incompatible with the given problem (typically because the problem definition does not allow for elements of type Taylor). If this happens, try one of SimpleInit, ForwardDiffInit (for low enough orders), ClassicSolverInit.\n\nReferences\n\n[7] Krämer et al, \"Stable Implementation of Probabilistic ODE Solvers\" (2020)\n\n\n\n\n\n","category":"type"},{"location":"initialization/#ProbNumDiffEq.ForwardDiffInit","page":"Initialization","title":"ProbNumDiffEq.ForwardDiffInit","text":"ForwardDiffInit(order)\n\nExact initialization via ForwardDiff.jl up to order order.\n\nWarning: This does not scale well to high orders! For orders > 3, TaylorModeInit most likely performs better.\n\n\n\n\n\n","category":"type"},{"location":"initialization/#ProbNumDiffEq.SimpleInit","page":"Initialization","title":"ProbNumDiffEq.SimpleInit","text":"SimpleInit()\n\nSimple initialization, only with the given initial value and derivative.\n\nThe remaining derivatives are set to zero with unit covariance (unless specified otherwise by setting a custom FixedDiffusion).\n\n\n\n\n\n","category":"type"},{"location":"initialization/#ProbNumDiffEq.ClassicSolverInit","page":"Initialization","title":"ProbNumDiffEq.ClassicSolverInit","text":"ClassicSolverInit(; alg=OrdinaryDiffEqCore.Tsit5(), init_on_ddu=false)\n\nInitialization via regression on a few steps of a classic ODE solver.\n\nIn a nutshell, instead of specifying mu_0 exactly and setting Sigma_0=0 (which is what TaylorModeInit does), use a classic ODE solver to compute a few steps of the solution, and then regress on the computed values (by running a smoother) to compute mu_0 and Sigma_0 as the mean and covariance of the smoothing posterior at time 0. See also [2].\n\nThe initial value and derivative are set directly from the given initial value problem; optionally the second derivative can also be set via automatic differentiation by setting init_on_ddu=true.\n\nArguments\n\nalg: The solver to be used. Can be any solver from OrdinaryDiffEqCore.jl.\ninit_on_ddu: If true, the second derivative is also initialized exactly via automatic differentiation with ForwardDiff.jl.\n\nReferences\n\n[7] Krämer et al, \"Stable Implementation of Probabilistic ODE Solvers\" (2020)\n[8] Schober et al, \"A probabilistic model for the numerical solution of initial value problems\", Statistics and Computing (2019)\n\n\n\n\n\n","category":"type"},{"location":"initialization/#initrefs","page":"Initialization","title":"References","text":"","category":"section"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"N. Krämer and P. Hennig. Stable Implementation of Probabilistic ODE Solvers. CoRR (2020), arXiv:2012.10106 [stat.ML].\n\n\n\nM. Schober, S. Särkkä and P. Hennig. A probabilistic model for the numerical solution of initial value problems. Statistics and Computing 29, 99–122 (2019).\n\n\n\n","category":"page"},{"location":"tutorials/dae/#Solving-DAEs-with-Probabilistic-Numerics","page":"Differential Algebraic Equations","title":"Solving DAEs with Probabilistic Numerics","text":"","category":"section"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"ProbNumDiffEq.jl provides probabilistic numerical solvers for differential algebraic equations (DAEs). Currently, we recommend using the semi-implicit EK1 algorithm.","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"note: Note\nFor a more general tutorial on DAEs check out the DifferentialEquations.jl DAE tutorial.","category":"page"},{"location":"tutorials/dae/#Solving-mass-matrix-DAEs-with-the-EK1","page":"Differential Algebraic Equations","title":"Solving mass-matrix DAEs with the EK1","text":"","category":"section"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"First, define the DAE (here the ROBER problem) as an ODE problem with singular mass matrix:","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"using ProbNumDiffEq, Plots, LinearAlgebra, OrdinaryDiffEq, ModelingToolkit, LinearAlgebra\n\nfunction rober(du, u, p, t)\n y₁, y₂, y₃ = u\n k₁, k₂, k₃ = p\n du[1] = -k₁ * y₁ + k₃ * y₂ * y₃\n du[2] = k₁ * y₁ - k₃ * y₂ * y₃ - k₂ * y₂^2\n du[3] = y₁ + y₂ + y₃ - 1\n nothing\nend\nM = [1 0 0\n 0 1 0\n 0 0 0]\nf = ODEFunction(rober, mass_matrix=M)\nprob_mm = ODEProblem(f, [1.0, 0.0, 0.0], (0.0, 1e5), (0.04, 3e7, 1e4))","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"We can solve this problem directly with the EK1:","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"sol = solve(prob_mm, EK1(), reltol=1e-8, abstol=1e-8)\nplot(\n sol,\n xscale=:log10,\n tspan=(1e-6, 1e5),\n layout=(3, 1),\n legend=false,\n ylabel=[\"u₁(t)\" \"u₂(t)\" \"u₃(t)\"],\n xlabel=[\"\" \"\" \"t\"],\n denseplot=false,\n)","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"Looks good!","category":"page"},{"location":"tutorials/dae/#Solving-an-Index-3-DAE-directly","page":"Differential Algebraic Equations","title":"Solving an Index-3 DAE directly","text":"","category":"section"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"The following is based on the \"Automatic Index Reduction of DAEs\" tutorial by ModelingToolkit.jl, which demonstrates how the classic Rodas4 solver fails to solve a DAE due to the fact that it is of index 3; which is why ModelingToolkit's automatic index reduction is so useful.","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"It turns out that our probabilistic numerical solvers can directly solve the index-3 DAE!","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"First, define the pendulum problem as in the tutorial:","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"function pendulum!(du, u, p, t)\n x, dx, y, dy, T = u\n g, L = p\n du[1] = dx\n du[2] = T * x\n du[3] = dy\n du[4] = T * y - g\n du[5] = x^2 + y^2 - L^2\nend\npendulum_fun! = ODEFunction(pendulum!, mass_matrix=Diagonal([1, 1, 1, 1, 0]))\nu0 = [1.0, 0, 0, 0, 0];\np = [9.8, 1];\ntspan = (0, 5.0);\npendulum_prob = ODEProblem(pendulum_fun!, u0, tspan, p)","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"We can try to solve it directly with one of the classic mass-matrix DAE solvers from OrdinaryDiffEq.jl:","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"solve(pendulum_prob, Rodas4())","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"It does not work! This is because of the index of the DAE; see for example this explanation from the tutorial.","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"Does this also hold for the EK1 solver? Let's find out:","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"sol = solve(pendulum_prob, EK1())","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"Nope! The EK1 is able to solve the index-3 DAE directly. Pretty cool!","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"plot(sol)","category":"page"},{"location":"tutorials/dae/#Is-index-reduction-still-worth-it?","page":"Differential Algebraic Equations","title":"Is index-reduction still worth it?","text":"","category":"section"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"The point of the \"Automatic Index Reduction of DAEs\" tutorial is to demonstrate ModelingToolkit's utility for automatic index reduction, which enables the classic implicit Runge-Kutta solvers such as Rodas5 to solve this DAE. Let's see if that still helps in this context here.","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"First, modelingtoolkitize the problem:","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"traced_sys = modelingtoolkitize(pendulum_prob)","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"(how cool is this latex output ?!?)","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"Next, lower the DAE index and simplify it with ModelingToolkit's dae_index_lowering and structural_simplify:","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"simplified_sys = structural_simplify(dae_index_lowering(traced_sys))","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"Let's build two different ODE problems, and check how well we can solve each:","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"prob_index3 = ODEProblem(complete(traced_sys), Pair[], tspan)\nprob_index1 = ODEProblem(simplified_sys, Pair[], tspan)\n\nsol3 = solve(prob_index3, EK1())\nsol1 = solve(prob_index1, EK1())\n\ntruesol = solve(prob_index1, Rodas4(), abstol=1e-10, reltol=1e-10)\n\nsol1_final_error = norm(sol1.u[end] - truesol.u[end])\nsol1_f_evals = sol1.stats.nf\nsol3_final_error = norm(sol3.u[end] - truesol.u[end])\nsol3_f_evals = sol3.stats.nf\n@info \"Results\" sol1_final_error sol1_f_evals sol3_final_error sol3_f_evals","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"The error for the index-1 DAE solve is much lower. Thus it seems that, even if the index-3 DAE could also be solved directly, index lowering might still be beneficial when solving DAEs with the EK1!","category":"page"},{"location":"tutorials/dae/#References","page":"Differential Algebraic Equations","title":"References","text":"","category":"section"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"N. Bosch, F. Tronarp and P. Hennig. Pick-and-Mix Information Operators for Probabilistic ODE Solvers. In: Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, Vol. 151 of Proceedings of Machine Learning Research, edited by G. Camps-Valls, F. J. Ruiz and I. Valera (PMLR, 28–30 Mar 2022); pp. 10015–10027.\n\n\n\n","category":"page"},{"location":"diffusions/#Diffusion-models-and-calibration","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":"","category":"section"},{"location":"diffusions/","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":"The notion of \"diffusion\" and \"calibration\" relates to the prior part of the model.","category":"page"},{"location":"diffusions/#Background:-The-prior","page":"Diffusion models and calibration","title":"Background: The prior","text":"","category":"section"},{"location":"diffusions/","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":"We model the ODE solution y(t) with a Gauss–Markov prior. More precisely, let","category":"page"},{"location":"diffusions/","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":"beginaligned\nY(t) = left Y^(0)(t) Y^(1)(t) dots Y^(q)(t) right\nendaligned","category":"page"},{"location":"diffusions/","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":"be the solution to the SDE","category":"page"},{"location":"diffusions/","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":"beginaligned\ntextd Y^(i)(t) = Y^(i+1)(t) textdt qquad i = 0 dots q-1 \ntextd Y^(q)(t) = textcolor389826A Y(t) textdt + textcolor4063D8Gamma textdW(t) \nY(0) sim textcolorpurple mathcalN left( mu_0 Sigma_0 right) \nendaligned","category":"page"},{"location":"diffusions/","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":"Then Y^(i)(t) models the i-th derivative of y(t). In this section, we consider choices relating to the \"diffusion\" textcolor4063D8Gamma. If you're more interested in the drift matrix textcolor389826A check out the Priors section, and for info on the initial distribution textcolorpurple mathcalN left( mu_0 Sigma_0 right) check out the Initialization section.","category":"page"},{"location":"diffusions/#Diffusion-and-calibration","page":"Diffusion models and calibration","title":"Diffusion and calibration","text":"","category":"section"},{"location":"diffusions/","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":"We call textcolor4063D8Gamma the \"diffusion\" parameter. Since it is typically not known we need to estimate it; this is called \"calibration\".","category":"page"},{"location":"diffusions/","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":"ProbNumDiffEq.jl provides a few different choices for how to model and estimate textcolor4063D8Gamma:","category":"page"},{"location":"diffusions/","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":"FixedDiffusion assumes an isotropic, time-fixed textcolor4063D8Gamma = sigma cdot I_d,\nDynamicDiffusion assumes an isotropic, time-varying textcolor4063D8Gamma(t) = sigma(t) cdot I_d (recommended),\nFixedMVDiffusion assumes a diagonal, time-fixed textcolor4063D8Gamma = operatornamediag(sigma_1 dots sigma_d),\nDynamicMVDiffusion assumes a diagonal, time-varying textcolor4063D8Gamma(t) = operatornamediag(sigma_1(t) dots sigma_d(t)).","category":"page"},{"location":"diffusions/","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":"Or more compactly:","category":"page"},{"location":"diffusions/","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":" Isotropic: Diagonal (only for the EK0)\nTime-varying DynamicDiffusion DynamicMVDiffusion\nTime-fixed FixedDiffusion FixedMVDiffusion","category":"page"},{"location":"diffusions/","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":"For more details on diffusions and calibration, check out this paper [9].","category":"page"},{"location":"diffusions/#API","page":"Diffusion models and calibration","title":"API","text":"","category":"section"},{"location":"diffusions/","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":"DynamicDiffusion\nFixedDiffusion\nDynamicMVDiffusion\nFixedMVDiffusion","category":"page"},{"location":"diffusions/#ProbNumDiffEq.DynamicDiffusion","page":"Diffusion models and calibration","title":"ProbNumDiffEq.DynamicDiffusion","text":"DynamicDiffusion()\n\nTime-varying, isotropic diffusion, which is quasi-maximum-likelihood-estimated at each step.\n\nThis is the recommended diffusion when using adaptive step-size selection, and in particular also when solving stiff systems.\n\n\n\n\n\n","category":"type"},{"location":"diffusions/#ProbNumDiffEq.FixedDiffusion","page":"Diffusion models and calibration","title":"ProbNumDiffEq.FixedDiffusion","text":"FixedDiffusion(; initial_diffusion=1.0, calibrate=true)\n\nTime-fixed, isotropic diffusion, which is (optionally) quasi-maximum-likelihood-estimated.\n\nThis is the recommended diffusion when using fixed steps.\n\nBy default with calibrate=true, all covariances are re-scaled at the end of the solve with the MLE diffusion. Set calibrate=false to skip this step, e.g. when setting the initial_diffusion and then estimating the diffusion outside of the solver (e.g. with Fenrir.jl).\n\n\n\n\n\n","category":"type"},{"location":"diffusions/#ProbNumDiffEq.DynamicMVDiffusion","page":"Diffusion models and calibration","title":"ProbNumDiffEq.DynamicMVDiffusion","text":"DynamicMVDiffusion()\n\nTime-varying, diagonal diffusion, which is quasi-maximum-likelihood-estimated at each step.\n\nOnly works with the EK0!\n\nA multi-variate version of DynamicDiffusion, where instead of an isotropic matrix, a diagonal matrix is estimated. This can be helpful to get more expressive posterior covariances when using the EK0, since the individual dimensions can be adjusted separately.\n\nReferences\n\n[9] Bosch et al, \"Calibrated Adaptive Probabilistic ODE Solvers\", AISTATS (2021)\n\n\n\n\n\n","category":"type"},{"location":"diffusions/#ProbNumDiffEq.FixedMVDiffusion","page":"Diffusion models and calibration","title":"ProbNumDiffEq.FixedMVDiffusion","text":"FixedMVDiffusion(; initial_diffusion=1.0, calibrate=true)\n\nTime-fixed, diagonal diffusion, which is quasi-maximum-likelihood-estimated at each step.\n\nOnly works with the EK0!\n\nA multi-variate version of FixedDiffusion, where instead of an isotropic matrix, a diagonal matrix is estimated. This can be helpful to get more expressive posterior covariances when using the EK0, since the individual dimensions can be adjusted separately.\n\nReferences\n\n[9] Bosch et al, \"Calibrated Adaptive Probabilistic ODE Solvers\", AISTATS (2021)\n\n\n\n\n\n","category":"type"},{"location":"diffusions/#diffusionrefs","page":"Diffusion models and calibration","title":"References","text":"","category":"section"},{"location":"diffusions/","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":"N. Bosch, P. Hennig and F. Tronarp. Calibrated Adaptive Probabilistic ODE Solvers. In: Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, Vol. 130 of Proceedings of Machine Learning Research, edited by A. Banerjee and K. Fukumizu (PMLR, 13–15 Apr 2021); pp. 3466–3474.\n\n\n\n","category":"page"},{"location":"priors/#Priors","page":"Priors","title":"Priors","text":"","category":"section"},{"location":"priors/","page":"Priors","title":"Priors","text":"TL;DR: If you're unsure which prior to use, just stick to the default integrated Wiener process prior IWP!","category":"page"},{"location":"priors/#Background","page":"Priors","title":"Background","text":"","category":"section"},{"location":"priors/","page":"Priors","title":"Priors","text":"We model the ODE solution y(t) with a Gauss–Markov prior. More precisely, let","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"beginaligned\nY(t) = left Y^(0)(t) Y^(1)(t) dots Y^(q)(t) right\nendaligned","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"be the solution to the SDE","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"beginaligned\ntextd Y^(i)(t) = Y^(i+1)(t) textdt qquad i = 0 dots q-1 \ntextd Y^(q)(t) = textcolor389826A Y(t) textdt + textcolor4063D8Gamma textdW(t) \nY(0) sim textcolorpurple mathcalN left( mu_0 Sigma_0 right) \nendaligned","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"Then Y^(i)(t) models the i-th derivative of y(t). In this section, we consider choices relating to the drift matrix textcolor389826A. If you're more interested in the diffusion textcolor4063D8Gamma check out the Diffusion models and calibration section, and for info on the initial distribution textcolorpurple mathcalN left( mu_0 Sigma_0 right) check out the Initialization section.","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"info: Info\nIf you're unsure which prior to use, just stick to the integrated Wiener process prior IWP! This is also the default choice for all solvers. The other priors are rather experimental / niche at the time of writing.","category":"page"},{"location":"priors/#Prior-choices","page":"Priors","title":"Prior choices","text":"","category":"section"},{"location":"priors/","page":"Priors","title":"Priors","text":"ProbNumDiffEq.jl currently supports three classes of priors for the solvers:","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"IWP: q-times integrated Wiener processes\nIOUP: q-times integrated Ornstein–Uhlenbeck processes\nMatern: Matérn processes","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"Let's look at each of them in turn and visualize some examples.","category":"page"},{"location":"priors/#Integrated-Wiener-process-([IWP](@ref))","page":"Priors","title":"Integrated Wiener process (IWP)","text":"","category":"section"},{"location":"priors/","page":"Priors","title":"Priors","text":"IWP","category":"page"},{"location":"priors/#ProbNumDiffEq.IWP","page":"Priors","title":"ProbNumDiffEq.IWP","text":"IWP([dim::Integer=1,] num_derivatives::Integer)\n\nq-times integrated Wiener process.\n\nThis is the recommended prior! The IWP is the most common prior choice in the probabilistic ODE solver literature (see the references) and is the default choice for the solvers in ProbNumDiffEq.jl. It is also the prior that has the most efficient implementation.\n\nThe IWP can be created without specifying the dimension of the Wiener process, in which case it will be one-dimensional. The ODE solver then assumes that each dimension of the ODE solution should be modeled with the same prior. This is typically the preferred usage.\n\nIn math\n\nThe IWP is a Gauss–Markov process, which we model with a state representation\n\nbeginaligned\nY(t) = left Y^(0)(t) Y^(1)(t) dots Y^(q)(t) right\nendaligned\n\ndefined as the solution of the stochastic differential equation\n\nbeginaligned\ntextd Y^(i)(t) = Y^(i+1)(t) textdt qquad i = 0 dots q-1 \ntextd Y^(q)(t) = Gamma textdW(t)\nendaligned\n\nThen, Y^(0)(t) is the q-times integrated Wiener process (IWP) and Y^(i)(t) is the i-th derivative of the IWP, for i = 1 dots q.\n\nExamples\n\njulia> solve(prob, EK1(prior=IWP(2)))\n\n\n\n\n\n","category":"type"},{"location":"priors/","page":"Priors","title":"Priors","text":"Here is how the IWP looks for varying smoothness parameters q:","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"using ProbNumDiffEq, Plots\nplotrange = range(0, 10, length=250)\nplot(\n plot(IWP(1), plotrange; title=\"q=1\"),\n plot(IWP(2), plotrange; title=\"q=2\"),\n plot(IWP(3), plotrange; title=\"q=3\"),\n plot(IWP(4), plotrange; title=\"q=4\");\n ylims=(-20,20),\n)","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"In the context of ODE solvers, the smoothness parameter q influences the convergence rate of the solver, and so it is typically chose similarly to the order of a Runge–Kutta method: lower order for low accuracy, higher order for high accuracy.","category":"page"},{"location":"priors/#Integrated-Ornstein–Uhlenbeck-process-([IOUP](@ref))","page":"Priors","title":"Integrated Ornstein–Uhlenbeck process (IOUP)","text":"","category":"section"},{"location":"priors/","page":"Priors","title":"Priors","text":"The q-times integrated Ornstein–Uhlenbeck process prior IOUP is a generalization of the IWP prior, where the drift matrix textcolor389826A is not zero:","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"IOUP","category":"page"},{"location":"priors/#ProbNumDiffEq.IOUP","page":"Priors","title":"ProbNumDiffEq.IOUP","text":"IOUP([dim::Integer=1,]\n num_derivatives::Integer,\n rate_parameter::Union{Number,AbstractVector,AbstractMatrix})\n\nIntegrated Ornstein–Uhlenbeck process.\n\nThis prior is mostly used in the context of Probabilistic Exponential Integrators to include the linear part of a semi-linear ODE in the prior, so it is used in the ExpEK and the RosenbrockExpEK.\n\nIn math\n\nThe IOUP is a Gauss–Markov process, which we model with a state representation\n\nbeginaligned\nY(t) = left Y^(0)(t) Y^(1)(t) dots Y^(q)(t) right\nendaligned\n\ndefined as the solution of the stochastic differential equation\n\nbeginaligned\ntextd Y^(i)(t) = Y^(i+1)(t) textdt qquad i = 0 dots q-1 \ntextd Y^(q)(t) = L Y^(q)(t) textdt + Gamma textdW(t)\nendaligned\n\nwhere L is called the rate_parameter. Then, Y^(0)(t) is the q-times integrated Ornstein–Uhlenbeck process (IOUP) and Y^(i)(t) is the i-th derivative of the IOUP, for i = 1 dots q.\n\nExamples\n\njulia> solve(prob, EK1(prior=IOUP(2, -1)))\n\n\n\n\n\n","category":"type"},{"location":"priors/","page":"Priors","title":"Priors","text":"Here is how the IOUP looks for varying rate parameters:","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"using ProbNumDiffEq, Plots\nplotrange = range(0, 10, length=250)\nplot(\n plot(IOUP(1, -1), plotrange; title=\"q=1,L=-1\", ylims=(-20,20)),\n plot(IOUP(1, 1), plotrange; title=\"q=1,L=1\", ylims=(-20,20)),\n plot(IOUP(4, -1), plotrange; title=\"q=4,L=-1\", ylims=(-50,50)),\n plot(IOUP(4, 1), plotrange; title=\"q=4,L=1\", ylims=(-50,50));\n)","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"In the context of Probabilistic Exponential Integrators, the rate parameter is often chosen according to the given ODE. Here is an example for a damped oscillator:","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"plot(IOUP(dim=2, num_derivatives=1, rate_parameter=[-0.2 -2π; 2π -0.2]),\n plotrange; plot_title=\"damped oscillator prior\")","category":"page"},{"location":"priors/#Matérn-process-([Matern](@ref))","page":"Priors","title":"Matérn process (Matern)","text":"","category":"section"},{"location":"priors/","page":"Priors","title":"Priors","text":"Matern","category":"page"},{"location":"priors/#ProbNumDiffEq.Matern","page":"Priors","title":"ProbNumDiffEq.Matern","text":"Matern([dim::Integer=1,]\n num_derivatives::Integer,\n lengthscale::Number)\n\nMatern process.\n\nThe class of Matern processes is well-known in the Gaussian process literature, and they also have a corresponding SDE representation similarly to the IWP and the IOUP. See also [6] for more details.\n\nIn math\n\nA Matern process is a Gauss–Markov process, which we model with a state representation\n\nbeginaligned\nY(t) = left Y^(0)(t) Y^(1)(t) dots Y^(q)(t) right\nendaligned\n\ndefined as the solution of the stochastic differential equation\n\nbeginaligned\ntextd Y^(i)(t) = Y^(i+1)(t) textdt qquad i = 0 dots q-1 \ntextd Y^(q)(t) = - sum_j=0^q left(\n beginpmatrix q+1 j endpmatrix\n left( fracsqrt2q - 1l right)^q-j\n Y^(j)(t) right) textdt + Gamma textdW(t)\nendaligned\n\nwhere l is called the lengthscale parameter. Then, Y^(0)(t) is a Matern process and Y^(i)(t) is the i-th derivative of this process, for i = 1 dots q.\n\nExamples\n\njulia> solve(prob, EK1(prior=Matern(2, 1)))\n\n\n\n\n\n","category":"type"},{"location":"priors/","page":"Priors","title":"Priors","text":"Here is how the Matern looks for varying smoothness parameters q:","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"using ProbNumDiffEq, Plots\nplotrange = range(0, 10, length=250)\nplot(\n plot(Matern(1, 1), plotrange; title=\"q=1\"),\n plot(Matern(2, 1), plotrange; title=\"q=2\"),\n plot(Matern(3, 1), plotrange; title=\"q=3\"),\n plot(Matern(4, 1), plotrange; title=\"q=4\");\n)","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"and for varying length scales ell:","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"plot(\n plot(Matern(2, 5), plotrange; title=\"l=5\"),\n plot(Matern(2, 2), plotrange; title=\"l=2\"),\n plot(Matern(2, 1), plotrange; title=\"l=1\"),\n plot(Matern(2, 0.5), plotrange; title=\"l=0.5\"),\n)","category":"page"},{"location":"priors/#API","page":"Priors","title":"API","text":"","category":"section"},{"location":"priors/","page":"Priors","title":"Priors","text":"ProbNumDiffEq.AbstractGaussMarkovProcess\nProbNumDiffEq.LTISDE\nProbNumDiffEq.dim\nProbNumDiffEq.num_derivatives\nProbNumDiffEq.to_sde\nProbNumDiffEq.discretize\nProbNumDiffEq.initial_distribution","category":"page"},{"location":"priors/#ProbNumDiffEq.AbstractGaussMarkovProcess","page":"Priors","title":"ProbNumDiffEq.AbstractGaussMarkovProcess","text":"AbstractGaussMarkovProcess{elType}\n\nAbstract type for Gauss-Markov processes.\n\nGauss-Markov processes are solutions to linear time-invariant stochastic differential equations (SDEs). Here we assume SDEs of the form\n\nbeginaligned\ndX_t = F X_t dt + L dW_t \nX_0 = mathcalN left( X_0 mu_0 Sigma_0 right)\nendaligned\n\nwhere X_t is the state, W_t is a Wiener process, and F and L are matrices.\n\nCurrently, ProbNumDiffEq.jl makes many assumptions about the structure of the SDEs that it can solve. In particular, it assumes that the state vector X_t contains a range of dervatives, and that the Wiener process only enters the highest one. It also assumes a certain ordering of dimensions and derivatives. This is not a limitation of the underlying mathematics, but rather a limitation of the current implementation. In the future, we hope to remove these limitations.\n\nwarning: Warning\nWe currently strongly recommended to not implement your own Gauss-Markov process by subtyping this type! The interface is not yet stable, and the implementation is not yet sufficiently documented. Proceed at your own risk.\n\n\n\n\n\n","category":"type"},{"location":"priors/#ProbNumDiffEq.LTISDE","page":"Priors","title":"ProbNumDiffEq.LTISDE","text":"LTISDE(F::AbstractMatrix, L::AbstractMatrix)\n\nLinear time-invariant stochastic differential equation.\n\nA LTI-SDE is a stochastic differential equation of the form\n\ndX_t = F X_t dt + L dW_t\n\nwhere X_t is the state, W_t is a Wiener process, and F and L are matrices. This LTISDE object holds the matrices F and L. It also provides some functionality to discretize the SDE via a matrix-fraction decomposition. See: discretize(::LTISDE, ::Real).\n\n\n\n\n\n","category":"type"},{"location":"priors/#ProbNumDiffEq.dim","page":"Priors","title":"ProbNumDiffEq.dim","text":"dim(p::AbstractGaussMarkovProcess)\n\nReturn the dimension of the process.\n\nThis is not the dimension of the \"state\" that is used to efficiently model the prior process as a state-space model, but it is the dimension of the process itself that we aim to model.\n\nSee AbstractGaussMarkovProcess for more details on Gauss-Markov processes in ProbNumDiffEq.\n\n\n\n\n\n","category":"function"},{"location":"priors/#ProbNumDiffEq.num_derivatives","page":"Priors","title":"ProbNumDiffEq.num_derivatives","text":"num_derivatives(p::AbstractGaussMarkovProcess)\n\nReturn the number of derivatives that are represented by the processes state.\n\nSee AbstractGaussMarkovProcess for more details on Gauss-Markov processes in ProbNumDiffEq.\n\n\n\n\n\n","category":"function"},{"location":"priors/#ProbNumDiffEq.to_sde","page":"Priors","title":"ProbNumDiffEq.to_sde","text":"to_sde(p::AbstractGaussMarkovProcess)\n\nConvert the prior to the corresponding SDE.\n\nGauss-Markov processes are solutions to linear time-invariant stochastic differential equations (SDEs) of the form\n\nbeginaligned\ndX_t = F X_t dt + L dW_t \nX_0 = mathcalN left( X_0 mu_0 Sigma_0 right)\nendaligned\n\nwhere X_t is the state, W_t is a Wiener process, and F and L are matrices. This function returns the corresponding SDE, i.e. the matrices F and L, as a LTISDE.\n\n\n\n\n\n","category":"function"},{"location":"priors/#ProbNumDiffEq.discretize","page":"Priors","title":"ProbNumDiffEq.discretize","text":"discretize(p::AbstractGaussMarkovProcess, step_size::Real)\n\nCompute the transition matrices of the process for a given step size.\n\n\n\n\n\ndiscretize(p::LTISDE, step_size::Real)\n\nCompute the transition matrices of the SDE solution for a given step size.\n\n\n\n\n\n","category":"function"},{"location":"priors/#ProbNumDiffEq.initial_distribution","page":"Priors","title":"ProbNumDiffEq.initial_distribution","text":"initial_distribution(p::AbstractGaussMarkovProcess)\n\nReturn the initial distribution of the process.\n\nCurrently this is always a Gaussian distribution with zero mean and unit variance, unless explicitly overwitten (e.g. for Matern processes to have the stationary distribution). This implementation is likely to change in the future to allow for more flexibility.\n\n\n\n\n\n","category":"function"},{"location":"priors/#Convenience-functions-to-analyze-and-visualize-priors","page":"Priors","title":"Convenience functions to analyze and visualize priors","text":"","category":"section"},{"location":"priors/","page":"Priors","title":"Priors","text":"ProbNumDiffEq.marginalize\nProbNumDiffEq.sample","category":"page"},{"location":"priors/#ProbNumDiffEq.marginalize","page":"Priors","title":"ProbNumDiffEq.marginalize","text":"marginalize(process::AbstractGaussMarkovProcess, times)\n\nCompute the marginal distributions of the process at the given time points.\n\nThis function computes the marginal distributions of the process at the given times. It does so by discretizing the process with the given step sizes (using ProbNumDiffEq.discretize), and then computing the marginal distributions of the resulting Gaussian distributions.\n\nSee also: sample.\n\n\n\n\n\n","category":"function"},{"location":"priors/#ProbNumDiffEq.sample","page":"Priors","title":"ProbNumDiffEq.sample","text":"sample(process::AbstractGaussMarkovProcess, times, N=1)\n\nSamples from the Gauss-Markov process on the given time grid.\n\nSee also: marginalize.\n\n\n\n\n\n","category":"function"},{"location":"tutorials/dynamical_odes/#Second-Order-ODEs-and-Energy-Preservation","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"","category":"section"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"In this tutorial we consider an energy-preserving, physical dynamical system, given by a second-order ODE.","category":"page"},{"location":"tutorials/dynamical_odes/#TL;DR:","page":"Second Order ODEs and Energy Preservation","title":"TL;DR:","text":"","category":"section"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"To efficiently solve second-order ODEs, just define the problem as a SecondOrderODEProblem.\nTo preserve constant quantities, use the ManifoldUpdate callback; same syntax as DiffEqCallback.jl's ManifoldProjection.","category":"page"},{"location":"tutorials/dynamical_odes/#Simulating-the-Hénon-Heiles-system","page":"Second Order ODEs and Energy Preservation","title":"Simulating the Hénon-Heiles system","text":"","category":"section"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"The Hénon-Heiles model describes the motion of a star around a galactic center, restricted to a plane. It is given by a second-order ODE","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"beginaligned\nddotx = - x - 2 x y \nddoty = y^2 - y - x^2\nendaligned","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"Our goal is to numerically simulate this system on a time span t in 0 T, starting with initial values x(0)=0, y(0) = 01, dotx(0) = 05, doty(0) = 0.","category":"page"},{"location":"tutorials/dynamical_odes/#Transforming-the-problem-into-a-first-order-ODE","page":"Second Order ODEs and Energy Preservation","title":"Transforming the problem into a first-order ODE","text":"","category":"section"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"A very common approach is to first transform the problem into a first-order ODE by introducing a new variable","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"u = dxdyxy","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"to obtain","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"beginaligned\ndotu_1(t) = - u_3 - 2 u_3 u_4 \ndotu_2(t) = u_4^2 - u_4 - u_4^2 \ndotu_3(t) = u_1 \ndotu_4(t) = u_2\nendaligned","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"This first-order ODE can then be solved using any conventional ODE solver - including our EK1:","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"using ProbNumDiffEq, Plots\n\nfunction Hénon_Heiles(du, u, p, t)\n du[1] = -u[3] - 2 * u[3] * u[4]\n du[2] = u[4]^2 - u[4] - u[3]^2\n du[3] = u[1]\n du[4] = u[2]\nend\nu0, du0 = [0.0, 0.1], [0.5, 0.0]\ntspan = (0.0, 100.0)\nprob = ODEProblem(Hénon_Heiles, [du0; u0], tspan)\nsol = solve(prob, EK1());\nplot(sol, idxs=(3, 4)) # where `idxs=(3,4)` is used to plot x agains y","category":"page"},{"location":"tutorials/dynamical_odes/#Solving-the-second-order-ODE-directly","page":"Second Order ODEs and Energy Preservation","title":"Solving the second-order ODE directly","text":"","category":"section"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"Instead of first transforming the problem, we can also solve it directly as a second-order ODE, by defining it as a SecondOrderODEProblem.","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"note: Note\nThe SecondOrderODEProblem type is not defined in ProbNumDiffEq.jl but is provided by SciMLBase.jl. For more information, check out the DifferentialEquations.jl documentation on Dynamical, Hamiltonian and 2nd Order ODE Problems.","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"function Hénon_Heiles2(ddu, du, u, p, t)\n ddu[1] = -u[1] - 2 * u[1] * u[2]\n ddu[2] = u[2]^2 - u[2] - u[1]^2\nend\nprob2 = SecondOrderODEProblem(Hénon_Heiles2, du0, u0, tspan)\nsol2 = solve(prob2, EK1());\nplot(sol2, idxs=(3, 4))","category":"page"},{"location":"tutorials/dynamical_odes/#Benchmark:-Solving-second-order-ODEs-is-*faster*","page":"Second Order ODEs and Energy Preservation","title":"Benchmark: Solving second order ODEs is faster","text":"","category":"section"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"Solving second-order ODEs is not just a matter of convenience - in fact, SciMLBase's SecondOrderODEProblem is neatly designed in such a way that all the classic solvers from OrdinaryDiffEq.jl can handle it by solving the corresponding first-order ODE. But, transforming the ODE to first order increases the dimensionality of the problem, and comes therefore at increased computational cost; this also motivates classic specialized solvers for second-order ODEs.","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"The probabilistic numerical solvers from ProbNumDiffEq.jl have the same internal state representation for first and second order ODEs; all that changes is the measurement model [1]. As a result, we can use the EK1 both for first and second order ODEs, but it automatically specializes on the latter to provide a 2x performance boost:","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"julia> using BenchmarkTools\n\njulia> @btime solve(prob, EK1(order=3), adaptive=false, dt=1e-2);\n 317.336 ms (140561 allocations: 140.41 MiB)\n\njulia> @btime solve(prob2, EK1(order=3), adaptive=false, dt=1e-2);\n 136.654 ms (180647 allocations: 47.95 MiB)","category":"page"},{"location":"tutorials/dynamical_odes/#Energy-preservation","page":"Second Order ODEs and Energy Preservation","title":"Energy preservation","text":"","category":"section"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"In addition to the ODE given above, we know that the solution of the Hénon-Heiles model has to preserve energy over time. The total energy can be expressed as the sum of the potential and kinetic energies, given by","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"beginaligned\noperatornamePotentialEnergy(xy) = frac12 left( x^2 + y^2 + 2 x^2 y - frac2y^33 right) \noperatornameKineticEnergy(dotx doty) = frac12 left( dotx^2 + doty^2 right)\nendaligned","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"In code:","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"PotentialEnergy(x, y) = 1 // 2 * (x^2 + y^2 + 2x^2 * y - 2 // 3 * y^3)\nKineticEnergy(dx, dy) = 1 // 2 * (dx^2 + dy^2)\nE(dx, dy, x, y) = PotentialEnergy(x, y) + KineticEnergy(dx, dy)\nE(u) = E(u...); # convenient shorthand","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"Let's have a look at how the total energy changes over time when we numerically simulate the Hénon-Heiles model over a long period of time: Standard solve","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"longprob = remake(prob2, tspan=(0.0, 1e3))\nlongsol = solve(longprob, EK1(smooth=false), dense=false)\nplot(longsol.t, E.(longsol.u))","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"It visibly loses energy over time, from an initial 0.12967 to a final 0.12899. Let's fix this to get a physically more meaningful solution.","category":"page"},{"location":"tutorials/dynamical_odes/#Energy-preservation-with-the-ManifoldUpdate-callback","page":"Second Order ODEs and Energy Preservation","title":"Energy preservation with the ManifoldUpdate callback","text":"","category":"section"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"In the language of ODE filters, preserving energy over time amounts to just another measurement model [1]. The most convenient way of updating on this additional zero measurement with ProbNumDiffEq.jl is with the ManifoldUpdate callback.","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"note: Note\nThe ManifoldUpdate callback can be thought of a probabilistic counterpart to the ManifoldProjection callback provided by DiffEqCallbacks.jl.","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"To do so, first define a (vector-valued) residual function, here chosen to be the difference between the current energy and the initial energy, and build a ManifoldUpdate callback","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"residual(u) = [E(u) - E(du0..., u0...)]\ncb = ManifoldUpdate(residual)","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"Then, solve the ODE with this callback","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"longsol_preserving = solve(longprob, EK1(smooth=false), dense=false, callback=cb)\nplot(longsol.t, E.(longsol.u))\nplot!(longsol_preserving.t, E.(longsol_preserving.u))","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"Voilà! With the ManifoldUpdate callback we could preserve the energy over time and obtain a more truthful probabilistic numerical long-term simulation of the Hénon-Heiles model.","category":"page"},{"location":"tutorials/dynamical_odes/#References","page":"Second Order ODEs and Energy Preservation","title":"References","text":"","category":"section"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"N. Bosch, F. Tronarp and P. Hennig. Pick-and-Mix Information Operators for Probabilistic ODE Solvers. In: Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, Vol. 151 of Proceedings of Machine Learning Research, edited by G. Camps-Valls, F. J. Ruiz and I. Valera (PMLR, 28–30 Mar 2022); pp. 10015–10027.\n\n\n\n","category":"page"},{"location":"benchmarks/multi-language-wrappers/#ProbNumDiffEq.jl-vs.-various-solver-packages","page":"Multi-Language Wrapper Benchmark","title":"ProbNumDiffEq.jl vs. various solver packages","text":"","category":"section"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"Adapted from SciMLBenchmarks.jl multi-language wrapper benchmark.","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
Code:","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"# Imports\nusing LinearAlgebra, Statistics\nusing StaticArrays, DiffEqDevTools, ParameterizedFunctions, Plots, SciMLBase, OrdinaryDiffEq\nusing ODEInterface, ODEInterfaceDiffEq, Sundials, SciPyDiffEq, deSolveDiffEq, MATLABDiffEq, LSODA\nusing LoggingExtras\n\nusing ProbNumDiffEq\n\nPlots.theme(\n :dao;\n markerstrokewidth=0.5,\n legend=:outertopright,\n margin=5Plots.mm,\n xticks=10.0 .^ (-16:1:16),\n yticks=10.0 .^ (-6:1:5),\n)\n\n# Constants used throughout this benchmark as we only consider final values\nconst DENSE = false # used to decide if we smooth or not\nconst SAVE_EVERYSTEP = false;\n\n# COLORS and a realted utility\nCOLORS = Dict(\n \"Julia\" => :LightGreen,\n \"Julia (static)\" => :DarkGreen,\n \"Hairer\" => :Red,\n \"MATLAB\" => :Orange,\n \"SciPy\" => :Yellow,\n \"deSolve\" => :Blue,\n \"Sundials\" => :Purple,\n \"liblsoda\" => :Purple,\n \"ProbNumDiffEq: EK0\" => :Gray30,\n \"ProbNumDiffEq: EK1\" => :Gray60,\n)\ntocolor(n) = if split(n, '(')[1] in keys(COLORS)\n COLORS[split(n, '(')[1]]\nelse\n COLORS[split(n, ':')[1]]\nend\n\n# Do not show \"deprecated warnings\"\ndeprecated_filter(log_args) = !contains(log_args.message, \"deprecated\")\nfiltered_logger = ActiveFilteredLogger(deprecated_filter, global_logger());","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
","category":"page"},{"location":"benchmarks/multi-language-wrappers/#Non-Stiff-Problem-1:-Lotka-Volterra","page":"Multi-Language Wrapper Benchmark","title":"Non-Stiff Problem 1: Lotka-Volterra","text":"","category":"section"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
Code:","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"f = @ode_def LotkaVolterra begin\n dx = a*x - b*x*y\n dy = -c*y + d*x*y\nend a b c d\np = [1.5, 1, 3, 1]\ntspan = (0.0, 10.0)\nu0 = [1.0, 1.0]\nprob = ODEProblem{true,SciMLBase.FullSpecialize()}(f,u0,tspan,p)\nstaticprob = ODEProblem{false,SciMLBase.FullSpecialize()}(f,SVector{2}(u0),tspan,SVector{4}(p))\n\nsol = solve(prob,Vern7(),abstol=1/10^14,reltol=1/10^14,dense=false)\ntest_sol = sol\nplot(sol, title=\"Lotka-Volterra Solution\", legend=false, xticks=:auto, yticks=:auto)","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"(Image: )","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
Code:","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"_setups = [\n \"Julia: DP5\" => Dict(:alg=>DP5())\n \"Julia: Tsit5\" => Dict(:alg=>Tsit5())\n \"Julia: Vern7\" => Dict(:alg=>Vern7())\n \"Hairer: dopri5\" => Dict(:alg=>ODEInterfaceDiffEq.dopri5())\n \"MATLAB: ode45\" => Dict(:alg=>MATLABDiffEq.ode45())\n \"MATLAB: ode113\" => Dict(:alg=>MATLABDiffEq.ode113())\n \"SciPy: RK45\" => Dict(:alg=>SciPyDiffEq.RK45())\n \"SciPy: LSODA\" => Dict(:alg=>SciPyDiffEq.LSODA())\n \"SciPy: odeint\" => Dict(:alg=>SciPyDiffEq.odeint())\n \"deSolve: lsoda\" => Dict(:alg=>deSolveDiffEq.lsoda())\n \"deSolve: ode45\" => Dict(:alg=>deSolveDiffEq.ode45())\n \"Sundials: Adams\" => Dict(:alg=>Sundials.CVODE_Adams())\n \"ProbNumDiffEq: EK0(3)\" => Dict(:alg=>EK0(order=3, smooth=DENSE))\n \"ProbNumDiffEq: EK0(5)\" => Dict(:alg=>EK0(order=5, smooth=DENSE))\n \"ProbNumDiffEq: EK1(3)\" => Dict(:alg=>EK1(order=3, smooth=DENSE))\n \"ProbNumDiffEq: EK1(5)\" => Dict(:alg=>EK1(order=5, smooth=DENSE))\n \"ProbNumDiffEq: EK1(8)\" => Dict(:alg=>EK1(order=8, smooth=DENSE))\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\ncolors = tocolor.(labels) |> permutedims\n\nabstols = 1.0 ./ 10.0 .^ (6:13)\nreltols = 1.0 ./ 10.0 .^ (3:10)\n\nwp = with_logger(filtered_logger) do\n WorkPrecisionSet(\n [prob, staticprob], abstols, reltols, setups;\n names = labels,\n appxsol = [test_sol, test_sol],\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n numruns = 10,\n maxiters = Int(1e7),\n timeseries_errors = false,\n verbose = false,\n )\nend\n\nplot(wp, title = \"Non-stiff 1: Lotka-Volterra\", color = colors)","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"(Image: )","category":"page"},{"location":"benchmarks/multi-language-wrappers/#Non-Stiff-Problem-2:-Rigid-Body","page":"Multi-Language Wrapper Benchmark","title":"Non-Stiff Problem 2: Rigid Body","text":"","category":"section"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
Code:","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"f = @ode_def RigidBodyBench begin\n dy1 = -2*y2*y3\n dy2 = 1.25*y1*y3\n dy3 = -0.5*y1*y2 + 0.25*sin(t)^2\nend\nu0 = [1.0;0.0;0.9]\ntspan = (0.0, 10.0)\nprob = ODEProblem{true,SciMLBase.FullSpecialize()}(f,u0,tspan)\nstaticprob = ODEProblem{false,SciMLBase.FullSpecialize()}(f,SVector{3}(u0),tspan)\nsol = solve(prob,Vern7(),abstol=1/10^14,reltol=1/10^14,dense=false)\ntest_sol = sol\nplot(sol, title=\"Rigid Body Solution\", legend=false, xticks=:auto, yticks=:auto)","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"(Image: )","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
Code:","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"_setups = [\n \"Julia: DP5\" => Dict(:alg=>DP5())\n \"Julia: Tsit5\" => Dict(:alg=>Tsit5())\n \"Julia: Vern7\" => Dict(:alg=>Vern7())\n \"Hairer: dopri5\" => Dict(:alg=>dopri5())\n \"MATLAB: ode45\" => Dict(:alg=>MATLABDiffEq.ode45())\n \"MATLAB: ode113\" => Dict(:alg=>MATLABDiffEq.ode113())\n \"SciPy: RK45\" => Dict(:alg=>SciPyDiffEq.RK45())\n \"SciPy: LSODA\" => Dict(:alg=>SciPyDiffEq.LSODA())\n \"SciPy: odeint\" => Dict(:alg=>SciPyDiffEq.odeint())\n \"deSolve: lsoda\" => Dict(:alg=>deSolveDiffEq.lsoda())\n \"deSolve: ode45\" => Dict(:alg=>deSolveDiffEq.ode45())\n \"Sundials: Adams\" => Dict(:alg=>CVODE_Adams())\n \"ProbNumDiffEq: EK0(3)\" => Dict(:alg=>EK0(order=3, smooth=DENSE))\n \"ProbNumDiffEq: EK0(5)\" => Dict(:alg=>EK0(order=5, smooth=DENSE))\n \"ProbNumDiffEq: EK1(3)\" => Dict(:alg=>EK1(order=3, smooth=DENSE))\n \"ProbNumDiffEq: EK1(5)\" => Dict(:alg=>EK1(order=5, smooth=DENSE))\n \"ProbNumDiffEq: EK1(8)\" => Dict(:alg=>EK1(order=8, smooth=DENSE))\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\ncolors = tocolor.(labels) |> permutedims\n\nabstols = 1.0 ./ 10.0 .^ (6:13)\nreltols = 1.0 ./ 10.0 .^ (3:10)\n\nwp = with_logger(filtered_logger) do\n WorkPrecisionSet(\n [prob,staticprob], abstols, reltols, setups;\n names = labels,\n appxsol = [test_sol, test_sol],\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n numruns = 10,\n maxiters = Int(1e7),\n timeseries_errors = false,\n verbose = false\n )\nend\n\nplot(wp, title = \"Non-stiff 2: Rigid-Body\", color = colors)","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"(Image: )","category":"page"},{"location":"benchmarks/multi-language-wrappers/#Stiff-Problem-1:-ROBER","page":"Multi-Language Wrapper Benchmark","title":"Stiff Problem 1: ROBER","text":"","category":"section"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
Code:","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"rober = @ode_def begin\n dy₁ = -k₁*y₁+k₃*y₂*y₃\n dy₂ = k₁*y₁-k₂*y₂^2-k₃*y₂*y₃\n dy₃ = k₂*y₂^2\nend k₁ k₂ k₃\nu0 = [1.0,0.0,0.0]\np = [0.04,3e7,1e4]\nprob = ODEProblem{true,SciMLBase.FullSpecialize()}(rober,u0,(0.0,1e5),p)\nstaticprob = ODEProblem{false,SciMLBase.FullSpecialize()}(rober,SVector{3}(u0),(0.0,1e5),SVector{3}(p))\nsol = solve(prob,CVODE_BDF(),abstol=1/10^14,reltol=1/10^14,dense=false)\ntest_sol = sol\nplot(sol, title=\"ROBER Solution\", legend=false, xlims=(1e0, 1e5), xticks=:auto, yticks=:auto)","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"(Image: )","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
Code:","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"_setups = [\n \"Julia: Rosenbrock23\" => Dict(:alg=>Rosenbrock23())\n \"Julia: Rodas4\" => Dict(:alg=>Rodas4())\n \"Julia: Rodas5\" => Dict(:alg=>Rodas5())\n \"Hairer: rodas\" => Dict(:alg=>rodas())\n \"Hairer: radau\" => Dict(:alg=>radau())\n \"MATLAB: ode23s\" => Dict(:alg=>MATLABDiffEq.ode23s())\n \"MATLAB: ode15s\" => Dict(:alg=>MATLABDiffEq.ode15s())\n \"SciPy: LSODA\" => Dict(:alg=>SciPyDiffEq.LSODA())\n \"SciPy: BDF\" => Dict(:alg=>SciPyDiffEq.BDF())\n \"SciPy: odeint\" => Dict(:alg=>SciPyDiffEq.odeint())\n \"deSolve: lsoda\" => Dict(:alg=>deSolveDiffEq.lsoda())\n \"Sundials: CVODE\" => Dict(:alg=>CVODE_BDF())\n \"ProbNumDiffEq: EK1(3)\" => Dict(:alg=>EK1(order=3, smooth=DENSE))\n \"ProbNumDiffEq: EK1(5)\" => Dict(:alg=>EK1(order=5, smooth=DENSE))\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\ncolors = tocolor.(labels) |> permutedims\n\nabstols = 1.0 ./ 10.0 .^ (5:12)\nreltols = 1.0 ./ 10.0 .^ (2:9)\n\nwp = with_logger(filtered_logger) do\n WorkPrecisionSet(\n [prob, staticprob], abstols, reltols, setups;\n names = labels,\n dense = DENSE,\n verbose = false,\n save_everystep = SAVE_EVERYSTEP,\n appxsol = [test_sol, test_sol],\n maxiters=Int(1e5)\n )\nend\n\nplot(wp, title = \"Stiff 1: ROBER\", color = colors)","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"(Image: )","category":"page"},{"location":"benchmarks/multi-language-wrappers/#Stiff-Problem-2:-HIRES","page":"Multi-Language Wrapper Benchmark","title":"Stiff Problem 2: HIRES","text":"","category":"section"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
Code:","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"f = @ode_def Hires begin\n dy1 = -1.71*y1 + 0.43*y2 + 8.32*y3 + 0.0007\n dy2 = 1.71*y1 - 8.75*y2\n dy3 = -10.03*y3 + 0.43*y4 + 0.035*y5\n dy4 = 8.32*y2 + 1.71*y3 - 1.12*y4\n dy5 = -1.745*y5 + 0.43*y6 + 0.43*y7\n dy6 = -280.0*y6*y8 + 0.69*y4 + 1.71*y5 -\n 0.43*y6 + 0.69*y7\n dy7 = 280.0*y6*y8 - 1.81*y7\n dy8 = -280.0*y6*y8 + 1.81*y7\nend\n\nu0 = zeros(8)\nu0[1] = 1\nu0[8] = 0.0057\nprob = ODEProblem{true,SciMLBase.FullSpecialize()}(f,u0,(0.0,321.8122))\nstaticprob = ODEProblem{false,SciMLBase.FullSpecialize()}(f,SVector{8}(u0),(0.0,321.8122))\n\nsol = solve(prob,Rodas5(),abstol=1/10^14,reltol=1/10^14, dense=false)\ntest_sol = sol\nplot(sol, title=\"HIRES Solution\", legend=false, xticks=:auto, yticks=:auto)","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"(Image: )","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
Code:","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"_setups = [\n \"Julia: Rosenbrock23\" => Dict(:alg=>Rosenbrock23())\n \"Julia: Rodas4\" => Dict(:alg=>Rodas4())\n \"Julia: radau\" => Dict(:alg=>RadauIIA5())\n \"Hairer: rodas\" => Dict(:alg=>rodas())\n \"Hairer: radau\" => Dict(:alg=>radau())\n \"MATLAB: ode23s\" => Dict(:alg=>MATLABDiffEq.ode23s())\n \"MATLAB: ode15s\" => Dict(:alg=>MATLABDiffEq.ode15s())\n \"SciPy: LSODA\" => Dict(:alg=>SciPyDiffEq.LSODA())\n \"SciPy: BDF\" => Dict(:alg=>SciPyDiffEq.BDF())\n \"SciPy: odeint\" => Dict(:alg=>SciPyDiffEq.odeint())\n \"deSolve: lsoda\" => Dict(:alg=>deSolveDiffEq.lsoda())\n \"Sundials: CVODE\" => Dict(:alg=>CVODE_BDF())\n \"ProbNumDiffEq: EK1(2)\" => Dict(:alg=>EK1(order=2, smooth=DENSE))\n \"ProbNumDiffEq: EK1(3)\" => Dict(:alg=>EK1(order=3, smooth=DENSE))\n \"ProbNumDiffEq: EK1(5)\" => Dict(:alg=>EK1(order=5, smooth=DENSE))\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\ncolors = tocolor.(labels) |> permutedims\n\nabstols = 1.0 ./ 10.0 .^ (5:10)\nreltols = 1.0 ./ 10.0 .^ (1:6)\n\nwp = with_logger(filtered_logger) do\n WorkPrecisionSet(\n [prob, staticprob], abstols, reltols, setups;\n names = labels,\n dense = false,\n verbose = false,\n save_everystep = false,\n appxsol = [test_sol, test_sol],\n maxiters = Int(1e5),\n numruns=100\n )\nend\n\nplot(wp, title = \"Stiff 2: Hires\", color=colors)","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"(Image: )","category":"page"},{"location":"benchmarks/multi-language-wrappers/#Appendix","page":"Multi-Language Wrapper Benchmark","title":"Appendix","text":"","category":"section"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
Computer information:","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"using InteractiveUtils\nInteractiveUtils.versioninfo()","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"Julia Version 1.10.0\nCommit 3120989f39b (2023-12-25 18:01 UTC)\nBuild Info:\n Official https://julialang.org/ release\nPlatform Info:\n OS: Linux (x86_64-linux-gnu)\n CPU: 12 × Intel(R) Core(TM) i7-6800K CPU @ 3.40GHz\n WORD_SIZE: 64\n LIBM: libopenlibm\n LLVM: libLLVM-15.0.7 (ORCJIT, broadwell)\n Threads: 17 on 12 virtual cores\nEnvironment:\n JULIA_NUM_THREADS = auto","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
Package information:","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"using Pkg\nPkg.status()","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Project.toml`\n [f3b72e0c] DiffEqDevTools v2.44.1\n [31c24e10] Distributions v0.25.107\n [7073ff75] IJulia v1.24.2\n [7f56f5a3] LSODA v0.7.5\n [e6f89c97] LoggingExtras v1.0.3\n [e2752cbe] MATLABDiffEq v1.2.0\n⌃ [961ee093] ModelingToolkit v8.73.2\n [54ca160b] ODEInterface v0.5.0\n [09606e27] ODEInterfaceDiffEq v3.13.3\n⌃ [1dea7af3] OrdinaryDiffEq v6.66.0\n [65888b18] ParameterizedFunctions v5.16.0\n [91a5bcdd] Plots v1.39.0\n [bf3e78b0] ProbNumDiffEq v0.14.0 `~/.julia/dev/ProbNumDiffEq`\n⌃ [0bca4576] SciMLBase v2.10.0\n [505e40e9] SciPyDiffEq v0.2.1\n [ce78b400] SimpleUnPack v1.1.0\n [90137ffa] StaticArrays v1.9.1\n [c3572dad] Sundials v4.23.1\n [44d3d7a6] Weave v0.10.12\n [0518478a] deSolveDiffEq v0.1.1\nInfo Packages marked with ⌃ have new versions available and may be upgradable.","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
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Full manifest:","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"Pkg.status(mode=Pkg.PKGMODE_MANIFEST)","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Manifest.toml`\n [47edcb42] ADTypes v0.2.6\n⌅ [c3fe647b] AbstractAlgebra v0.34.7\n [621f4979] AbstractFFTs v1.5.0\n [1520ce14] AbstractTrees v0.4.4\n [7d9f7c33] Accessors v0.1.35\n⌅ [79e6a3ab] Adapt v3.7.2\n [ec485272] ArnoldiMethod v0.2.0\n [c9d4266f] ArrayAllocators v0.3.0\n [4fba245c] ArrayInterface v7.7.0\n [4c555306] ArrayLayouts v1.5.2\n [6e4b80f9] BenchmarkTools v1.4.0\n [e2ed5e7c] Bijections v0.1.6\n [d1d4a3ce] BitFlags v0.1.8\n [62783981] BitTwiddlingConvenienceFunctions v0.1.5\n [fa961155] CEnum v0.5.0\n [2a0fbf3d] CPUSummary v0.2.4\n [00ebfdb7] CSTParser 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Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
","category":"page"},{"location":"benchmarks/pleiades/#Pleiades-benchmark","page":"Pleiades","title":"Pleiades benchmark","text":"","category":"section"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"note: Summary\nPleiades is a medium-dimensional, non-stiff, second-order ODE. We see that:The EK0 is much faster than the EK1 as it scales linearly with the ODE dimension.\nIf the problem is a second-order ODE, implement it as a second-order ODE!","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"
Code:","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"using LinearAlgebra, Statistics, Distributions\nusing DiffEqDevTools, ParameterizedFunctions, SciMLBase, OrdinaryDiffEq, Sundials, Plots, ODEInterfaceDiffEq\nusing ModelingToolkit\nusing ProbNumDiffEq\n\nPlots.theme(\n :dao;\n markerstrokewidth=0.5,\n legend=:outertopright,\n margin=5Plots.mm,\n xticks=10.0 .^ (-16:1:16),\n)\n\nfunction plot_chisq_interval!(df, q=0.01)\n dist = Chisq(df)\n low, high, mid = quantile(dist, [q, 1-q])..., mean(dist)\n hline!([low, high], linestyle=:dash, color=:black, label=\"\",\n fill_between=[high nothing], fillcolor=:green, fillalpha=0.15)\n hline!([mid], linestyle=:solid, color=:black, label=\"\")\nend","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"
","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"
Code:","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"# first-order ODE\n@fastmath function pleiades(du, u, p, t)\n v = view(u, 1:7) # x\n w = view(u, 8:14) # y\n x = view(u, 15:21) # x′\n y = view(u, 22:28) # y′\n du[15:21] .= v\n du[22:28] .= w\n @inbounds @simd ivdep for i in 1:14\n du[i] = zero(eltype(u))\n end\n @inbounds @simd ivdep for i in 1:7\n @inbounds @simd ivdep for j in 1:7\n if i != j\n r = ((x[i] - x[j])^2 + (y[i] - y[j])^2)^(3 / 2)\n du[i] += j * (x[j] - x[i]) / r\n du[7+i] += j * (y[j] - y[i]) / r\n end\n end\n end\nend\nx0 = [3.0, 3.0, -1.0, -3.0, 2.0, -2.0, 2.0]\ny0 = [3.0, -3.0, 2.0, 0, 0, -4.0, 4.0]\ndx0 = [0, 0, 0, 0, 0, 1.75, -1.5]\ndy0 = [0, 0, 0, -1.25, 1, 0, 0]\nu0 = [dx0; dy0; x0; y0]\ntspan = (0.0, 3.0)\nprob1 = ODEProblem(pleiades, u0, tspan)\n\n# second-order ODE\nfunction pleiades2(ddu, du, u, p, t)\n x = view(u, 1:7)\n y = view(u, 8:14)\n for i in 1:14\n ddu[i] = zero(eltype(u))\n end\n for i in 1:7, j in 1:7\n if i != j\n r = ((x[i] - x[j])^2 + (y[i] - y[j])^2)^(3 / 2)\n ddu[i] += j * (x[j] - x[i]) / r\n ddu[7+i] += j * (y[j] - y[i]) / r\n end\n end\nend\nu0 = [x0; y0]\ndu0 = [dx0; dy0]\nprob2 = SecondOrderODEProblem(pleiades2, du0, u0, tspan)\nprobs = [prob1, prob2]\n\nref_sol1 = solve(prob1, Vern9(), abstol=1/10^14, reltol=1/10^14)\nref_sol2 = solve(prob2, Vern9(), abstol=1/10^14, reltol=1/10^14)\nref_sols = [ref_sol1, ref_sol2]\n\nplot(ref_sol1, idxs=[(14+i,21+i) for i in 1:7], title=\"Pleiades Solution\", legend=false,\n xticks=:auto, yticks=:auto)\nscatter!(ref_sol1.u[end][15:21], ref_sol1.u[end][22:end], color=1:7)","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"
","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"(Image: )","category":"page"},{"location":"benchmarks/pleiades/#pleiades_results","page":"Pleiades","title":"EK0 vs EK1 & first-order vs. second-order","text":"","category":"section"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"
Code:","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"DENSE = false;\nSAVE_EVERYSTEP = false;\n\n_setups = [\n \"EK0(3) (1st order ODE)\" => Dict(:alg => EK0(order=3, smooth=DENSE), :prob_choice => 1)\n \"EK0(5) (1st order ODE)\" => Dict(:alg => EK0(order=5, smooth=DENSE), :prob_choice => 1)\n \"EK0(4) (2nd order ODE)\" => Dict(:alg => EK0(order=4, smooth=DENSE), :prob_choice => 2)\n \"EK0(6) (2nd order ODE)\" => Dict(:alg => EK0(order=6, smooth=DENSE), :prob_choice => 2)\n \"EK1(3) (1st order ODE)\" => Dict(:alg => EK1(order=3, smooth=DENSE), :prob_choice => 1)\n \"EK1(5) (1st order ODE)\" => Dict(:alg => EK1(order=5, smooth=DENSE), :prob_choice => 1)\n \"EK1(4) (2nd order ODE)\" => Dict(:alg => EK1(order=4, smooth=DENSE), :prob_choice => 2)\n \"EK1(6) (2nd order ODE)\" => Dict(:alg => EK1(order=6, smooth=DENSE), :prob_choice => 2)\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nabstols = 1.0 ./ 10.0 .^ (6:11)\nreltols = 1.0 ./ 10.0 .^ (3:8)\n\nwp = WorkPrecisionSet(\n probs, abstols, reltols, setups;\n names = labels,\n appxsol = ref_sols,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n numruns = 5,\n maxiters = Int(1e7),\n)\n\ncolor = [1 1 1 1 2 2 2 2]\nlinestyle = [:solid :solid :dash :dash :solid :solid :dash :dash]\nplot(wp; color, linestyle)\n\n_ref_setups = [\n \"Classic: Tsit5\" => Dict(:alg => Tsit5(), :prob_choice => 1)\n \"Classic: RadauIIA5\" => Dict(:alg => RadauIIA5(), :prob_choice => 1)\n \"Classic: DPRKN6\" => Dict(:alg => DPRKN6(), :prob_choice => 2)\n]\nref_labels = first.(_ref_setups)\nref_setups = last.(_ref_setups)\nref_wp = WorkPrecisionSet(\n probs, abstols ./ 1000, reltols ./ 1000, ref_setups;\n names = ref_labels,\n appxsol = ref_sols,\n dense = false,\n save_everystep = false,\n maxiters = Int(1e7),\n)\nplot!(ref_wp, x=:final, color=:gray, alpha=0.7, linestyle=[:solid :solid :dash])","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"
","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"(Image: )","category":"page"},{"location":"benchmarks/pleiades/#Calibration","page":"Pleiades","title":"Calibration","text":"","category":"section"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"
Code:","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"plot(wp; x=:final, y=:chi2_final, color, linestyle)\nplot_chisq_interval!(length(u0)*2)","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"
","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"(Image: )","category":"page"},{"location":"benchmarks/pleiades/#Appendix","page":"Pleiades","title":"Appendix","text":"","category":"section"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"
Computer information:","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"using InteractiveUtils\nInteractiveUtils.versioninfo()","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"Julia Version 1.10.0\nCommit 3120989f39b (2023-12-25 18:01 UTC)\nBuild Info:\n Official https://julialang.org/ release\nPlatform Info:\n OS: Linux (x86_64-linux-gnu)\n CPU: 12 × Intel(R) Core(TM) i7-6800K CPU @ 3.40GHz\n WORD_SIZE: 64\n LIBM: libopenlibm\n LLVM: libLLVM-15.0.7 (ORCJIT, broadwell)\n Threads: 17 on 12 virtual cores\nEnvironment:\n JULIA_NUM_THREADS = auto","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"
","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"
Package information:","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"using Pkg\nPkg.status()","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Project.toml`\n [f3b72e0c] DiffEqDevTools v2.44.1\n [31c24e10] Distributions v0.25.107\n [7073ff75] IJulia v1.24.2\n [7f56f5a3] LSODA v0.7.5\n [e6f89c97] LoggingExtras v1.0.3\n [e2752cbe] MATLABDiffEq v1.2.0\n⌃ [961ee093] ModelingToolkit v8.73.2\n [54ca160b] ODEInterface v0.5.0\n [09606e27] ODEInterfaceDiffEq v3.13.3\n⌃ [1dea7af3] OrdinaryDiffEq v6.66.0\n [65888b18] ParameterizedFunctions v5.16.0\n [91a5bcdd] Plots v1.39.0\n [bf3e78b0] ProbNumDiffEq v0.14.0 `~/.julia/dev/ProbNumDiffEq`\n⌃ [0bca4576] SciMLBase v2.10.0\n [505e40e9] SciPyDiffEq v0.2.1\n [ce78b400] SimpleUnPack v1.1.0\n [90137ffa] StaticArrays v1.9.1\n [c3572dad] Sundials v4.23.1\n [44d3d7a6] Weave v0.10.12\n [0518478a] deSolveDiffEq v0.1.1\nInfo Packages marked with ⌃ have new versions available and may be upgradable.","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"
","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"
Full manifest:","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"Pkg.status(mode=Pkg.PKGMODE_MANIFEST)","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Manifest.toml`\n [47edcb42] ADTypes v0.2.6\n⌅ [c3fe647b] AbstractAlgebra v0.34.7\n [621f4979] AbstractFFTs v1.5.0\n [1520ce14] AbstractTrees v0.4.4\n [7d9f7c33] Accessors v0.1.35\n⌅ [79e6a3ab] Adapt v3.7.2\n [ec485272] ArnoldiMethod v0.2.0\n [c9d4266f] ArrayAllocators v0.3.0\n [4fba245c] ArrayInterface v7.7.0\n [4c555306] ArrayLayouts v1.5.2\n [6e4b80f9] BenchmarkTools v1.4.0\n [e2ed5e7c] Bijections v0.1.6\n [d1d4a3ce] BitFlags v0.1.8\n [62783981] BitTwiddlingConvenienceFunctions v0.1.5\n [fa961155] CEnum v0.5.0\n [2a0fbf3d] CPUSummary v0.2.4\n [00ebfdb7] CSTParser v3.4.0\n [49dc2e85] Calculus v0.5.1\n [324d7699] CategoricalArrays v0.10.8\n [d360d2e6] ChainRulesCore v1.19.1\n [fb6a15b2] 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Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"
","category":"page"},{"location":"tutorials/getting_started/#Solving-ODEs-with-Probabilistic-Numerics","page":"Getting Started","title":"Solving ODEs with Probabilistic Numerics","text":"","category":"section"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"In this tutorial we solve a simple non-linear ordinary differential equation (ODE) with the probabilistic numerical ODE solvers implemented in this package.","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"note: Note\nIf you never used DifferentialEquations.jl, check out their \"Getting Started with Differential Equations in Julia\" tutorial. It explains how to define and solve ODE problems and how to analyze the solution, so it's a great starting point. Most of ProbNumDiffEq.jl works exactly as you would expect from DifferentialEquations.jl – just with some added uncertainties and related functionality on top!","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"In this tutorial, we consider a Fitzhugh-Nagumo model described by an ODE of the form","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"beginaligned\ndoty_1 = c (y_1 - fracy_1^33 + y_2) \ndoty_2 = -frac1c (y_1 - a - b y_2)\nendaligned","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"on a time span t in 0 T, with initial value y(0) = y_0. In the following, we","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"define the problem with explicit choices of initial values, integration domains, and parameters,\nsolve the problem with our ODE filters, and\nvisualize the results and the corresponding uncertainties.","category":"page"},{"location":"tutorials/getting_started/#TL;DR:-Just-use-DifferentialEquations.jl-with-the-EK1-algorithm","page":"Getting Started","title":"TL;DR: Just use DifferentialEquations.jl with the EK1 algorithm","text":"","category":"section"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"using ProbNumDiffEq, Plots\n\nfunction fitz(du, u, p, t)\n a, b, c = p\n du[1] = c * (u[1] - u[1]^3 / 3 + u[2])\n du[2] = -(1 / c) * (u[1] - a - b * u[2])\nend\nu0 = [-1.0; 1.0]\ntspan = (0.0, 20.0)\np = (0.2, 0.2, 3.0)\nprob = ODEProblem(fitz, u0, tspan, p)\n\nusing Logging; Logging.disable_logging(Logging.Warn); # hide\nsol = solve(prob, EK1())\nLogging.disable_logging(Logging.Debug) # hide\nplot(sol)","category":"page"},{"location":"tutorials/getting_started/#Step-1:-Define-the-problem","page":"Getting Started","title":"Step 1: Define the problem","text":"","category":"section"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"First, import ProbNumDiffEq.jl","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"using ProbNumDiffEq","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"Then, set up the ODEProblem exactly as you would in DifferentialEquations.jl. Define the vector field","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"function fitz(du, u, p, t)\n a, b, c = p\n du[1] = c * (u[1] - u[1]^3 / 3 + u[2])\n du[2] = -(1 / c) * (u[1] - a - b * u[2])\nend\nnothing # hide","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"and then the ODEProblem, with initial value u0, time span tspan, and parameters p","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"u0 = [-1.0; 1.0]\ntspan = (0.0, 20.0)\np = (0.2, 0.2, 3.0)\nprob = ODEProblem(fitz, u0, tspan, p)\nnothing # hide","category":"page"},{"location":"tutorials/getting_started/#Step-2:-Solve-the-problem","page":"Getting Started","title":"Step 2: Solve the problem","text":"","category":"section"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"To solve the ODE we just use DifferentialEquations.jl's solve interface, together with one of the algorithms implemented in this package. For now, let's use EK1:","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"sol = solve(prob, EK1())\n# nothing # hide","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"That's it! we just computed a probabilistic numerical ODE solution!","category":"page"},{"location":"tutorials/getting_started/#Step-3:-Analyze-the-solution","page":"Getting Started","title":"Step 3: Analyze the solution","text":"","category":"section"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"Let's plot the result with Plots.jl.","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"using Plots\nplot(sol)","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"Looks good! Looks like the EK1 managed to solve the Fitzhugh-Nagumo problem quite well.","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"tip: Tip\nTo learn more about plotting ODE solutions, check out the plotting tutorial for DifferentialEquations.jl + Plots.jl provided here. Most of that works exactly as expected with ProbNumDiffEq.jl.","category":"page"},{"location":"tutorials/getting_started/#Plot-the-probabilistic-error-estimates","page":"Getting Started","title":"Plot the probabilistic error estimates","text":"","category":"section"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"The plot above looks like a standard ODE solution – but it's not! The numerical errors are just so small that we can't see them in the plot, and the probabilistic error estimates are too. We can visualize them by plotting the errors and error estimates directly:","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"using OrdinaryDiffEq, Statistics\nreference = solve(prob, Vern9(), abstol=1e-9, reltol=1e-9, saveat=sol.t)\nerrors = reduce(hcat, mean.(sol.pu) .- reference.u)'\nerror_estimates = reduce(hcat, std.(sol.pu))'\nplot(sol.t, errors, label=\"error\", color=[1 2], xlabel=\"t\", ylabel=\"err\")\nplot!(sol.t, zero(errors), ribbon=3error_estimates, label=\"error estimate\",\n color=[1 2], alpha=0.2)","category":"page"},{"location":"tutorials/getting_started/#More-about-the-ProbabilisticODESolution","page":"Getting Started","title":"More about the ProbabilisticODESolution","text":"","category":"section"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"The solution object returned by ProbNumDiffEq.jl mostly behaves just like any other ODESolution in DifferentialEquations.jl – with some added uncertainties and related functionality on top. The ProbabilisticODESolution can be indexed with","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"sol.u[1]\nsol.u[end]\nsol.t[end]","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"But since sol is a probabilistic numerical ODE solution, it contains a Gaussian distributions over solution values. The marginals of this posterior are stored in sol.pu:","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"sol.pu[end]","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"You can compute means, covariances, and standard deviations via Statistics.jl:","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"using Statistics\nmean(sol.pu[5])\ncov(sol.pu[5])\nstd(sol.pu[5])","category":"page"},{"location":"tutorials/getting_started/#Dense-output","page":"Getting Started","title":"Dense output","text":"","category":"section"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"Probabilistic numerical ODE solvers approximate the posterior distribution","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"p Big( y(t) big y(0) = y_0 doty(t_i) = f_theta(y(t_i) t_i) Big)","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"which describes a posterior not just for the discrete steps but for any t in the continuous space t in 0 T; in classic ODE solvers, this is also known as \"interpolation\" or \"dense output\". The probabilistic solutions returned by our solvers can be interpolated as usual by treating them as functions, but they return Gaussian distributions","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"sol(0.45)\nmean(sol(0.45))","category":"page"},{"location":"tutorials/getting_started/#Next-steps","page":"Getting Started","title":"Next steps","text":"","category":"section"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"Check out one of the other tutorials:","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"\"Second Order ODEs and Energy Preservation\" explains how to solve second-order ODEs more efficiently while also better preserving energy or other conserved quantities;\n\"Solving DAEs with Probabilistic Numerics\" demonstrates how to solve differential algebraic equations in a probabilistic numerical way.","category":"page"},{"location":"tutorials/ode_parameter_inference/#Parameter-Inference-with-ProbNumDiffEq.jl","page":"Parameter Inference","title":"Parameter Inference with ProbNumDiffEq.jl","text":"","category":"section"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"Let's assume we have an initial value problem (IVP)","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"beginaligned\ndoty = f_theta(y t) qquad y(t_0) = y_0\nendaligned","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"which we observe through a set mathcalD = u(t_n)_n=1^N of noisy data points","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"beginaligned\nu(t_n) = H y(t_n) + v_n qquad v_n sim mathcalN(0 R)\nendaligned","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"The question of interest is: How can we compute the marginal likelihood p(mathcalD mid theta)? Short answer: We can't. It's intractable, because computing the true IVP solution exactly y(t) is intractable. What we can do however is compute an approximate marginal likelihood. This is what ProbNumDiffEq.DataLikelihoods provides.","category":"page"},{"location":"tutorials/ode_parameter_inference/#The-specific-problem,-in-code","page":"Parameter Inference","title":"The specific problem, in code","text":"","category":"section"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"Let's assume that the true underlying dynamics are given by a FitzHugh-Nagumo model","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"using ProbNumDiffEq, LinearAlgebra, OrdinaryDiffEq, Plots\nPlots.theme(:default; markersize=2, markerstrokewidth=0.1)\n\nfunction f(du, u, p, t)\n a, b, c = p\n du[1] = c*(u[1] - u[1]^3/3 + u[2])\n du[2] = -(1/c)*(u[1] - a - b*u[2])\nend\nu0 = [-1.0, 1.0]\ntspan = (0.0, 20.0)\np = (0.2, 0.2, 3.0)\ntrue_prob = ODEProblem(f, u0, tspan, p)","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"from which we generate some artificial noisy data","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"true_sol = solve(true_prob, Vern9(), abstol=1e-10, reltol=1e-10)\n\ntimes = 1:0.5:20\nσ = 1e-1\nH = [1 0;]\nodedata = [H*true_sol(t) .+ σ * randn() for t in times]\n\nplot(true_sol, color=:black, linestyle=:dash, label=[\"True Solution\" \"\"])\nscatter!(times, stack(odedata)', color=1, label=[\"Noisy Data\" \"\"])","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"Our goal is then to recover the true parameter p (and thus also the true trajectory plotted above) the noisy data.","category":"page"},{"location":"tutorials/ode_parameter_inference/#Computing-the-negative-log-likelihood","page":"Parameter Inference","title":"Computing the negative log-likelihood","text":"","category":"section"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"To do parameter inference - be it maximum-likelihod, maximum a posteriori, or full Bayesian inference with MCMC - we need to evaluate the likelihood of given a parameter estimate theta_textest, which corresponds to the probability of the data under the trajectory returned by the ODE solver","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"θ_est = (0.1, 0.1, 2.0)\nprob = remake(true_prob, p=θ_est)\nplot(true_sol, color=:black, linestyle=:dash, label=[\"True Solution\" \"\"])\nscatter!(times, stack(odedata)', color=1, label=[\"Noisy Data\" \"\"])\nsol = solve(prob, EK1(), adaptive=false, dt=1e-1)\nplot!(sol, color=2, label=[\"Numerical solution for θ_est\" \"\"])","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"This quantity can be computed in multiple ways; see Data Likelihoods. Here we use ProbNumDiffEq.DataLikelihoods.fenrir_data_loglik:","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"using ProbNumDiffEq.DataLikelihoods\n\ndata = (t=times, u=odedata)\nnll = -fenrir_data_loglik(\n prob, EK1(smooth=true); \n data, observation_noise_cov=σ^2, observation_matrix=H, \n adaptive=false, dt=1e-1)","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"This is the negative marginal log-likelihood of the parameter θ_est. You can use it as any other NLL: Optimize it to compute maximum-likelihood estimates or MAPs, or plug it into MCMC to sample from the posterior. In our paper [3] we compute MLEs by pairing Fenrir with Optimization.jl and ForwardDiff.jl. Let's quickly explore how to do this next.","category":"page"},{"location":"tutorials/ode_parameter_inference/#Maximum-likelihood-parameter-inference","page":"Parameter Inference","title":"Maximum-likelihood parameter inference","text":"","category":"section"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"To compute a maximum-likelihood estimate (MLE), we just need to maximize theta to p(mathcalD mid theta) - that is, minimize the nll from above. We use Optimization.jl for this. First, define a loss function and create an OptimizationProblem","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"using Optimization, OptimizationOptimJL\n\nfunction loss(x, _)\n ode_params = x[begin:end-1]\n prob = remake(true_prob, p=ode_params)\n κ² = exp(x[end]) # we also optimize the diffusion parameter of the EK1\n return -fenrir_data_loglik(\n prob, EK1(smooth=true, diffusionmodel=FixedDiffusion(κ², false));\n data, observation_noise_cov=σ^2, observation_matrix=H,\n adaptive=false, dt=1e-1\n )\nend\n\nfun = OptimizationFunction(loss, Optimization.AutoForwardDiff())\noptprob = OptimizationProblem(\n fun, [θ_est..., 1e0];\n lb=[0.0, 0.0, 0.0, -10], ub=[1.0, 1.0, 5.0, 20] # lower and upper bounds\n)","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"Then, just solve it! Here we use LBFGS:","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"optsol = solve(optprob, LBFGS())\np_mle = optsol.u[1:3]\np_mle # hide","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"Success! The computed MLE is quite close to the true parameter which we used to generate the data. As a final step, let's plot the true solution, the data, and the result of the MLE:","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"plot(true_sol, color=:black, linestyle=:dash, label=[\"True Solution\" \"\"])\nscatter!(times, stack(odedata)', color=1, label=[\"Noisy Data\" \"\"])\nmle_sol = solve(remake(true_prob, p=p_mle), EK1())\nplot!(mle_sol, color=3, label=[\"MLE-parameter Solution\" \"\"])","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"Looks good!","category":"page"},{"location":"tutorials/ode_parameter_inference/#API-Documentation","page":"Parameter Inference","title":"API Documentation","text":"","category":"section"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"For more details, see the API documentation of ProbNumDiffEq.DataLikelihoods at Data Likelihoods.","category":"page"},{"location":"tutorials/ode_parameter_inference/#References","page":"Parameter Inference","title":"References","text":"","category":"section"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"F. Tronarp, N. Bosch and P. Hennig. Fenrir: Physics-Enhanced Regression for Initial Value Problems. In: Proceedings of the 39th International Conference on Machine Learning, Vol. 162 of Proceedings of Machine Learning Research, edited by K. Chaudhuri, S. Jegelka, L. Song, C. Szepesvari, G. Niu and S. Sabato (PMLR, 17–23 Jul 2022); pp. 21776–21794.\n\n\n\nM. Wu and M. Lysy. Data-Adaptive Probabilistic Likelihood Approximation for Ordinary Differential Equations. CoRR (2023), arXiv:2306.05566 [stat.ML].\n\n\n\n","category":"page"},{"location":"#Probabilistic-Numerical-Differential-Equation-Solvers","page":"Home","title":"Probabilistic Numerical Differential Equation Solvers","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"(Image: Banner)","category":"page"},{"location":"","page":"Home","title":"Home","text":"ProbNumDiffEq.jl provides probabilistic numerical solvers to the DifferentialEquations.jl ecosystem. The implemented ODE filters solve differential equations via Bayesian filtering and smoothing and compute not just a single point estimate of the true solution, but a posterior distribution that contains an estimate of its numerical approximation error.","category":"page"},{"location":"","page":"Home","title":"Home","text":"For a short intro video, check out our poster presentation at JuliaCon2021.","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Run Julia, enter ] to bring up Julia's package manager, and add the ProbNumDiffEq.jl package:","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> ]\n(v1.10) pkg> add ProbNumDiffEq","category":"page"},{"location":"#Getting-Started","page":"Home","title":"Getting Started","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"For a quick introduction check out the \"Solving ODEs with Probabilistic Numerics\" tutorial.","category":"page"},{"location":"#Features","page":"Home","title":"Features","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Two extended Kalman filtering-based probabilistic solvers: the explicit EK0 and semi-implicit EK1.\nAdaptive step-size selection with PI control; fully compatible with DifferentialEquations.jl's timestepping options\nOnline uncertainty calibration for multiple different diffusion models (see \"Diffusion models and calibration\")\nDense output\nSampling from the solution\nCallback support\nConvenient plotting through a Plots.jl recipe\nAutomatic differentiation via ForwardDiff.jl\nArbitrary precision via Julia's built-in arbitrary precision arithmetic\nSpecialized solvers for second-order ODEs (see Second Order ODEs and Energy Preservation)\nCompatible with DAEs in mass-matrix ODE form (see Solving DAEs with Probabilistic Numerics)\nData likelihoods for parameter-inference in ODEs (see Parameter Inference with ProbNumDiffEq.jl)","category":"page"},{"location":"#Related-packages","page":"Home","title":"Related packages","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"probdiffeq: Fast and feature-rich filtering-based probabilistic ODE solvers in JAX.\nProbNum: Probabilistic numerics in Python. It has not only probabilistic ODE solvers, but also probabilistic linear solvers, Bayesian quadrature, and many filtering and smoothing implementations.","category":"page"},{"location":"implementation/#Solver-Implementation-via-OrdinaryDiffEq.jl","page":"Implementation via OrdinaryDiffEq.jl","title":"Solver Implementation via OrdinaryDiffEq.jl","text":"","category":"section"},{"location":"implementation/","page":"Implementation via OrdinaryDiffEq.jl","title":"Implementation via OrdinaryDiffEq.jl","text":"ProbNumDiffEq.jl builds directly on OrdinaryDiffEq.jl to benefit from its iterator interface, flexible step-size control, and efficient Jacobian calculations. But, this requires extending non-public APIs. This page is meant to provide an overview on which parts exactly ProbNumDiffEq.jl builds on.","category":"page"},{"location":"implementation/","page":"Implementation via OrdinaryDiffEq.jl","title":"Implementation via OrdinaryDiffEq.jl","text":"For more discussion on the pros and cons of building on OrdinaryDiffEq.jl, see this thread on discourse.","category":"page"},{"location":"implementation/#Building-on-OrdinaryDiffEq.jl","page":"Implementation via OrdinaryDiffEq.jl","title":"Building on OrdinaryDiffEq.jl","text":"","category":"section"},{"location":"implementation/","page":"Implementation via OrdinaryDiffEq.jl","title":"Implementation via OrdinaryDiffEq.jl","text":"ProbNumDiffEq.jl shares most of OrdinaryDiffEq.jl's implementation. In particular:","category":"page"},{"location":"implementation/","page":"Implementation via OrdinaryDiffEq.jl","title":"Implementation via OrdinaryDiffEq.jl","text":"OrdinaryDiffEq.__init builds the cache and the integrator, and calls OrdinaryDiffEq.initialize!\nOrdinaryDiffEq.solve! implements the actual iterator structure, with\nOrdinaryDiffEq.loopheader!\nOrdinaryDiffEq.perform_step!\nOrdinaryDiffEq.loopfooter!\nOrdinaryDiffEq.postamble!","category":"page"},{"location":"implementation/","page":"Implementation via OrdinaryDiffEq.jl","title":"Implementation via OrdinaryDiffEq.jl","text":"ProbNumDiffEq.jl builds around this structure and overloads some of the parts:","category":"page"},{"location":"implementation/","page":"Implementation via OrdinaryDiffEq.jl","title":"Implementation via OrdinaryDiffEq.jl","text":"Algorithms: EK0/EK1 <: AbstractEK <: OrdinaryDiffEq.OrdinaryDiffEqAdaptiveAlgorithm\n./src/algorithms.jl provides the algorithms themselves\n./src/alg_utils.jl implements many traits (relating to automatic differentiation, implicitness, step-size control, etc)\nCache: EKCache <: AbstractODEFilterCache <: OrdinaryDiffEq.OrdinaryDiffEqCache\n./src/caches.jl implements the cache and its main constructor: OrdinaryDiffEq.alg_cache\nInitialization and perform_step!: via OrdinaryDiffEq.initialize! and OrdinaryDiffEq.perform_step!. Implemented in ./src/perform_step.jl.\nCustom postamble by overloading OrdinaryDiffEq.postamble! (which should always call OrdinaryDiffEq._postamble!). This is where we do the \"smoothing\" of the solution. Implemented in ./src/integrator_utils.jl.\nCustom saving by overloading OrdinaryDiffEq.savevalues! (which should always call OrdinaryDiffEq._savevalues!). Implemented in ./src/integrator_utils.jl.","category":"page"},{"location":"implementation/#Building-on-DiffEqBase.jl","page":"Implementation via OrdinaryDiffEq.jl","title":"Building on DiffEqBase.jl","text":"","category":"section"},{"location":"implementation/","page":"Implementation via OrdinaryDiffEq.jl","title":"Implementation via OrdinaryDiffEq.jl","text":"DiffEqBase.__init is currently overloaded to transform OOP problems into IIP problems (in ./src/solve.jl).\nThe solution object: ProbODESolution <: AbstractProbODESolution <: DiffEqBase.AbstractODESolution\n./src/solution.jl implements the main parts. Note that the main constructor DiffEqBase.build_solution is called by OrdinaryDiffEq.__init, so OrdinaryDiffEq.jl has control over its inputs.\nMeanProbODESolution <: DiffEqBase.AbstractODESolution is a wrapper that allows handling the mean of a probabilistic ODE solution the same way one would handle any \"standard\" ODE solution, by just ignoring the covariances.\nAbstractODEFilterPosterior <: DiffEqBase.AbstractDiffEqInterpolation handles the interpolation.\nPlot recipe in ./ext/RecipesBaseExt.jl\nSampling in ./src/solution_sampling.jl\nDiffEqBase.prepare_alg(::EK1{0}); closely follows a similar function implemented in OrdinaryDiffEq.jl ./src/alg_utils.jl\nthis also required DiffEqBase.remake(::EK1)","category":"page"},{"location":"implementation/#Other-packages","page":"Implementation via OrdinaryDiffEq.jl","title":"Other packages","text":"","category":"section"},{"location":"implementation/","page":"Implementation via OrdinaryDiffEq.jl","title":"Implementation via OrdinaryDiffEq.jl","text":"DiffEqDevTools.appxtrue: We extend this function to work with ProbODESolution. This also enables DiffEqDevTools.WorkPrecision to work out of the box.","category":"page"}] +[{"location":"references/#references","page":"References","title":"References","text":"","category":"section"},{"location":"references/","page":"References","title":"References","text":"N. Bosch, F. Tronarp and P. Hennig. Pick-and-Mix Information Operators for Probabilistic ODE Solvers. In: Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, Vol. 151 of Proceedings of Machine Learning Research, edited by G. Camps-Valls, F. J. Ruiz and I. Valera (PMLR, 28–30 Mar 2022); pp. 10015–10027.\n\n\n\nN. Bosch, P. Hennig and F. Tronarp. Probabilistic Exponential Integrators. In: Thirty-seventh Conference on Neural Information Processing Systems (2023).\n\n\n\nF. Tronarp, N. Bosch and P. Hennig. Fenrir: Physics-Enhanced Regression for Initial Value Problems. In: Proceedings of the 39th International Conference on Machine Learning, Vol. 162 of Proceedings of Machine Learning Research, edited by K. Chaudhuri, S. Jegelka, L. Song, C. Szepesvari, G. Niu and S. Sabato (PMLR, 17–23 Jul 2022); pp. 21776–21794.\n\n\n\nF. Tronarp, H. Kersting, S. Särkkä and P. Hennig. Probabilistic solutions to ordinary differential equations as nonlinear Bayesian filtering: a new perspective. Statistics and Computing 29, 1297–1315 (2019).\n\n\n\nN. Krämer, N. Bosch, J. Schmidt and P. Hennig. Probabilistic ODE Solutions in Millions of Dimensions. In: Proceedings of the 39th International Conference on Machine Learning, Vol. 162 of Proceedings of Machine Learning Research, edited by K. Chaudhuri, S. Jegelka, L. Song, C. Szepesvari, G. Niu and S. Sabato (PMLR, 17–23 Jul 2022); pp. 11634–11649.\n\n\n\nS. Särkkä and A. Solin. Applied Stochastic Differential Equations. Institute of Mathematical Statistics Textbooks (Cambridge University Press, 2019).\n\n\n\nN. Krämer and P. Hennig. Stable Implementation of Probabilistic ODE Solvers. CoRR (2020), arXiv:2012.10106 [stat.ML].\n\n\n\nM. Schober, S. Särkkä and P. Hennig. A probabilistic model for the numerical solution of initial value problems. Statistics and Computing 29, 99–122 (2019).\n\n\n\nN. Bosch, P. Hennig and F. Tronarp. Calibrated Adaptive Probabilistic ODE Solvers. In: Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, Vol. 130 of Proceedings of Machine Learning Research, edited by A. Banerjee and K. Fukumizu (PMLR, 13–15 Apr 2021); pp. 3466–3474.\n\n\n\nM. Wu and M. Lysy. Data-Adaptive Probabilistic Likelihood Approximation for Ordinary Differential Equations. CoRR (2023), arXiv:2306.05566 [stat.ML].\n\n\n\nP. Hennig, M. A. Osborne and H. P. Kersting. Probabilistic Numerics: Computation as Machine Learning (Cambridge University Press, 2022).\n\n\n\nH. Kersting and P. Hennig. Active Uncertainty Calibration in Bayesian ODE Solvers. In: Proceedings of the Thirty-Second Conference on Uncertainty in Artificial Intelligence, UAI'16 (AUAI Press, 2016); pp. 309–318.\n\n\n\nH. Kersting, T. J. Sullivan and P. Hennig. Convergence rates of Gaussian ODE filters. Statistics and Computing 30, 1791–1816 (2020).\n\n\n\nF. Tronarp, S. Särkkä and P. Hennig. Bayesian ODE solvers: the maximum a posteriori estimate. Statistics and Computing 31, 23 (2021).\n\n\n\n","category":"page"},{"location":"benchmarks/vanderpol/#Van-der-Pol-benchmark","page":"Van der Pol","title":"Van der Pol benchmark","text":"","category":"section"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"note: Summary\nVan der Pol is a low-dimensional, stiff, second-order ODE. We see that:The EK1 is very well able to solve stiff problems.\nSince Van der Pol is actually a second-order ODE, do solve it as a second-order ODE.\nUse the TaylorInit or ForwardDiffInit initialization. While SimpleInit works well for lower orders, it fails for higher orders. And since Taylor-mode initialization is fast and works well, there is no reason not to use it.","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Code:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"using LinearAlgebra, Statistics, Distributions\nusing DiffEqDevTools, ParameterizedFunctions, SciMLBase, OrdinaryDiffEq, Plots\nusing ProbNumDiffEq\n\nPlots.theme(\n :dao;\n markerstrokewidth=0.5,\n legend=:outertopright,\n margin=5Plots.mm,\n xticks = 10.0 .^ (-16:1:16)\n)\n\nfunction plot_chisq_interval!(df, q=0.01)\n dist = Chisq(df)\n low, high, mid = quantile(dist, [q, 1-q])..., mean(dist)\n hline!([low, high], linestyle=:dash, color=:black, label=\"\",\n fill_between=[high nothing], fillcolor=:green, fillalpha=0.15)\n hline!([mid], linestyle=:solid, color=:black, label=\"\")\nend","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Code:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"function vanderpol!(du, u, p, t)\n du[1] = u[2]\n du[2] = p[1] * ((1 - u[1]^2) * u[2] - u[1])\nend\np = [1e5]\ntspan = (0.0, 2.0)\nu0 = [2.0, 0.0]\nprob = ODEProblem(vanderpol!, u0, tspan, p)\n\ntest_sol = solve(prob, RadauIIA5(), abstol=1/10^14, reltol=1/10^14)\nplot(test_sol, title=\"Van der Pol Solution\", legend=false, ylims=(-5, 5), xticks=:auto)","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"(Image: )","category":"page"},{"location":"benchmarks/vanderpol/#vdp_main_results","page":"Van der Pol","title":"EK1 across orders","text":"","category":"section"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Code:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"DENSE = false;\nSAVE_EVERYSTEP = false;\n\n_setups = [\n \"EK1($order)\" => Dict(:alg => EK1(order=order, smooth=DENSE))\n for order in 3:7\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nabstols = 1.0 ./ 10.0 .^ (6:11)\nreltols = 1.0 ./ 10.0 .^ (3:8)\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n numruns = 5,\n)\n\nplot(wp, palette=Plots.palette([:blue, :red], length(_setups)))\n\n_ref_setups = [\n \"Rosenbrock23\" => Dict(:alg => Rosenbrock23())\n \"Rodas4P\" => Dict(:alg => Rodas4P())\n \"RadauIIA5\" => Dict(:alg => RadauIIA5())\n]\nref_labels = first.(_ref_setups)\nref_setups = last.(_ref_setups)\nref_wp_final = WorkPrecisionSet(\n prob, abstols, reltols, ref_setups;\n names = ref_labels,\n appxsol = test_sol,\n dense = false,\n save_everystep = false,\n maxiters = Int(1e7),\n)\nref_wp_dense = WorkPrecisionSet(\n prob, abstols, reltols, ref_setups;\n names = ref_labels,\n appxsol = test_sol,\n dense = true,\n save_everystep = true,\n maxiters = Int(1e7),\n)\n\nplot!(ref_wp_final, x=:final, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"(Image: )","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Discrete time-series errors (l2):","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Code:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"DENSE = true;\nSAVE_EVERYSTEP = true;\n\n_setups = [\n \"EK1($order)\" => Dict(:alg => EK1(order=order, smooth=DENSE))\n for order in 3:7\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nabstols = 1.0 ./ 10.0 .^ (6:11)\nreltols = 1.0 ./ 10.0 .^ (3:8)\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n numruns = 5,\n)\n\nplot(wp, x=:l2, palette=Plots.palette([:blue, :red], length(_setups)))\nplot!(ref_wp_dense, x=:l2, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"(Image: )","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Interpolation errors (L2):","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Code:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"plot(wp, x=:L2, palette=Plots.palette([:blue, :red], length(_setups)))\nplot!(ref_wp_dense, x=:L2, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"(Image: )","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/#Calibration","page":"Van der Pol","title":"Calibration","text":"","category":"section"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Code:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"plot(wp, x=:final, y=:chi2_final, yguide=\"Chi-squared (final)\",\n palette=Plots.palette([:blue, :red], length(_setups)))\nplot_chisq_interval!(2)","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"(Image: )","category":"page"},{"location":"benchmarks/vanderpol/#vdp_initialization","page":"Van der Pol","title":"Comparison of the different initialization schemes","text":"","category":"section"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Code:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"DENSE = false;\nSAVE_EVERYSTEP = false;\n\nabstols = 1.0 ./ 10.0 .^ (6:11)\nreltols = 1.0 ./ 10.0 .^ (3:8)\n\norders = (3, 5, 8)\nps = []\nfor o in orders\n _setups = [\n \"EK1($o) TaylorInit\" => Dict(:alg => EK1(order=o, smooth=DENSE, initialization=TaylorModeInit(o)))\n \"EK1($o) ForwardDiffInit\" => Dict(:alg => EK1(order=o, smooth=DENSE, initialization=ForwardDiffInit(o)))\n \"EK1($o) SimpleInit\" => Dict(:alg => EK1(order=o, smooth=DENSE, initialization=SimpleInit()))\n # \"EK1($o) ClassicSolverInit\" => Dict(:alg => EK1(order=o, smooth=DENSE, initialization=ClassicSolverInit())) # unstable\n ]\n\n labels = first.(_setups)\n setups = last.(_setups)\n\n wp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n numruns = 5,\n )\n\n p = plot(wp, color=[2 4 5 6], xticks = 10.0 .^ (-16:1:5))\n push!(ps, p)\nend\nplot(\n ps...,\n layout=(length(orders), 1),\n size = (1000, length(orders)*300),\n xlabel=[\"\" \"\" \"\" \"Error\"],\n)","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"(Image: )","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Code:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"DENSE = false;\nSAVE_EVERYSTEP = false;\n\nabstols = 1.0 ./ 10.0 .^ (5:8)\nreltols = 1.0 ./ 10.0 .^ (2:5)\n\n_setups = [\n \"EK1(8) TaylorInit(8)\" => Dict(:alg => EK1(order=8, smooth=DENSE, initialization=TaylorModeInit(8)))\n \"EK1(8) TaylorInit(7)\" => Dict(:alg => EK1(order=8, smooth=DENSE, initialization=TaylorModeInit(7)))\n \"EK1(8) TaylorInit(6)\" => Dict(:alg => EK1(order=8, smooth=DENSE, initialization=TaylorModeInit(6)))\n \"EK1(8) TaylorInit(5)\" => Dict(:alg => EK1(order=8, smooth=DENSE, initialization=TaylorModeInit(5)))\n \"EK1(8) TaylorInit(4)\" => Dict(:alg => EK1(order=8, smooth=DENSE, initialization=TaylorModeInit(4)))\n \"EK1(8) TaylorInit(3)\" => Dict(:alg => EK1(order=8, smooth=DENSE, initialization=TaylorModeInit(3)))\n \"EK1(8) TaylorInit(2)\" => Dict(:alg => EK1(order=8, smooth=DENSE, initialization=TaylorModeInit(2)))\n # \"EK1(8) TaylorInit(1)\" => Dict(:alg => EK1(order=8, smooth=DENSE, initialization=TaylorModeInit(1))) # fails, see above\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n numruns = 5,\n)\n\nplot(wp, palette=Plots.palette([:blue, :red], length(_setups)), xticks = 10.0 .^ (-16:1:5))","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"(Image: )","category":"page"},{"location":"benchmarks/vanderpol/#vdp_second_order","page":"Van der Pol","title":"Solving the first- vs second-order ODE","text":"","category":"section"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Code:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"function vanderpol2!(ddu, du, u, p, t)\n ddu[1] = p[1] * ((1 - u[1]^2) * du[1] - u[1])\nend\np = [1e5]\ntspan = (0.0, 2.0)\nu0 = [2.0]\ndu0 = [0.0]\nprob2 = SecondOrderODEProblem(vanderpol2!, du0, u0, tspan, p)\n\ntest_sol2 = solve(prob2, RadauIIA5(), abstol=1/10^14, reltol=1/10^14)\n# plot(test_sol2, title=\"Van der Pol Solution (2nd order)\", legend=false, ylims=(-5, 5), xticks=:auto)\nnothing","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Code:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"DENSE = true;\nSAVE_EVERYSTEP = true;\n\n_setups = [\n \"EK1(3) 1st order\" => Dict(:alg => EK1(order=3, smooth=DENSE))\n \"EK1(4) 1st order\" => Dict(:alg => EK1(order=4, smooth=DENSE))\n \"EK1(5) 1st order\" => Dict(:alg => EK1(order=5, smooth=DENSE))\n \"EK1(6) 1st order\" => Dict(:alg => EK1(order=6, smooth=DENSE))\n \"EK1(4) 2nd order\" => Dict(:prob_choice => 2, :alg => EK1(order=4, smooth=DENSE))\n \"EK1(5) 2nd order\" => Dict(:prob_choice => 2, :alg => EK1(order=5, smooth=DENSE))\n \"EK1(6) 2nd order\" => Dict(:prob_choice => 2, :alg => EK1(order=6, smooth=DENSE))\n \"EK1(7) 2nd order\" => Dict(:prob_choice => 2, :alg => EK1(order=7, smooth=DENSE))\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nabstols = 1.0 ./ 10.0 .^ (6:11)\nreltols = 1.0 ./ 10.0 .^ (3:8)\n\nwp = WorkPrecisionSet(\n [prob, prob2], abstols, reltols, setups;\n names = labels,\n appxsol = [test_sol, test_sol2],\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n numruns = 5,\n)\n\ncolor = [1 1 1 1 2 2 2 2]\nplot(wp; x=:final, color)\nplot!(ref_wp_dense, x=:final, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"(Image: )","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Interpolation errors (L2):","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Code:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"plot(wp; x=:L2, color)\nplot!(ref_wp_dense, x=:L2, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"(Image: )","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/#Calibration-2","page":"Van der Pol","title":"Calibration","text":"","category":"section"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Code:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"plot(wp; x=:final, y=:chi2_final, yguide=\"Chi-squared (final)\", color)\nplot_chisq_interval!(2)","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"(Image: )","category":"page"},{"location":"benchmarks/vanderpol/#Appendix","page":"Van der Pol","title":"Appendix","text":"","category":"section"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Computer information:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"using InteractiveUtils\nInteractiveUtils.versioninfo()","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"Julia Version 1.10.0\nCommit 3120989f39b (2023-12-25 18:01 UTC)\nBuild Info:\n Official https://julialang.org/ release\nPlatform Info:\n OS: Linux (x86_64-linux-gnu)\n CPU: 12 × Intel(R) Core(TM) i7-6800K CPU @ 3.40GHz\n WORD_SIZE: 64\n LIBM: libopenlibm\n LLVM: libLLVM-15.0.7 (ORCJIT, broadwell)\n Threads: 17 on 12 virtual cores\nEnvironment:\n JULIA_NUM_THREADS = auto","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Package information:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"using Pkg\nPkg.status()","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Project.toml`\n [f3b72e0c] DiffEqDevTools v2.44.1\n [31c24e10] Distributions v0.25.107\n [7073ff75] IJulia v1.24.2\n [7f56f5a3] LSODA v0.7.5\n [e6f89c97] LoggingExtras v1.0.3\n [e2752cbe] MATLABDiffEq v1.2.0\n⌃ [961ee093] ModelingToolkit v8.73.2\n [54ca160b] ODEInterface v0.5.0\n [09606e27] ODEInterfaceDiffEq v3.13.3\n⌃ [1dea7af3] OrdinaryDiffEq v6.66.0\n [65888b18] ParameterizedFunctions v5.16.0\n [91a5bcdd] Plots v1.39.0\n [bf3e78b0] ProbNumDiffEq v0.14.0 `~/.julia/dev/ProbNumDiffEq`\n⌃ [0bca4576] SciMLBase v2.10.0\n [505e40e9] SciPyDiffEq v0.2.1\n [ce78b400] SimpleUnPack v1.1.0\n [90137ffa] StaticArrays v1.9.1\n [c3572dad] Sundials v4.23.1\n [44d3d7a6] Weave v0.10.12\n [0518478a] deSolveDiffEq v0.1.1\nInfo Packages marked with ⌃ have new versions available and may be upgradable.","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
Full manifest:","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"Pkg.status(mode=Pkg.PKGMODE_MANIFEST)","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Manifest.toml`\n [47edcb42] ADTypes v0.2.6\n⌅ [c3fe647b] AbstractAlgebra v0.34.7\n [621f4979] AbstractFFTs v1.5.0\n [1520ce14] AbstractTrees v0.4.4\n [7d9f7c33] Accessors v0.1.35\n⌅ [79e6a3ab] Adapt v3.7.2\n [ec485272] ArnoldiMethod v0.2.0\n [c9d4266f] ArrayAllocators v0.3.0\n [4fba245c] ArrayInterface v7.7.0\n [4c555306] ArrayLayouts v1.5.2\n [6e4b80f9] BenchmarkTools v1.4.0\n [e2ed5e7c] Bijections v0.1.6\n [d1d4a3ce] BitFlags v0.1.8\n [62783981] BitTwiddlingConvenienceFunctions v0.1.5\n [fa961155] CEnum v0.5.0\n [2a0fbf3d] CPUSummary v0.2.4\n [00ebfdb7] CSTParser v3.4.0\n [49dc2e85] Calculus v0.5.1\n [324d7699] CategoricalArrays v0.10.8\n [d360d2e6] ChainRulesCore v1.19.1\n 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Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`","category":"page"},{"location":"benchmarks/vanderpol/","page":"Van der Pol","title":"Van der Pol","text":"
","category":"page"},{"location":"benchmarks/rober/#ROBER-benchmark","page":"ROBER","title":"ROBER benchmark","text":"","category":"section"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"note: Summary\nThe EK1 can solve mass-matrix DAEs. But for this problem, it only works well for low tolerances.\nFor this problem it only works well for low tolerances, but the order-to-error-tolerance heuristic should in principle still hold: lower tolerance level rightarrow higher order.","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"Adapted from SciMLBenchmarks.jl.","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"
Code:","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"using LinearAlgebra, Statistics, Distributions\nusing DiffEqDevTools, ParameterizedFunctions, SciMLBase, OrdinaryDiffEq, Sundials, Plots\nusing ModelingToolkit\nusing ProbNumDiffEq\n\nPlots.theme(\n :dao;\n markerstrokewidth=0.5,\n legend=:outertopright,\n margin=5Plots.mm,\n xticks=10.0 .^ (-16:1:16),\n)\n\nfunction plot_chisq_interval!(df, q=0.01)\n dist = Chisq(df)\n low, high, mid = quantile(dist, [q, 1-q])..., mean(dist)\n hline!([low, high], linestyle=:dash, color=:black, label=\"\",\n fill_between=[high nothing], fillcolor=:green, fillalpha=0.15)\n hline!([mid], linestyle=:solid, color=:black, label=\"\")\nend","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"
","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"
Code:","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"@variables t y₁(t)=1.0 y₂(t)=0.0 y₃(t)=0.0\n@parameters k₁=0.04 k₂=3e7 k₃=1e4\nD = Differential(t)\neqs = [\n D(y₁) ~ -k₁*y₁ + k₃*y₂*y₃\n D(y₂) ~ k₁*y₁ - k₃*y₂*y₃ - k₂*y₂^2\n 0 ~ y₁ + y₂ + y₃ - 1\n]\n@named sys = ODESystem(eqs)\nmmprob = ODEProblem(sys,[],(0.0,1e5))\ndaeprob = DAEProblem(sys,[D(y₁)=>-0.04, D(y₂)=>0.04, D(y₃)=>0.0],[],(0.0,1e5)) # can't handle this yet\nodaeprob = ODAEProblem(structural_simplify(sys),[],(0.0,1e5)) # can't handle this yet\n\nref_sol = solve(daeprob,IDA(),abstol=1/10^14,reltol=1/10^14)\nplot(ref_sol, idxs=[y₁,y₂,y₃], title=\"ROBER Solution\", legend=false, ylims=(0, 1), xticks=:auto)","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"
","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"(Image: )","category":"page"},{"location":"benchmarks/rober/#rober_results","page":"ROBER","title":"EK1 across orders","text":"","category":"section"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"
Code:","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"DENSE = false;\nSAVE_EVERYSTEP = false;\n\n_setups = [\n \"EK1($order)\" => Dict(:alg => EK1(order=order, smooth=DENSE))\n for order in 2:4\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nabstols = 1.0 ./ 10.0 .^ (4:7)\nreltols = 1.0 ./ 10.0 .^ (1:4)\n\nwp = WorkPrecisionSet(\n mmprob, abstols, reltols, setups;\n names = labels,\n appxsol = ref_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n numruns = 10,\n maxiters = Int(1e7),\n)\n\nplot(wp, palette=Plots.palette([:blue, :red], length(_setups)))\n\n\n_ref_setups = [\n \"Rosenbrock23\" => Dict(:alg => Rosenbrock23())\n \"Rodas4P\" => Dict(:alg => Rodas4P())\n \"RadauIIA\" => Dict(:alg => RadauIIA5())\n]\nref_labels = first.(_ref_setups)\nref_setups = last.(_ref_setups)\nref_wp = WorkPrecisionSet(\n mmprob, abstols ./ 10000, reltols ./ 10000, ref_setups;\n names = ref_labels,\n appxsol = ref_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n)\nplot!(ref_wp, x=:final, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"
","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"(Image: )","category":"page"},{"location":"benchmarks/rober/#Calibration","page":"ROBER","title":"Calibration","text":"","category":"section"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"
Code:","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"plot(wp; x=:final, y=:chi2_final, yguide=\"Chi-squared (final)\",\n palette=Plots.palette([:blue, :red], length(_setups)))\nplot_chisq_interval!(3)","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"
","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"(Image: )","category":"page"},{"location":"benchmarks/rober/#Appendix","page":"ROBER","title":"Appendix","text":"","category":"section"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"
Computer information:","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"using InteractiveUtils\nInteractiveUtils.versioninfo()","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"Julia Version 1.10.0\nCommit 3120989f39b (2023-12-25 18:01 UTC)\nBuild Info:\n Official https://julialang.org/ release\nPlatform Info:\n OS: Linux (x86_64-linux-gnu)\n CPU: 12 × Intel(R) Core(TM) i7-6800K CPU @ 3.40GHz\n WORD_SIZE: 64\n LIBM: libopenlibm\n LLVM: libLLVM-15.0.7 (ORCJIT, broadwell)\n Threads: 17 on 12 virtual cores\nEnvironment:\n JULIA_NUM_THREADS = auto","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"
","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"
Package information:","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"using Pkg\nPkg.status()","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Project.toml`\n [f3b72e0c] DiffEqDevTools v2.44.1\n [31c24e10] Distributions v0.25.107\n [7073ff75] IJulia v1.24.2\n [7f56f5a3] LSODA v0.7.5\n [e6f89c97] LoggingExtras v1.0.3\n [e2752cbe] MATLABDiffEq v1.2.0\n⌃ [961ee093] ModelingToolkit v8.73.2\n [54ca160b] ODEInterface v0.5.0\n [09606e27] ODEInterfaceDiffEq v3.13.3\n⌃ [1dea7af3] OrdinaryDiffEq v6.66.0\n [65888b18] ParameterizedFunctions v5.16.0\n [91a5bcdd] Plots v1.39.0\n [bf3e78b0] ProbNumDiffEq v0.14.0 `~/.julia/dev/ProbNumDiffEq`\n⌃ [0bca4576] SciMLBase v2.10.0\n [505e40e9] SciPyDiffEq v0.2.1\n [ce78b400] SimpleUnPack v1.1.0\n [90137ffa] StaticArrays v1.9.1\n [c3572dad] Sundials v4.23.1\n [44d3d7a6] Weave v0.10.12\n [0518478a] deSolveDiffEq v0.1.1\nInfo Packages marked with ⌃ have new versions available and may be upgradable.","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"
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Full manifest:","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"Pkg.status(mode=Pkg.PKGMODE_MANIFEST)","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Manifest.toml`\n [47edcb42] ADTypes v0.2.6\n⌅ [c3fe647b] AbstractAlgebra v0.34.7\n [621f4979] AbstractFFTs v1.5.0\n [1520ce14] AbstractTrees v0.4.4\n [7d9f7c33] Accessors v0.1.35\n⌅ [79e6a3ab] Adapt v3.7.2\n [ec485272] ArnoldiMethod v0.2.0\n [c9d4266f] ArrayAllocators v0.3.0\n [4fba245c] ArrayInterface v7.7.0\n [4c555306] ArrayLayouts v1.5.2\n [6e4b80f9] BenchmarkTools v1.4.0\n [e2ed5e7c] Bijections v0.1.6\n [d1d4a3ce] BitFlags v0.1.8\n [62783981] BitTwiddlingConvenienceFunctions v0.1.5\n [fa961155] CEnum v0.5.0\n [2a0fbf3d] CPUSummary v0.2.4\n [00ebfdb7] CSTParser v3.4.0\n [49dc2e85] Calculus v0.5.1\n [324d7699] CategoricalArrays v0.10.8\n [d360d2e6] ChainRulesCore v1.19.1\n [fb6a15b2] CloseOpenIntervals 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Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`","category":"page"},{"location":"benchmarks/rober/","page":"ROBER","title":"ROBER","text":"
","category":"page"},{"location":"likelihoods/#Data-Likelihoods","page":"Data Likelihoods","title":"Data Likelihoods","text":"","category":"section"},{"location":"likelihoods/","page":"Data Likelihoods","title":"Data Likelihoods","text":"ProbNumDiffEq.DataLikelihoods.fenrir_data_loglik\nProbNumDiffEq.DataLikelihoods.dalton_data_loglik","category":"page"},{"location":"likelihoods/#ProbNumDiffEq.fenrir_data_loglik","page":"Data Likelihoods","title":"ProbNumDiffEq.fenrir_data_loglik","text":"fenrir_data_loglik(\n prob::SciMLBase.AbstractODEProblem,\n alg::ProbNumDiffEq.AbstractEK,\n args...;\n observation_matrix,\n observation_noise_cov,\n data,\n kwargs...\n)\n\n\nCompute the Fenrir [3] approximate negative log-likelihood (NLL) of the data.\n\nThis is a convenience function that\n\nSolves the ODE with a ProbNumDiffEq.EK1 of the specified order and with a diffusion as provided by the diffusion_var argument, and\nFits the ODE posterior to the data via Kalman filtering and thereby computes the log-likelihood of the data on the way.\n\nYou can control the step-size behaviour of the solver as you would for a standard ODE solve, but additionally the solver always steps through the data.t locations by adding them to tstops.\n\nYou can also choose steps adaptively by setting adaptive=true, but this is not well-tested so use at your own risk!\n\nArguments\n\nprob::SciMLBase.AbstractODEProblem: the initial value problem of interest\nalg::AbstractEK: the probabilistic ODE solver to be used; use EK1 for best results.\ndata::NamedTuple{(:t, :u)}: the data to be fitted\nobservation_matrix::Union{AbstractMatrix,UniformScaling}: the matrix which maps the ODE state to the measurements; typically a projection matrix\nobservation_noise_cov::Union{Number,AbstractMatrix}: the scalar observation noise variance\n\nReference\n\n[3] Tronarp et al, \"Fenrir: Physics-Enhanced Regression for Initial Value Problems\", ICML (2022)\n\n\n\n\n\n","category":"function"},{"location":"likelihoods/#ProbNumDiffEq.dalton_data_loglik","page":"Data Likelihoods","title":"ProbNumDiffEq.dalton_data_loglik","text":"dalton_data_loglik(\n prob::SciMLBase.AbstractODEProblem,\n alg::ProbNumDiffEq.AbstractEK,\n args...;\n observation_matrix,\n observation_noise_cov,\n data,\n kwargs...\n)\n\n\nCompute the DALTON [10] approximate negative log-likelihood (NLL) of the data.\n\nYou can control the step-size behaviour of the solver as you would for a standard ODE solve, but additionally the solver always steps through the data.t locations by adding them to tstops. You can also choose steps adaptively by setting adaptive=true, but this is not well-tested so use at your own risk!\n\nArguments\n\nprob::SciMLBase.AbstractODEProblem: the initial value problem of interest\nalg::AbstractEK: the probabilistic ODE solver to be used; use EK1 for best results.\ndata::NamedTuple{(:t, :u)}: the data to be fitted\nobservation_matrix::Union{AbstractMatrix,UniformScaling}: the matrix which maps the ODE state to the measurements; typically a projection matrix\nobservation_noise_cov::Union{Number,AbstractMatrix}: the scalar observation noise variance\n\nReference\n\n[10] Wu et al, \"Data-Adaptive Probabilistic Likelihood Approximation for Ordinary Differential Equations\", arXiv (2023)\n\n\n\n\n\n","category":"function"},{"location":"filtering/#Gaussian-Filtering-and-Smoothing","page":"Filtering and Smoothing","title":"Gaussian Filtering and Smoothing","text":"","category":"section"},{"location":"filtering/#Predict","page":"Filtering and Smoothing","title":"Predict","text":"","category":"section"},{"location":"filtering/","page":"Filtering and Smoothing","title":"Filtering and Smoothing","text":"ProbNumDiffEq.predict\nProbNumDiffEq.predict!","category":"page"},{"location":"filtering/#ProbNumDiffEq.predict","page":"Filtering and Smoothing","title":"ProbNumDiffEq.predict","text":"predict(x::Gaussian, A::AbstractMatrix, Q::AbstractMatrix)\n\nPrediction step in Kalman filtering for linear dynamics models.\n\nGiven a Gaussian x = mathcalN(μ Σ), compute and return mathcalN(A μ A Σ A^T + Q).\n\nSee also the non-allocating square-root version predict!.\n\n\n\n\n\n","category":"function"},{"location":"filtering/#ProbNumDiffEq.predict!","page":"Filtering and Smoothing","title":"ProbNumDiffEq.predict!","text":"predict!(x_out, x_curr, Ah, Qh, cachemat)\n\nIn-place and square-root implementation of predict which saves the result into x_out.\n\nOnly works with PSDMatrices.PSDMatrix types as Ah, Qh, and in the covariances of x_curr and x_out (both of type Gaussian). To prevent allocations, a cache matrix cachemat of size D times 2D (where D times D is the size of Ah and Qh) needs to be passed.\n\nSee also: predict.\n\n\n\n\n\n","category":"function"},{"location":"filtering/#Update","page":"Filtering and Smoothing","title":"Update","text":"","category":"section"},{"location":"filtering/","page":"Filtering and Smoothing","title":"Filtering and Smoothing","text":"ProbNumDiffEq.update\nProbNumDiffEq.update!","category":"page"},{"location":"filtering/#ProbNumDiffEq.update","page":"Filtering and Smoothing","title":"ProbNumDiffEq.update","text":"update(x, measurement, H)\n\nUpdate step in Kalman filtering for linear dynamics models.\n\nGiven a Gaussian x = mathcalN(μ Σ) and a measurement z = mathcalN(hatz S), with S = H Σ H^T, compute\n\nbeginaligned\nK = Σ^P H^T S^-1 \nμ^F = μ + K (0 - hatz) \nΣ^F = Σ - K S K^T\nendaligned\n\nand return an updated state \\mathcal{N}(μ^F, Σ^F). Note that this assumes zero-measurements. When called with ProbNumDiffEq.SquarerootMatrix type arguments it performs the update in Joseph / square-root form.\n\nFor better performance, we recommend to use the non-allocating update!.\n\n\n\n\n\n","category":"function"},{"location":"filtering/#ProbNumDiffEq.update!","page":"Filtering and Smoothing","title":"ProbNumDiffEq.update!","text":"update!(x_out, x_pred, measurement, H, K_cache, M_cache, S_cache)\n\nIn-place and square-root implementation of update which saves the result into x_out.\n\nImplemented in Joseph Form to retain the PSDMatrix covariances:\n\nbeginaligned\nK = Σ^P H^T S^-1 \nμ^F = μ + K (0 - hatz) \nsqrtΣ^F = (I - KH) sqrt(Σ)\nendaligned\n\nwhere sqrtM denotes the left square-root of a matrix M, i.e. M = sqrtM sqrtM^T.\n\nTo prevent allocations, write into caches K_cache and M_cache, both of size D × D, and S_cache of same type as measurement.Σ.\n\nSee also: update.\n\n\n\n\n\n","category":"function"},{"location":"filtering/#Smooth","page":"Filtering and Smoothing","title":"Smooth","text":"","category":"section"},{"location":"filtering/","page":"Filtering and Smoothing","title":"Filtering and Smoothing","text":"ProbNumDiffEq.smooth","category":"page"},{"location":"filtering/#ProbNumDiffEq.smooth","page":"Filtering and Smoothing","title":"ProbNumDiffEq.smooth","text":"smooth(x_curr, x_next_smoothed, A, Q)\n\nUpdate step of the Kalman smoother, aka. Rauch-Tung-Striebel smoother, for linear dynamics models.\n\nGiven Gaussians x_n = mathcalN(μ_n Σ_n) and x_n+1 = mathcalN(μ_n+1^S Σ_n+1^S), compute\n\nbeginaligned\nμ_n+1^P = A μ_n^F \nP_n+1^P = A Σ_n^F A + Q \nG = Σ_n^S A^T (Σ_n+1^P)^-1 \nμ_n^S = μ_n^F + G (μ_n+1^S - μ_n+1^P) \nΣ_n^S = (I - G A) Σ_n^F (I - G A)^T + G Q G^T + G Σ_n+1^S G^T\nendaligned\n\nand return a smoothed state \\mathcal{N}(μ_n^S, Σ_n^S). When called with ProbNumDiffEq.SquarerootMatrix type arguments it performs the update in Joseph / square-root form.\n\n\n\n\n\n","category":"function"},{"location":"filtering/#Markov-Kernels","page":"Filtering and Smoothing","title":"Markov Kernels","text":"","category":"section"},{"location":"filtering/","page":"Filtering and Smoothing","title":"Filtering and Smoothing","text":"ProbNumDiffEq.AffineNormalKernel\nProbNumDiffEq.marginalize!\nProbNumDiffEq.compute_backward_kernel!","category":"page"},{"location":"filtering/#ProbNumDiffEq.AffineNormalKernel","page":"Filtering and Smoothing","title":"ProbNumDiffEq.AffineNormalKernel","text":"AffineNormalKernel(A[, b], C)\n\nStructure to represent affine Normal Markov kernels, i.e. conditional distributions of the form\n\nbeginaligned\ny mid x sim mathcalN left( y A x + b C right)\nendaligned\n\nAt the point of writing, AffineNormalKernels are only used to precompute and store the backward representation of the posterior (via compute_backward_kernel!) and for smoothing (via marginalize!).\n\n\n\n\n\n","category":"type"},{"location":"filtering/#ProbNumDiffEq.marginalize!","page":"Filtering and Smoothing","title":"ProbNumDiffEq.marginalize!","text":"marginalize!(\n xout::Gaussian{Vector{T},PSDMatrix{T,S}}\n x::Gaussian{Vector{T},PSDMatrix{T,S}},\n K::AffineNormalKernel{<:AbstractMatrix,Union{<:Number,<:AbstractVector,Missing},<:PSDMatrix};\n C_DxD, C_3DxD\n)\n\nBasically the same as predict!), but in kernel language and with support for affine transitions. At the time of writing, this is only used to smooth the posterior using it's backward representation, where the kernels are precomputed with compute_backward_kernel!.\n\nNote that this function assumes certain shapes:\n\nsize(x.μ) == (D, D)\nsize(x.Σ) == (D, D)\nsize(K.A) == (D, D)\nsize(K.b) == (D,), or missing\nsize(K.C) == (D, D), _but with a tall square-root size(K.C.R) == (3D, D)\n\nxout is assumes to have the same shapes as x.\n\n\n\n\n\n","category":"function"},{"location":"filtering/#ProbNumDiffEq.compute_backward_kernel!","page":"Filtering and Smoothing","title":"ProbNumDiffEq.compute_backward_kernel!","text":"compute_backward_kernel!(Kout, xpred, x, K; C_DxD[, diffusion=1])\n\nCompute the backward representation of the posterior, i.e. the conditional distribution of the current state given the next state and the transition kernel.\n\nMore precisely, given a distribution (x)\n\nbeginaligned\nx sim mathcalN left( x μ Σ right)\nendaligned\n\na kernel (K)\n\nbeginaligned\ny mid x sim mathcalN left( y A x + b C right)\nendaligned\n\nand a distribution (xpred) obtained via marginalization\n\nbeginaligned\ny sim mathcalN left( y μ^P Σ^P right) \nμ^P = A μ + b \nΣ^P = A Σ A^top + C\nendaligned\n\nthis function computes the conditional distribution\n\nbeginaligned\nx mid y sim mathcalN left( x G y + d Λ right)\nendaligned\n\nwhere\n\nbeginaligned\nG = Σ A^top (Σ^P)^-1 \nd = μ - G μ^P \nΛ = Σ - G Σ^P G^top\nendaligned\n\nEverything is computed in square-root form and with minimal allocations (thus the cache C_DxD), so the actual formulas implemented here differ a bit.\n\nThe resulting backward kernels are used to smooth the posterior, via marginalize!.\n\n\n\n\n\n","category":"function"},{"location":"benchmarks/orego/#OREGO-benchmark","page":"OREGO","title":"OREGO benchmark","text":"","category":"section"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"note: Summary\nThe EK1 is able to solve mass-matrix DAEs. To achieve low error, use order 4 or higher.\nThe order-to-error-tolerance heuristic holds: lower tolerance level rightarrow higher order.","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"Adapted from SciMLBenchmarks.jl.","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"
Code:","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"using LinearAlgebra, Statistics, Distributions\nusing DiffEqDevTools, ParameterizedFunctions, SciMLBase, OrdinaryDiffEq, Sundials, Plots\nusing ModelingToolkit\nusing ProbNumDiffEq\n\nPlots.theme(\n :dao;\n markerstrokewidth=0.5,\n legend=:outertopright,\n margin=5Plots.mm,\n xticks=10.0 .^ (-16:1:16),\n)\n\nfunction plot_chisq_interval!(df, q=0.01)\n dist = Chisq(df)\n low, high, mid = quantile(dist, [q, 1-q])..., mean(dist)\n hline!([low, high], linestyle=:dash, color=:black, label=\"\",\n fill_between=[high nothing], fillcolor=:green, fillalpha=0.15)\n hline!([mid], linestyle=:solid, color=:black, label=\"\")\nend","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"
","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"
Code:","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"@variables t y1(t)=1.0 y2(t)=2.0 y3(t)=3.0\n@parameters p1=77.27 p2=8.375e-6 p3=0.161\nD = Differential(t)\neqs = [\n D(y1) ~ p1*(y2+y1*(1-p2*y1-y2))\n D(y2) ~ (y3-(1+y1)*y2)/p1\n D(y3) ~ p3*(y1-y3)\n]\n@named sys = ODESystem(eqs)\nsimpsys = structural_simplify(sys)\nmmprob = ODEProblem(sys,[],(0.0,30.0))\ndaeprob = DAEProblem(sys,[D(y1)=>77.26935286375,\n D(y2)=>-0.012941633234114146,\n D(y3)=>-0.322],[],(0.0,30.0))\nodaeprob = ODAEProblem(simpsys,[],(0.0,30.0))\n\nref_sol = solve(daeprob,IDA(),abstol=1/10^14,reltol=1/10^14)\n\nplot(ref_sol, title=\"OREGO Solution\", legend=false, xticks=:auto)","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"
","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"(Image: )","category":"page"},{"location":"benchmarks/orego/#orego_results","page":"OREGO","title":"EK1 across orders","text":"","category":"section"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"
Code:","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"DENSE = false;\nSAVE_EVERYSTEP = false;\n\n_setups = [\n \"EK1($order)\" => Dict(:alg => EK1(order=order, smooth=DENSE))\n for order in 2:6\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nabstols = 1.0 ./ 10.0 .^ (6:10)\nreltols = 1.0 ./ 10.0 .^ (3:7)\n\nwp = WorkPrecisionSet(\n mmprob, abstols, reltols, setups;\n names = labels,\n appxsol = ref_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n numruns = 10,\n maxiters = Int(1e7),\n)\n\nplot(wp, palette=Plots.palette([:blue, :red], length(_setups)))\n\n\n_ref_setups = [\n \"Rosenbrock23\" => Dict(:alg => Rosenbrock23())\n \"Rodas4P\" => Dict(:alg => Rodas4P())\n \"RadauIIA\" => Dict(:alg => RadauIIA5())\n]\nref_labels = first.(_ref_setups)\nref_setups = last.(_ref_setups)\nref_wp = WorkPrecisionSet(\n mmprob, abstols, reltols, ref_setups;\n names = ref_labels,\n appxsol = ref_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n)\nplot!(ref_wp, x=:final, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"
","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"(Image: )","category":"page"},{"location":"benchmarks/orego/#Calibration","page":"OREGO","title":"Calibration","text":"","category":"section"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"
Code:","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"plot(wp; x=:final, y=:chi2_final, yguide=\"Chi-squared (final)\",\n palette=Plots.palette([:blue, :red], length(_setups)))\nplot_chisq_interval!(3)","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"
","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"(Image: )","category":"page"},{"location":"benchmarks/orego/#Appendix","page":"OREGO","title":"Appendix","text":"","category":"section"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"
Computer information:","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"using InteractiveUtils\nInteractiveUtils.versioninfo()","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"Julia Version 1.10.0\nCommit 3120989f39b (2023-12-25 18:01 UTC)\nBuild Info:\n Official https://julialang.org/ release\nPlatform Info:\n OS: Linux (x86_64-linux-gnu)\n CPU: 12 × Intel(R) Core(TM) i7-6800K CPU @ 3.40GHz\n WORD_SIZE: 64\n LIBM: libopenlibm\n LLVM: libLLVM-15.0.7 (ORCJIT, broadwell)\n Threads: 17 on 12 virtual cores\nEnvironment:\n JULIA_NUM_THREADS = auto","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"
","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"
Package information:","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"using Pkg\nPkg.status()","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Project.toml`\n [f3b72e0c] DiffEqDevTools v2.44.1\n [31c24e10] Distributions v0.25.107\n [7073ff75] IJulia v1.24.2\n [7f56f5a3] LSODA v0.7.5\n [e6f89c97] LoggingExtras v1.0.3\n [e2752cbe] MATLABDiffEq v1.2.0\n⌃ [961ee093] ModelingToolkit v8.73.2\n [54ca160b] ODEInterface v0.5.0\n [09606e27] ODEInterfaceDiffEq v3.13.3\n⌃ [1dea7af3] OrdinaryDiffEq v6.66.0\n [65888b18] ParameterizedFunctions v5.16.0\n [91a5bcdd] Plots v1.39.0\n [bf3e78b0] ProbNumDiffEq v0.14.0 `~/.julia/dev/ProbNumDiffEq`\n⌃ [0bca4576] SciMLBase v2.10.0\n [505e40e9] SciPyDiffEq v0.2.1\n [ce78b400] SimpleUnPack v1.1.0\n [90137ffa] StaticArrays v1.9.1\n [c3572dad] Sundials v4.23.1\n [44d3d7a6] Weave v0.10.12\n [0518478a] deSolveDiffEq v0.1.1\nInfo Packages marked with ⌃ have new versions available and may be upgradable.","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"
","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"
Full manifest:","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"Pkg.status(mode=Pkg.PKGMODE_MANIFEST)","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Manifest.toml`\n [47edcb42] ADTypes v0.2.6\n⌅ [c3fe647b] AbstractAlgebra v0.34.7\n [621f4979] AbstractFFTs v1.5.0\n [1520ce14] AbstractTrees v0.4.4\n [7d9f7c33] Accessors v0.1.35\n⌅ [79e6a3ab] Adapt v3.7.2\n [ec485272] ArnoldiMethod v0.2.0\n [c9d4266f] ArrayAllocators v0.3.0\n [4fba245c] ArrayInterface v7.7.0\n [4c555306] ArrayLayouts v1.5.2\n [6e4b80f9] BenchmarkTools v1.4.0\n [e2ed5e7c] Bijections v0.1.6\n [d1d4a3ce] BitFlags v0.1.8\n [62783981] BitTwiddlingConvenienceFunctions v0.1.5\n [fa961155] CEnum v0.5.0\n [2a0fbf3d] CPUSummary v0.2.4\n [00ebfdb7] CSTParser v3.4.0\n [49dc2e85] Calculus v0.5.1\n [324d7699] CategoricalArrays v0.10.8\n [d360d2e6] ChainRulesCore v1.19.1\n [fb6a15b2] CloseOpenIntervals 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Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`","category":"page"},{"location":"benchmarks/orego/","page":"OREGO","title":"OREGO","text":"
","category":"page"},{"location":"solvers/#Solvers","page":"Solvers","title":"Solvers","text":"","category":"section"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"ProbNumDiffEq.jl provides two solvers: the EK1 and the EK0. Both based on extended Kalman filtering and smoothing, but the latter relies on evaluating the Jacobian of the vector field.","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"Which solver should I use?","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"Use the EK1 to get the best uncertainty quantification and to solve stiff problems.\nUse the EK0 to get the fastest runtimes and to solve high-dimensional problems.","category":"page"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"All solvers are compatible with DAEs in mass-matrix ODE form. They also specialize on second-order ODEs: If the problem is of type SecondOrderODEProblem, it solves the second-order problem directly; this is more efficient than solving the transformed first-order problem and provides more meaningful posteriors [1].","category":"page"},{"location":"solvers/#API","page":"Solvers","title":"API","text":"","category":"section"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"EK1\nEK0","category":"page"},{"location":"solvers/#ProbNumDiffEq.EK1","page":"Solvers","title":"ProbNumDiffEq.EK1","text":"EK1(; order=3,\n smooth=true,\n prior=IWP(order),\n diffusionmodel=DynamicDiffusion(),\n initialization=TaylorModeInit(num_derivatives(prior)),\n kwargs...)\n\nGaussian ODE filter with first-order vector field linearization.\n\nThis is a semi-implicit, L-stable ODE solver so it can handle stiffness quite well [4], and it generally produces more expressive posterior covariances than the EK0. However, as typical implicit ODE solvers it scales cubically with the ODE dimension [5], so if you're solving a high-dimensional non-stiff problem you might want to give the EK0 a try.\n\nArguments\n\norder::Integer: Order of the integrated Wiener process (IWP) prior.\nsmooth::Bool: Turn smoothing on/off; smoothing is required for dense output.\nprior::AbstractGaussMarkovProcess: Prior to be used by the ODE filter. By default, uses a 3-times integrated Wiener process prior IWP(3). See also: Priors.\ndiffusionmodel::ProbNumDiffEq.AbstractDiffusion: See Diffusion models and calibration.\ninitialization::ProbNumDiffEq.InitializationScheme: See Initialization.\n\nSome additional kwargs relating to implicit solvers are supported; check out DifferentialEquations.jl's Extra Options page. Right now, we support autodiff, chunk_size, and diff_type. In particular, autodiff=false can come in handy to use finite differences instead of ForwardDiff.jl to compute Jacobians.\n\nExamples\n\njulia> solve(prob, EK1())\n\nReferences\n\n\n\n\n\n","category":"type"},{"location":"solvers/#ProbNumDiffEq.EK0","page":"Solvers","title":"ProbNumDiffEq.EK0","text":"EK0(; order=3,\n smooth=true,\n prior=IWP(order),\n diffusionmodel=DynamicDiffusion(),\n initialization=TaylorModeInit(num_derivatives(prior)))\n\nGaussian ODE filter with zeroth-order vector field linearization.\n\nThis is an explicit ODE solver. It is fast and scales well to high-dimensional problems [5], but it is not L-stable [4]. So for stiff problems, use the EK1.\n\nWhenever possible this solver will use a Kronecker-factored implementation to achieve its linear scaling and to get the best runtimes. This can currently be done only with an IWP prior (default), with a scalar diffusion model (either DynamicDiffusion or FixedDiffusion). For other configurations the solver falls back to a dense implementation which scales cubically with the problem size.\n\nArguments\n\norder::Integer: Order of the integrated Wiener process (IWP) prior.\nsmooth::Bool: Turn smoothing on/off; smoothing is required for dense output.\nprior::AbstractGaussMarkovProcess: Prior to be used by the ODE filter. By default, uses a 3-times integrated Wiener process prior IWP(3). See also: Priors.\ndiffusionmodel::ProbNumDiffEq.AbstractDiffusion: See Diffusion models and calibration.\ninitialization::ProbNumDiffEq.InitializationScheme: See Initialization.\n\nExamples\n\njulia> solve(prob, EK0())\n\nReferences\n\n\n\n\n\n","category":"type"},{"location":"solvers/#Probabilistic-Exponential-Integrators","page":"Solvers","title":"Probabilistic Exponential Integrators","text":"","category":"section"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"ExpEK\nRosenbrockExpEK","category":"page"},{"location":"solvers/#ProbNumDiffEq.ExpEK","page":"Solvers","title":"ProbNumDiffEq.ExpEK","text":"ExpEK(; L, order=3, kwargs...)\n\nProbabilistic exponential integrator\n\nProbabilistic exponential integrators are a class of integrators for semi-linear stiff ODEs that provide improved stability by essentially solving the linear part of the ODE exactly. In probabilistic numerics, this amounts to including the linear part into the prior model of the solver.\n\nExpEK is therefore just a short-hand for EK0 with IOUP prior:\n\nExpEK(; order=3, L, kwargs...) = EK0(; prior=IOUP(order, L), kwargs...)\n\nSee also RosenbrockExpEK, EK0, EK1.\n\nArguments\n\nSee EK0 for available keyword arguments.\n\nExamples\n\njulia> prob = ODEProblem((du, u, p, t) -> (@. du = - u + sin(u)), [1.0], (0.0, 10.0))\njulia> solve(prob, ExpEK(L=-1))\n\nReference\n\n[2] Bosch et al, \"Probabilistic Exponential Integrators\", arXiv (2021)\n\n\n\n\n\n","category":"function"},{"location":"solvers/#ProbNumDiffEq.RosenbrockExpEK","page":"Solvers","title":"ProbNumDiffEq.RosenbrockExpEK","text":"RosenbrockExpEK(; order=3, kwargs...)\n\nProbabilistic Rosenbrock-type exponential integrator\n\nA probabilistic exponential integrator similar to ExpEK, but with automatic linearization along the mean numerical solution. This brings the advantage that the linearity does not need to be specified manually, and the more accurate local linearization can sometimes also improve stability; but since the \"prior\" is adjusted at each step the probabilistic interpretation becomes more complicated.\n\nRosenbrockExpEK is just a short-hand for EK1 with locally-updated IOUP prior:\n\nRosenbrockExpEK(; order=3, kwargs...) = EK1(; prior=IOUP(order, update_rate_parameter=true), kwargs...)\n\nSee also ExpEK, EK0, EK1.\n\nArguments\n\nSee EK1 for available keyword arguments.\n\nExamples\n\njulia> prob = ODEProblem((du, u, p, t) -> (@. du = - u + sin(u)), [1.0], (0.0, 10.0))\njulia> solve(prob, RosenbrockExpEK())\n\nReference\n\n[2] Bosch et al, \"Probabilistic Exponential Integrators\", NeurIPS (2023)\n\n\n\n\n\n","category":"function"},{"location":"solvers/#solversrefs","page":"Solvers","title":"References","text":"","category":"section"},{"location":"solvers/","page":"Solvers","title":"Solvers","text":"F. Tronarp, H. Kersting, S. Särkkä and P. Hennig. Probabilistic solutions to ordinary differential equations as nonlinear Bayesian filtering: a new perspective. Statistics and Computing 29, 1297–1315 (2019).\n\n\n\nN. Krämer, N. Bosch, J. Schmidt and P. Hennig. Probabilistic ODE Solutions in Millions of Dimensions. In: Proceedings of the 39th International Conference on Machine Learning, Vol. 162 of Proceedings of Machine Learning Research, edited by K. Chaudhuri, S. Jegelka, L. Song, C. Szepesvari, G. Niu and S. Sabato (PMLR, 17–23 Jul 2022); pp. 11634–11649.\n\n\n\nN. Bosch, P. Hennig and F. Tronarp. Probabilistic Exponential Integrators. In: Thirty-seventh Conference on Neural Information Processing Systems (2023).\n\n\n\n","category":"page"},{"location":"benchmarks/hodgkinhuxley/#Hodgkin-Huxley-benchmark","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley benchmark","text":"","category":"section"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"note: Summary\nHodgkin-Huxley is a four-dimensional ODE, which can be stiff or non-stiff depending on the parameters; here we consider a non-stiff version. We see that:EK0 is the fastest solver.\nRosenbrockExpEK is slowest; but suffers less from smoothing than EK0 and EK1.\nResults are similar for fixed time steps.","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
Code:","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"using LinearAlgebra, Statistics, Distributions\nusing DiffEqDevTools, SciMLBase, OrdinaryDiffEq, Plots, SimpleUnPack\nusing ProbNumDiffEq\n\nPlots.theme(\n :dao;\n markerstrokewidth=0.5,\n legend=:outertopright,\n margin=5Plots.mm,\n xticks=10.0 .^ (-16:1:16),\n yticks=10.0 .^ (-6:1:5),\n)","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
Code:","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"αm(V, VT) = -0.32 * (V - VT - 13) / (exp(-(V - VT - 13) / 4) - 1)\nβm(V, VT) = 0.28 * (V - VT - 40) / (exp((V - VT - 40) / 5) - 1)\n\nαn(V, VT) = -0.032 * (V - VT - 15) / (exp(-(V - VT - 15) / 5) - 1)\nβn(V, VT) = 0.5 * exp(-(V - VT - 10) / 40)\n\nαh(V, VT) = 0.128 * exp(-(V - VT - 17) / 18)\nβh(V, VT) = 4 / (1 + exp(-(V - VT - 40) / 5))\n\nconst current_tspan = (5, 40)\nInj(t) = (current_tspan[1] <= t <= current_tspan[2]) ? 500one(t) : zero(t)\n\nfunction f(du, u, p, t)\n @unpack gNa, gK, ENa, EK, area, C, Eleak, VT, gleak = p\n\n V, m, n, h = u\n\n I_inj = Inj(t) * 1e-6\n\n du[2] = dmdt = (αm(V, VT) * (1 - m) - βm(V, VT) * m)\n du[3] = dndt = (αn(V, VT) * (1 - n) - βn(V, VT) * n)\n du[4] = dhdt = (αh(V, VT) * (1 - h) - βh(V, VT) * h)\n\n INa = gNa * m^3 * h * (V - ENa) * area\n IK = gK * n^4 * (V - EK) * area\n Ileak = gleak * (V - Eleak) * area\n Cm = C * area\n du[1] = dVdt = -(Ileak + INa + IK - I_inj) / Cm\nend\n\np = (gNa=20.0, gK=15.0, ENa = 53, EK = -107, area = 15e-5, C = 1, Eleak = -70, VT = -60, gleak = 0.1, V0 = -70)\n\nm_inf(V, VT) = 1 / (1 + βm(V, VT) / αm(V, VT))\nn_inf(V, VT) = 1 / (1 + βn(V, VT) / αn(V, VT))\nh_inf(V, VT) = 1 / (1 + βh(V, VT) / αh(V, VT))\nu0 = [p.V0, m_inf(p.V0, p.VT), n_inf(p.V0, p.VT), h_inf(p.V0, p.VT)]\n\nprob = ODEProblem{true,SciMLBase.FullSpecialize()}(f, u0, (0.0, 50.0), p)\n\ntest_sol = solve(prob, Vern7(), abstol=1/10^14, reltol=1/10^14)\nplot(test_sol,\n legend=false,\n layout=(4,1),\n title=[\"Hodgkin-Huxley Solution\" \"\" \"\" \"\"],\n ylabel=[\"V(t)\" \"m(t)\" \"n(t)\" \"h(t)\"],\n xlabel=[\"\" \"\" \"\" \"t\"],\n size = (1000, 600),\n color=[1 2 3 4],\n xticks=:auto, yticks=:auto\n)","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"(Image: )","category":"page"},{"location":"benchmarks/hodgkinhuxley/#hh_solver_comparison","page":"Hodgkin-Huxley","title":"Solver comparison: EK0 vs. EK1 vs RosenbrockExpEK","text":"","category":"section"},{"location":"benchmarks/hodgkinhuxley/#Without-smoothing","page":"Hodgkin-Huxley","title":"Without smoothing","text":"","category":"section"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
Code:","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"DENSE = SAVE_EVERYSTEP = false\n\n_setups = [\n \"EK0(2)\" => Dict(:alg=>EK0(order=2, smooth=DENSE))\n \"EK0(3)\" => Dict(:alg=>EK0(order=3, smooth=DENSE))\n \"EK1(2)\" => Dict(:alg=>EK1(order=2, smooth=DENSE))\n \"EK1(3)\" => Dict(:alg=>EK1(order=3, smooth=DENSE))\n \"RosenbrockExpEK1(2)\" => Dict(:alg=>RosenbrockExpEK(order=2, smooth=DENSE))\n \"RosenbrockExpEK1(3)\" => Dict(:alg=>RosenbrockExpEK(order=3, smooth=DENSE))\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\ncolors = [1 1 2 2 3 3]\n\nabstols = 1.0 ./ 10.0 .^ (6:10)\nreltols = 1.0 ./ 10.0 .^ (3:7)\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n numruns = 5,\n tstops = current_tspan,\n)\n\nplot(wp, title=\"Adaptive steps - no smoothing\", color=colors)\n\n\n_ref_setups = [\n \"Tsit5\" => Dict(:alg => Tsit5())\n \"Vern7\" => Dict(:alg => Vern7())\n \"RadauIIA5\" => Dict(:alg => RadauIIA5())\n]\nref_labels = first.(_ref_setups)\nref_setups = last.(_ref_setups)\nref_wp_final = WorkPrecisionSet(\n prob, abstols ./ 1000, reltols ./ 1000, ref_setups;\n names = ref_labels,\n appxsol = test_sol,\n dense = false,\n save_everystep = false,\n maxiters = Int(1e7),\n tstops = current_tspan,\n)\nref_wp_dense = WorkPrecisionSet(\n prob, abstols ./ 1000, reltols ./ 1000, ref_setups;\n names = ref_labels,\n appxsol = test_sol,\n dense = true,\n save_everystep = true,\n maxiters = Int(1e7),\n tstops = current_tspan,\n)\n\nplot!(ref_wp_final, x=:final, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"(Image: )","category":"page"},{"location":"benchmarks/hodgkinhuxley/#With-smoothing","page":"Hodgkin-Huxley","title":"With smoothing","text":"","category":"section"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
Code:","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"DENSE = SAVE_EVERYSTEP = true\n\n_setups = [\n \"EK0(2)\" => Dict(:alg=>EK0(order=2, smooth=DENSE))\n \"EK0(3)\" => Dict(:alg=>EK0(order=3, smooth=DENSE))\n \"EK1(2)\" => Dict(:alg=>EK1(order=2, smooth=DENSE))\n \"EK1(3)\" => Dict(:alg=>EK1(order=3, smooth=DENSE))\n \"RosenbrockExpEK1(2)\" => Dict(:alg=>RosenbrockExpEK(order=2, smooth=DENSE))\n \"RosenbrockExpEK1(3)\" => Dict(:alg=>RosenbrockExpEK(order=3, smooth=DENSE))\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\ncolors = [1 1 2 2 3 3]\n\nabstols = 1.0 ./ 10.0 .^ (6:10)\nreltols = 1.0 ./ 10.0 .^ (3:7)\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n numruns = 5,\n tstops = current_tspan,\n)\n\nplot(wp, title=\"Adaptive steps - with smoothing\", color=colors)\nplot!(ref_wp_dense, x=:final, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"(Image: )","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
Interoplation errors (L2):","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
Code:","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"plot(wp, x=:L2, title=\"Adaptive steps - with smoothing\", color=colors)\nplot!(ref_wp_dense, x=:L2, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"(Image: )","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
","category":"page"},{"location":"benchmarks/hodgkinhuxley/#Calibration","page":"Hodgkin-Huxley","title":"Calibration","text":"","category":"section"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
Code:","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"plot(wp; x=:final, y=:chi2_final, yguide=\"Chi-squared (final)\", color=colors)\n\n# Should be distributed according to a Chi-squared distribution:\nfunction plot_chisq_interval!(df, q=0.01)\n dist = Chisq(df)\n low, high, mid = quantile(dist, [q, 1-q])..., mean(dist)\n hline!([low, high], linestyle=:dash, color=:black, label=\"\",\n fill_between=[high nothing], fillcolor=:green, fillalpha=0.15)\n hline!([mid], linestyle=:solid, color=:black, label=\"\")\nend\nplot_chisq_interval!(4)","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"(Image: )","category":"page"},{"location":"benchmarks/hodgkinhuxley/#hh_fixed_steps","page":"Hodgkin-Huxley","title":"Fixed-step solver comparison","text":"","category":"section"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"Without smothing:","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
Code:","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"DENSE = SAVE_EVERYSTEP = false\n\ndts = 10.0 .^ range(-2, -3, length=10)[begin:end-1]\nabstols = reltols = repeat([missing], length(dts))\n\nDM = FixedDiffusion()\n_setups = [\n \"EK0(3)\" => Dict(:alg=>EK0(order=3, diffusionmodel=DM, smooth=DENSE), :dts=>dts)\n \"EK1(3)\" => Dict(:alg=>EK1(order=3, diffusionmodel=DM, smooth=DENSE), :dts=>dts)\n \"RosenbrockExpEK1(3)\" => Dict(:alg=>RosenbrockExpEK(order=3, diffusionmodel=DM, smooth=DENSE), :dts=>dts)\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\ncolors = [1 2 3]\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n adaptive = false,\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n numruns = 5,\n)\n\nplot(wp, title=\"Fixed steps - no smoothing\", color=colors)","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"(Image: )","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
With smoothing:","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
Code:","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"DENSE = SAVE_EVERYSTEP = true\n\ndts = 10.0 .^ range(-2, -3, length=10)[begin:end-1]\nabstols = reltols = repeat([missing], length(dts))\n\nDM = FixedDiffusion()\n_setups = [\n \"EK0(3)\" => Dict(:alg=>EK0(order=3, diffusionmodel=DM, smooth=DENSE), :dts=>dts)\n \"EK1(3)\" => Dict(:alg=>EK1(order=3, diffusionmodel=DM, smooth=DENSE), :dts=>dts)\n \"RosenbrockExpEK1(3)\" => Dict(:alg=>RosenbrockExpEK(order=3, diffusionmodel=DM, smooth=DENSE), :dts=>dts)\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\ncolors = [1 2 3]\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n adaptive = false,\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n numruns = 5,\n)\n\nplot(wp, title=\"Fixed steps - with smoothing\", color=colors)","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"(Image: )","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
","category":"page"},{"location":"benchmarks/hodgkinhuxley/#Appendix","page":"Hodgkin-Huxley","title":"Appendix","text":"","category":"section"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
Computer information:","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"using InteractiveUtils\nInteractiveUtils.versioninfo()","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"Julia Version 1.10.0\nCommit 3120989f39b (2023-12-25 18:01 UTC)\nBuild Info:\n Official https://julialang.org/ release\nPlatform Info:\n OS: Linux (x86_64-linux-gnu)\n CPU: 12 × Intel(R) Core(TM) i7-6800K CPU @ 3.40GHz\n WORD_SIZE: 64\n LIBM: libopenlibm\n LLVM: libLLVM-15.0.7 (ORCJIT, broadwell)\n Threads: 17 on 12 virtual cores\nEnvironment:\n JULIA_NUM_THREADS = auto","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
Package information:","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"using Pkg\nPkg.status()","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Project.toml`\n [f3b72e0c] DiffEqDevTools v2.44.1\n [31c24e10] Distributions v0.25.107\n [7073ff75] IJulia v1.24.2\n [7f56f5a3] LSODA v0.7.5\n [e6f89c97] LoggingExtras v1.0.3\n [e2752cbe] MATLABDiffEq v1.2.0\n⌃ [961ee093] ModelingToolkit v8.73.2\n [54ca160b] ODEInterface v0.5.0\n [09606e27] ODEInterfaceDiffEq v3.13.3\n⌃ [1dea7af3] OrdinaryDiffEq v6.66.0\n [65888b18] ParameterizedFunctions v5.16.0\n [91a5bcdd] Plots v1.39.0\n [bf3e78b0] ProbNumDiffEq v0.14.0 `~/.julia/dev/ProbNumDiffEq`\n⌃ [0bca4576] SciMLBase v2.10.0\n [505e40e9] SciPyDiffEq v0.2.1\n [ce78b400] SimpleUnPack v1.1.0\n [90137ffa] StaticArrays v1.9.1\n [c3572dad] Sundials v4.23.1\n [44d3d7a6] Weave v0.10.12\n [0518478a] deSolveDiffEq v0.1.1\nInfo Packages marked with ⌃ have new versions available and may be upgradable.","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
Full manifest:","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"Pkg.status(mode=Pkg.PKGMODE_MANIFEST)","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Manifest.toml`\n [47edcb42] ADTypes v0.2.6\n⌅ [c3fe647b] AbstractAlgebra v0.34.7\n [621f4979] AbstractFFTs v1.5.0\n [1520ce14] AbstractTrees v0.4.4\n [7d9f7c33] Accessors v0.1.35\n⌅ [79e6a3ab] Adapt v3.7.2\n [ec485272] ArnoldiMethod v0.2.0\n [c9d4266f] ArrayAllocators v0.3.0\n [4fba245c] ArrayInterface v7.7.0\n [4c555306] ArrayLayouts v1.5.2\n [6e4b80f9] BenchmarkTools v1.4.0\n [e2ed5e7c] Bijections v0.1.6\n [d1d4a3ce] BitFlags v0.1.8\n [62783981] BitTwiddlingConvenienceFunctions v0.1.5\n [fa961155] CEnum v0.5.0\n [2a0fbf3d] CPUSummary v0.2.4\n [00ebfdb7] CSTParser v3.4.0\n [49dc2e85] Calculus v0.5.1\n [324d7699] CategoricalArrays v0.10.8\n [d360d2e6] 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available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`","category":"page"},{"location":"benchmarks/hodgkinhuxley/","page":"Hodgkin-Huxley","title":"Hodgkin-Huxley","text":"
","category":"page"},{"location":"tutorials/exponential_integrators/#probexpinttutorial","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"","category":"section"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"Exponential integrators are a class of numerical methods for solving semi-linear ordinary differential equations (ODEs) of the form","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"beginaligned\ndoty(t) = L y(t) + f(y(t) t) quad y(0) = y_0\nendaligned","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"where L is a linear operator and f is a nonlinear function. In a nutshell, exponential integrators solve the linear part of the ODE exactly, and only approximate the nonlinear part. Probabilistic exponential integrators [2] are the probabilistic numerics approach to exponential integrators.","category":"page"},{"location":"tutorials/exponential_integrators/#Example","page":"Probabilistic Exponential Integrators","title":"Example","text":"","category":"section"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"Let's consider a simple semi-linear ODE","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"beginaligned\ndoty(t) = - y(t) + sin(y(t)) quad y(0) = 10\nendaligned","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"We can solve this ODE reasonably well with the standard EK1 and adaptive steps (the default):","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"using ProbNumDiffEq, Plots, LinearAlgebra\ntheme(:default; palette=[\"#4063D8\", \"#389826\", \"#9558B2\", \"#CB3C33\"])\n\nf(du, u, p, t) = (@. du = -u + sin(u))\nu0 = [1.0]\ntspan = (0.0, 20.0)\nprob = ODEProblem(f, u0, tspan)\n\nref = solve(prob, EK1(), abstol=1e-10, reltol=1e-10)\nplot(ref, color=:black, linestyle=:dash, label=\"Reference\")","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"But for fixed (large) step sizes this ODE is more challenging: The explicit EK0 method oscillates and diverges due to the stiffness of the ODE, and the semi-implicit EK1 method is stable but the solution is not very accurate.","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"STEPSIZE = 4\nDM = FixedDiffusion() # recommended for fixed steps\n\n# we don't smooth the EK0 here to show the oscillations more clearly\nsol0 = solve(prob, EK0(smooth=false, diffusionmodel=DM), adaptive=false, dt=STEPSIZE, dense=false)\nsol1 = solve(prob, EK1(diffusionmodel=DM), adaptive=false, dt=STEPSIZE)\n\nplot(ylims=(0.3, 1.05))\nplot!(ref, color=:black, linestyle=:dash, label=\"Reference\")\nplot!(sol0, denseplot=false, marker=:o, markersize=2, label=\"EK0\", color=1)\nplot!(sol1, denseplot=false, marker=:o, markersize=2, label=\"EK1\", color=2)","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"Probabilistic exponential integrators leverage the semi-linearity of the ODE to compute more accurate solutions for the same fixed step size. You can use either the ExpEK method and provide the linear part (with the keyword argument L), or the RosenbrockExpEK to automatically linearize along the mean of the numerical solution:","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"sol_exp = solve(prob, ExpEK(L=-1, diffusionmodel=DM), adaptive=false, dt=STEPSIZE)\nsol_ros = solve(prob, RosenbrockExpEK(diffusionmodel=DM), adaptive=false, dt=STEPSIZE)\n\nplot(ylims=(0.3, 1.05))\nplot!(ref, color=:black, linestyle=:dash, label=\"Reference\")\nplot!(sol_exp, denseplot=false, marker=:o, markersize=2, label=\"ExpEK\", color=3)\nplot!(sol_ros, denseplot=false, marker=:o, markersize=2, label=\"RosenbrockExpEK\", color=4)","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"The solutions are indeed much more accurate than those of the standard EK1, for the same fixed step size!","category":"page"},{"location":"tutorials/exponential_integrators/#Background:-Integrated-Ornstein-Uhlenbeck-priors","page":"Probabilistic Exponential Integrators","title":"Background: Integrated Ornstein-Uhlenbeck priors","text":"","category":"section"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"Probabilistic exponential integrators \"solve the linear part exactly\" by including it into the prior model of the solver. Namely, the solver chooses a (q-times) integrated Ornstein-Uhlenbeck prior with rate parameter equal to the linearity. The ExpEK solver is just a short-hand for an EK0 with appropriate prior:","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"ExpEK(order=3, L=-1) == EK0(prior=IOUP(3, -1))","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"Similarly, the RosenbrockExpEK solver is also just a short-hand:","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"RosenbrockExpEK(order=3) == EK1(prior=IOUP(3, update_rate_parameter=true))","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"This means that you can also construct other probabilistic exponential integrators by hand! In this example the EK1 with IOUP prior with rate parameter -1 performs extremely well:","category":"page"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"sol_expek1 = solve(prob, EK1(prior=IOUP(3, -1), diffusionmodel=DM), adaptive=false, dt=STEPSIZE)\n\nplot(ylims=(0.3, 1.05))\nplot!(ref, color=:black, linestyle=:dash, label=\"Reference\")\nplot!(sol_expek1, denseplot=false, marker=:o, markersize=2, label=\"EK1 + IOUP\")","category":"page"},{"location":"tutorials/exponential_integrators/#References","page":"Probabilistic Exponential Integrators","title":"References","text":"","category":"section"},{"location":"tutorials/exponential_integrators/","page":"Probabilistic Exponential Integrators","title":"Probabilistic Exponential Integrators","text":"N. Bosch, P. Hennig and F. Tronarp. Probabilistic Exponential Integrators. In: Thirty-seventh Conference on Neural Information Processing Systems (2023).\n\n\n\n","category":"page"},{"location":"benchmarks/lotkavolterra/#Lotka-Volterra-benchmark","page":"Lotka-Volterra","title":"Lotka-Volterra benchmark","text":"","category":"section"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"note: Summary\nLotka-Volterra is a simple, low-dimensional, non-stiff ODE. We see that:EK0 and EK1 have a very similar runtime. But note that Lotka-Volterra is a non-stiff low-dimensional problem: if it were stiff, EK1 would be better; if it were high-dimensional, EK0 would be faster.\nOrders behave as in classic solvers: Use low order for low accuracy, medium order for medium accuracy, high order for high accuracy.\nDo not use EK0 with order > 5: The adaptive step size selection apparently does not work well for high orders right now.\nUse diffusionmodel=DynamicDiffusion: Error-wise, the performance of the diffusion models is similar, but the calibration of FixedDiffusion and FixedMVDiffusion with adaptive steps is currently broken.\nInitialization schemes are all similar, but initialization=TaylorModeInit performs best.\nIf you only need to solve for the last time point, set smooth=false, dense=false, and save_everystep=false. This greatly reduces the run time of the solver.","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"Benchmark adapted from SciMLBenchmarks.jl.","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"using LinearAlgebra, Statistics, Distributions\nusing DiffEqDevTools, ParameterizedFunctions, SciMLBase, OrdinaryDiffEq, Plots\nusing ProbNumDiffEq\n\nPlots.theme(\n :dao;\n markerstrokewidth=0.5,\n legend=:outertopright,\n margin=5Plots.mm,\n xticks=10.0 .^ (-16:1:16),\n)\n\nfunction plot_chisq_interval!(df, q=0.01)\n dist = Chisq(df)\n low, high, mid = quantile(dist, [q, 1-q])..., mean(dist)\n hline!([low, high], linestyle=:dash, color=:black, label=\"\",\n fill_between=[high nothing], fillcolor=:green, fillalpha=0.15)\n hline!([mid], linestyle=:solid, color=:black, label=\"\")\nend","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"f = @ode_def LotkaVolterra begin\n dx = a*x - b*x*y\n dy = -c*y + d*x*y\nend a b c d\np = [1.5, 1, 3, 1]\ntspan = (0.0, 10.0)\nu0 = [1.0, 1.0]\nprob = ODEProblem{true, SciMLBase.FullSpecialize}(f, u0, tspan, p)\n\ntest_sol = solve(prob, Vern7(), abstol=1/10^14, reltol=1/10^14)\nplot(test_sol, title=\"Lotka-Volterra Solution\", legend=false, xticks=:auto)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/#lv_ek0_comparison","page":"Lotka-Volterra","title":"EK0 Benchmark Across Orders","text":"","category":"section"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"DENSE = false;\nSAVE_EVERYSTEP = false;\n\n_setups = [\n \"EK0(order=$order)\" => Dict(:alg => EK0(order=order, smooth=DENSE))\n for order in 2:7\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nabstols = 1.0 ./ 10.0 .^ (4:12)\nreltols = 1.0 ./ 10.0 .^ (1:9)\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n)\n\nplot(wp, palette=Plots.palette([:blue, :red], length(_setups)))\n\n_ref_setups = [\n \"Tsit5\" => Dict(:alg => Tsit5())\n \"Vern7\" => Dict(:alg => Vern7())\n \"RadauIIA5\" => Dict(:alg => RadauIIA5())\n]\nref_labels = first.(_ref_setups)\nref_setups = last.(_ref_setups)\nref_wp_final = WorkPrecisionSet(\n prob, abstols, reltols, ref_setups;\n names = ref_labels,\n appxsol = test_sol,\n dense = false,\n save_everystep = false,\n maxiters = Int(1e7),\n)\nref_wp_dense = WorkPrecisionSet(\n prob, abstols, reltols, ref_setups;\n names = ref_labels,\n appxsol = test_sol,\n dense = true,\n save_everystep = true,\n maxiters = Int(1e7),\n)\n\nplot!(ref_wp_final, x=:final, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Discrete time-series errors (l2):","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"DENSE = true;\nSAVE_EVERYSTEP = true;\n\n_setups = [\n \"EK0(order=$order)\" => Dict(:alg => EK0(order=order, smooth=DENSE))\n for order in 2:7\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nabstols = 1.0 ./ 10.0 .^ (4:12)\nreltols = 1.0 ./ 10.0 .^ (1:9)\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n)\n\nplot(wp, x=:l2, palette=Plots.palette([:blue, :red], length(_setups)))\nplot!(ref_wp_dense, x=:l2, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Interoplation errors (L2):","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"plot(wp, x=:L2, palette=Plots.palette([:blue, :red], length(_setups)))\nplot!(ref_wp_dense, x=:L2, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/#lv_ek1_comparison","page":"Lotka-Volterra","title":"EK1 Benchmark Across Orders","text":"","category":"section"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"DENSE = false;\nSAVE_EVERYSTEP = false;\n\n_setups = [\n \"EK1(order=$order)\" => Dict(:alg => EK1(order=order, smooth=DENSE))\n for order in 2:7\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nabstols = 1.0 ./ 10.0 .^ (4:12)\nreltols = 1.0 ./ 10.0 .^ (1:9)\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n)\n\nplot(wp, palette=Plots.palette([:blue, :red], length(_setups)))\nplot!(ref_wp_final, x=:final, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Discrete time-series errors (l2):","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"DENSE = true;\nSAVE_EVERYSTEP = true;\n\n_setups = [\n \"EK1(order=$order)\" => Dict(:alg => EK1(order=order, smooth=DENSE))\n for order in 2:7\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nabstols = 1.0 ./ 10.0 .^ (4:12)\nreltols = 1.0 ./ 10.0 .^ (1:9)\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n)\n\nplot(wp, x=:l2, palette=Plots.palette([:blue, :red], length(_setups)))\nplot!(ref_wp_dense, x=:l2, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Interoplation errors (L2):","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"plot(wp, x=:L2, palette=Plots.palette([:blue, :red], length(_setups)))\nplot!(ref_wp_dense, x=:L2, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/#lv_ek0_vs_ek1","page":"Lotka-Volterra","title":"EK0 vs. EK1: Work-Precision","text":"","category":"section"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"DENSE = false;\nSAVE_EVERYSTEP = false;\n\n_setups = [\n \"EK0(order=2)\" => Dict(:alg => EK0(order=2, smooth=DENSE))\n \"EK0(order=3)\" => Dict(:alg => EK0(order=3, smooth=DENSE))\n \"EK0(order=4)\" => Dict(:alg => EK0(order=4, smooth=DENSE))\n \"EK0(order=5)\" => Dict(:alg => EK0(order=5, smooth=DENSE))\n \"EK1(order=2)\" => Dict(:alg => EK1(order=2, smooth=DENSE))\n \"EK1(order=3)\" => Dict(:alg => EK1(order=3, smooth=DENSE))\n \"EK1(order=4)\" => Dict(:alg => EK1(order=4, smooth=DENSE))\n \"EK1(order=5)\" => Dict(:alg => EK1(order=5, smooth=DENSE))\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nabstols = 1.0 ./ 10.0 .^ (4:12)\nreltols = 1.0 ./ 10.0 .^ (1:9)\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n)\n\nplot(wp, color=[1 1 1 1 2 2 2 2])\nplot!(ref_wp_final, x=:final, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Interoplation errors (L2):","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"DENSE = true;\nSAVE_EVERYSTEP = true;\n\n_setups = [\n \"EK0(order=2)\" => Dict(:alg => EK0(order=2, smooth=DENSE))\n \"EK0(order=3)\" => Dict(:alg => EK0(order=3, smooth=DENSE))\n \"EK0(order=4)\" => Dict(:alg => EK0(order=4, smooth=DENSE))\n \"EK0(order=5)\" => Dict(:alg => EK0(order=5, smooth=DENSE))\n \"EK1(order=2)\" => Dict(:alg => EK1(order=2, smooth=DENSE))\n \"EK1(order=3)\" => Dict(:alg => EK1(order=3, smooth=DENSE))\n \"EK1(order=4)\" => Dict(:alg => EK1(order=4, smooth=DENSE))\n \"EK1(order=5)\" => Dict(:alg => EK1(order=5, smooth=DENSE))\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nabstols = 1.0 ./ 10.0 .^ (4:12)\nreltols = 1.0 ./ 10.0 .^ (1:9)\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n)\n\nplot(wp, x=:L2, color=[1 1 1 1 2 2 2 2])\nplot!(ref_wp_dense, x=:L2, color=:gray, alpha=0.7, linestyle=:dash)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/#[EK0](@ref)-vs.-[EK1](@ref):-Calibration","page":"Lotka-Volterra","title":"EK0 vs. EK1: Calibration","text":"","category":"section"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"plot(wp, x=:final, y=:chi2_final, color=[1 1 1 1 2 2 2 2], yguide=\"Chi-squared (final)\")\nplot_chisq_interval!(2)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/#lv_diffusion","page":"Lotka-Volterra","title":"Diffusion model comparison","text":"","category":"section"},{"location":"benchmarks/lotkavolterra/#[EK0](@ref)-with-different-diffusions","page":"Lotka-Volterra","title":"EK0 with different diffusions","text":"","category":"section"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"DENSE = false;\nSAVE_EVERYSTEP = false;\n\n_setups = [\n \"EK0(3) Dynamic\" => Dict(:alg => EK0(order=3, smooth=DENSE, diffusionmodel=DynamicDiffusion()))\n \"EK0(5) Dynamic\" => Dict(:alg => EK0(order=5, smooth=DENSE, diffusionmodel=DynamicDiffusion()))\n \"EK0(3) Fixed\" => Dict(:alg => EK0(order=3, smooth=DENSE, diffusionmodel=FixedDiffusion()))\n \"EK0(5) Fixed\" => Dict(:alg => EK0(order=5, smooth=DENSE, diffusionmodel=FixedDiffusion()))\n \"EK0(3) DynamicMV\" => Dict(:alg => EK0(order=3, smooth=DENSE, diffusionmodel=DynamicMVDiffusion()))\n \"EK0(5) DynamicMV\" => Dict(:alg => EK0(order=5, smooth=DENSE, diffusionmodel=DynamicMVDiffusion()))\n \"EK0(3) FixedMV\" => Dict(:alg => EK0(order=3, smooth=DENSE, diffusionmodel=FixedMVDiffusion()))\n \"EK0(5) FixedMV\" => Dict(:alg => EK0(order=5, smooth=DENSE, diffusionmodel=FixedMVDiffusion()))\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nabstols = 1.0 ./ 10.0 .^ (4:12)\nreltols = 1.0 ./ 10.0 .^ (1:9)\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n)\n\ncolor = [2 2 3 3 4 4 5 5]\nlinestyle = [:solid :dash :solid :dash :solid :dash :solid :dash]\nplot(wp; color, linestyle)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"plot(wp; x=:final, y=:chi2_final, color, linestyle, yguide=\"Chi-squared (final)\")\nplot_chisq_interval!(2)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/#[EK1](@ref)-with-different-diffusions","page":"Lotka-Volterra","title":"EK1 with different diffusions","text":"","category":"section"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"DENSE = false;\nSAVE_EVERYSTEP = false;\n\n_setups = [\n \"EK1(3) Dynamic\" => Dict(:alg => EK1(order=3, smooth=DENSE, diffusionmodel=DynamicDiffusion()))\n \"EK1(5) Dynamic\" => Dict(:alg => EK1(order=5, smooth=DENSE, diffusionmodel=DynamicDiffusion()))\n \"EK1(3) Fixed\" => Dict(:alg => EK1(order=3, smooth=DENSE, diffusionmodel=FixedDiffusion()))\n \"EK1(5) Fixed\" => Dict(:alg => EK1(order=5, smooth=DENSE, diffusionmodel=FixedDiffusion()))\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nabstols = 1.0 ./ 10.0 .^ (4:12)\nreltols = 1.0 ./ 10.0 .^ (1:9)\n\nwp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n)\n\ncolor = [2 2 3 3]\nlinestyle = [:solid :dash :solid :dash]\nplot(wp; color, linestyle)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"plot(wp; x=:final, y=:chi2_final, color, linestyle, yguide=\"Chi-squared (final)\")\nplot_chisq_interval!(2)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/#lv_initialization","page":"Lotka-Volterra","title":"Initialization scheme comparison","text":"","category":"section"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Code:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"DENSE = false;\nSAVE_EVERYSTEP = false;\n\nabstols = 1.0 ./ 10.0 .^ (4:12)\nreltols = 1.0 ./ 10.0 .^ (1:9)\n\norders = (2, 3, 5, 8)\nps = []\nfor o in orders\n _setups = [\n \"EK1($o) TaylorInit\" => Dict(:alg => EK1(order=o, smooth=DENSE, initialization=TaylorModeInit(o)))\n \"EK1($o) ForwardDiffInit\" => Dict(:alg => EK1(order=o, smooth=DENSE, initialization=ForwardDiffInit(o)))\n \"EK1($o) SimpleInit\" => Dict(:alg => EK1(order=o, smooth=DENSE, initialization=SimpleInit()))\n \"EK1($o) ClassicSolverInit\" => Dict(:alg => EK1(order=o, smooth=DENSE, initialization=ClassicSolverInit()))\n ]\n\n labels = first.(_setups)\n setups = last.(_setups)\n\n wp = WorkPrecisionSet(\n prob, abstols, reltols, setups;\n names = labels,\n appxsol = test_sol,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n maxiters = Int(1e7),\n )\n\n p = plot(wp, color=[2 4 5 6], xticks = 10.0 .^ (-16:1:5), title = \"Order $o\")\n push!(ps, p)\nend\nplot(\n ps...,\n layout=(length(orders), 1),\n size = (800, length(orders)*300),\n xlabel=[\"\" \"\" \"\" \"Error (final)\"],\n)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"(Image: )","category":"page"},{"location":"benchmarks/lotkavolterra/#Appendix","page":"Lotka-Volterra","title":"Appendix","text":"","category":"section"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Computer information:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"using InteractiveUtils\nInteractiveUtils.versioninfo()","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"Julia Version 1.10.0\nCommit 3120989f39b (2023-12-25 18:01 UTC)\nBuild Info:\n Official https://julialang.org/ release\nPlatform Info:\n OS: Linux (x86_64-linux-gnu)\n CPU: 12 × Intel(R) Core(TM) i7-6800K CPU @ 3.40GHz\n WORD_SIZE: 64\n LIBM: libopenlibm\n LLVM: libLLVM-15.0.7 (ORCJIT, broadwell)\n Threads: 17 on 12 virtual cores\nEnvironment:\n JULIA_NUM_THREADS = auto","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Package information:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"using Pkg\nPkg.status()","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Project.toml`\n [f3b72e0c] DiffEqDevTools v2.44.1\n [31c24e10] Distributions v0.25.107\n [7073ff75] IJulia v1.24.2\n [7f56f5a3] LSODA v0.7.5\n [e6f89c97] LoggingExtras v1.0.3\n [e2752cbe] MATLABDiffEq v1.2.0\n⌃ [961ee093] ModelingToolkit v8.73.2\n [54ca160b] ODEInterface v0.5.0\n [09606e27] ODEInterfaceDiffEq v3.13.3\n⌃ [1dea7af3] OrdinaryDiffEq v6.66.0\n [65888b18] ParameterizedFunctions v5.16.0\n [91a5bcdd] Plots v1.39.0\n [bf3e78b0] ProbNumDiffEq v0.14.0 `~/.julia/dev/ProbNumDiffEq`\n⌃ [0bca4576] SciMLBase v2.10.0\n [505e40e9] SciPyDiffEq v0.2.1\n [ce78b400] SimpleUnPack v1.1.0\n [90137ffa] StaticArrays v1.9.1\n [c3572dad] Sundials v4.23.1\n [44d3d7a6] Weave v0.10.12\n [0518478a] deSolveDiffEq v0.1.1\nInfo Packages marked with ⌃ have new versions available and may be upgradable.","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
Full manifest:","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"Pkg.status(mode=Pkg.PKGMODE_MANIFEST)","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Manifest.toml`\n [47edcb42] ADTypes v0.2.6\n⌅ [c3fe647b] AbstractAlgebra v0.34.7\n [621f4979] AbstractFFTs v1.5.0\n [1520ce14] AbstractTrees v0.4.4\n [7d9f7c33] Accessors v0.1.35\n⌅ [79e6a3ab] Adapt v3.7.2\n [ec485272] ArnoldiMethod v0.2.0\n [c9d4266f] ArrayAllocators v0.3.0\n [4fba245c] ArrayInterface v7.7.0\n [4c555306] ArrayLayouts v1.5.2\n [6e4b80f9] BenchmarkTools v1.4.0\n [e2ed5e7c] Bijections v0.1.6\n [d1d4a3ce] BitFlags v0.1.8\n [62783981] BitTwiddlingConvenienceFunctions v0.1.5\n [fa961155] CEnum v0.5.0\n [2a0fbf3d] CPUSummary v0.2.4\n [00ebfdb7] CSTParser v3.4.0\n [49dc2e85] Calculus v0.5.1\n [324d7699] CategoricalArrays v0.10.8\n [d360d2e6] 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Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`","category":"page"},{"location":"benchmarks/lotkavolterra/","page":"Lotka-Volterra","title":"Lotka-Volterra","text":"
","category":"page"},{"location":"initialization/#Initialization","page":"Initialization","title":"Initialization","text":"","category":"section"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"The notion of \"initialization\" relates to the prior part of the model.","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"TL;DR: In the vast majority of cases, just stick to the exact Taylor-mode initialization TaylorModeInit!","category":"page"},{"location":"initialization/#Background:-The-prior","page":"Initialization","title":"Background: The prior","text":"","category":"section"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"We model the ODE solution y(t) with a Gauss–Markov prior. More precisely, let","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"beginaligned\nY(t) = left Y^(0)(t) Y^(1)(t) dots Y^(q)(t) right\nendaligned","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"be the solution to the SDE","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"beginaligned\ntextd Y^(i)(t) = Y^(i+1)(t) textdt qquad i = 0 dots q-1 \ntextd Y^(q)(t) = textcolor389826A Y(t) textdt + textcolor4063D8Gamma textdW(t) \nY(0) sim textcolor9558B2 mathcalN left( mu_0 Sigma_0 right) \nendaligned","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"Then Y^(i)(t) models the i-th derivative of y(t). In this section, we consider the initial distribution textcolorpurple mathcalN left( mu_0 Sigma_0 right) . If you're more interested in the drift matrix textcolor389826A check out the Priors section, and for more info on the diffusion textcolor4063D8Gamma check out the Diffusion models and calibration section.","category":"page"},{"location":"initialization/#Setting-the-initial-distribution","page":"Initialization","title":"Setting the initial distribution","text":"","category":"section"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"Let's assume an initial value problem of the form","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"beginaligned\ndoty(t) = f(y(t) t) qquad 0 T \ny(0) = y_0\nendaligned","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"It is clear that this contains quite some information for Y(0): The initial value y_0 and the vector field f imply","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"beginaligned\nY^(0)(0) = y_0 \nY^(1)(0) = f(y_0 0)\nendaligned","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"It turns out that we can also compute higher-order derivatives of y with the chain rule, and then use these to better initialize Y^(i)(0). This, done efficiently with Taylor-mode automatic differentiation by using TaylorIntegration.jl, is what TaylorModeInit does. See also [1].","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"In the vast majority of cases, just stick to the exact Taylor-mode initialization TaylorModeInit!","category":"page"},{"location":"initialization/#API","page":"Initialization","title":"API","text":"","category":"section"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"TaylorModeInit\nForwardDiffInit\nSimpleInit\nClassicSolverInit","category":"page"},{"location":"initialization/#ProbNumDiffEq.TaylorModeInit","page":"Initialization","title":"ProbNumDiffEq.TaylorModeInit","text":"TaylorModeInit(order)\n\nExact initialization via Taylor-mode automatic differentiation up to order order.\n\nThis is the recommended initialization method!\n\nIt uses TaylorIntegration.jl to efficiently compute the higher-order derivatives of the solution at the initial value, via Taylor-mode automatic differentiation.\n\nIn some special cases it can happen that TaylorIntegration.jl is incompatible with the given problem (typically because the problem definition does not allow for elements of type Taylor). If this happens, try one of SimpleInit, ForwardDiffInit (for low enough orders), ClassicSolverInit.\n\nReferences\n\n[7] Krämer et al, \"Stable Implementation of Probabilistic ODE Solvers\" (2020)\n\n\n\n\n\n","category":"type"},{"location":"initialization/#ProbNumDiffEq.ForwardDiffInit","page":"Initialization","title":"ProbNumDiffEq.ForwardDiffInit","text":"ForwardDiffInit(order)\n\nExact initialization via ForwardDiff.jl up to order order.\n\nWarning: This does not scale well to high orders! For orders > 3, TaylorModeInit most likely performs better.\n\n\n\n\n\n","category":"type"},{"location":"initialization/#ProbNumDiffEq.SimpleInit","page":"Initialization","title":"ProbNumDiffEq.SimpleInit","text":"SimpleInit()\n\nSimple initialization, only with the given initial value and derivative.\n\nThe remaining derivatives are set to zero with unit covariance (unless specified otherwise by setting a custom FixedDiffusion).\n\n\n\n\n\n","category":"type"},{"location":"initialization/#ProbNumDiffEq.ClassicSolverInit","page":"Initialization","title":"ProbNumDiffEq.ClassicSolverInit","text":"ClassicSolverInit(; alg=OrdinaryDiffEqCore.Tsit5(), init_on_ddu=false)\n\nInitialization via regression on a few steps of a classic ODE solver.\n\nIn a nutshell, instead of specifying mu_0 exactly and setting Sigma_0=0 (which is what TaylorModeInit does), use a classic ODE solver to compute a few steps of the solution, and then regress on the computed values (by running a smoother) to compute mu_0 and Sigma_0 as the mean and covariance of the smoothing posterior at time 0. See also [2].\n\nThe initial value and derivative are set directly from the given initial value problem; optionally the second derivative can also be set via automatic differentiation by setting init_on_ddu=true.\n\nArguments\n\nalg: The solver to be used. Can be any solver from OrdinaryDiffEqCore.jl.\ninit_on_ddu: If true, the second derivative is also initialized exactly via automatic differentiation with ForwardDiff.jl.\n\nReferences\n\n[7] Krämer et al, \"Stable Implementation of Probabilistic ODE Solvers\" (2020)\n[8] Schober et al, \"A probabilistic model for the numerical solution of initial value problems\", Statistics and Computing (2019)\n\n\n\n\n\n","category":"type"},{"location":"initialization/#initrefs","page":"Initialization","title":"References","text":"","category":"section"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"N. Krämer and P. Hennig. Stable Implementation of Probabilistic ODE Solvers. CoRR (2020), arXiv:2012.10106 [stat.ML].\n\n\n\nM. Schober, S. Särkkä and P. Hennig. A probabilistic model for the numerical solution of initial value problems. Statistics and Computing 29, 99–122 (2019).\n\n\n\n","category":"page"},{"location":"tutorials/dae/#Solving-DAEs-with-Probabilistic-Numerics","page":"Differential Algebraic Equations","title":"Solving DAEs with Probabilistic Numerics","text":"","category":"section"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"ProbNumDiffEq.jl provides probabilistic numerical solvers for differential algebraic equations (DAEs). Currently, we recommend using the semi-implicit EK1 algorithm.","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"note: Note\nFor a more general tutorial on DAEs check out the DifferentialEquations.jl DAE tutorial.","category":"page"},{"location":"tutorials/dae/#Solving-mass-matrix-DAEs-with-the-EK1","page":"Differential Algebraic Equations","title":"Solving mass-matrix DAEs with the EK1","text":"","category":"section"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"First, define the DAE (here the ROBER problem) as an ODE problem with singular mass matrix:","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"using ProbNumDiffEq, Plots, LinearAlgebra, OrdinaryDiffEq, ModelingToolkit, LinearAlgebra\n\nfunction rober(du, u, p, t)\n y₁, y₂, y₃ = u\n k₁, k₂, k₃ = p\n du[1] = -k₁ * y₁ + k₃ * y₂ * y₃\n du[2] = k₁ * y₁ - k₃ * y₂ * y₃ - k₂ * y₂^2\n du[3] = y₁ + y₂ + y₃ - 1\n nothing\nend\nM = [1 0 0\n 0 1 0\n 0 0 0]\nf = ODEFunction(rober, mass_matrix=M)\nprob_mm = ODEProblem(f, [1.0, 0.0, 0.0], (0.0, 1e5), (0.04, 3e7, 1e4))","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"We can solve this problem directly with the EK1:","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"sol = solve(prob_mm, EK1(), reltol=1e-8, abstol=1e-8)\nplot(\n sol,\n xscale=:log10,\n tspan=(1e-6, 1e5),\n layout=(3, 1),\n legend=false,\n ylabel=[\"u₁(t)\" \"u₂(t)\" \"u₃(t)\"],\n xlabel=[\"\" \"\" \"t\"],\n denseplot=false,\n)","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"Looks good!","category":"page"},{"location":"tutorials/dae/#Solving-an-Index-3-DAE-directly","page":"Differential Algebraic Equations","title":"Solving an Index-3 DAE directly","text":"","category":"section"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"The following is based on the \"Automatic Index Reduction of DAEs\" tutorial by ModelingToolkit.jl, which demonstrates how the classic Rodas4 solver fails to solve a DAE due to the fact that it is of index 3; which is why ModelingToolkit's automatic index reduction is so useful.","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"It turns out that our probabilistic numerical solvers can directly solve the index-3 DAE!","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"First, define the pendulum problem as in the tutorial:","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"function pendulum!(du, u, p, t)\n x, dx, y, dy, T = u\n g, L = p\n du[1] = dx\n du[2] = T * x\n du[3] = dy\n du[4] = T * y - g\n du[5] = x^2 + y^2 - L^2\nend\npendulum_fun! = ODEFunction(pendulum!, mass_matrix=Diagonal([1, 1, 1, 1, 0]))\nu0 = [1.0, 0, 0, 0, 0];\np = [9.8, 1];\ntspan = (0, 5.0);\npendulum_prob = ODEProblem(pendulum_fun!, u0, tspan, p)","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"We can try to solve it directly with one of the classic mass-matrix DAE solvers from OrdinaryDiffEq.jl:","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"solve(pendulum_prob, Rodas4())","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"It does not work! This is because of the index of the DAE; see for example this explanation from the tutorial.","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"Does this also hold for the EK1 solver? Let's find out:","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"sol = solve(pendulum_prob, EK1())","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"Nope! The EK1 is able to solve the index-3 DAE directly. Pretty cool!","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"plot(sol)","category":"page"},{"location":"tutorials/dae/#Is-index-reduction-still-worth-it?","page":"Differential Algebraic Equations","title":"Is index-reduction still worth it?","text":"","category":"section"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"The point of the \"Automatic Index Reduction of DAEs\" tutorial is to demonstrate ModelingToolkit's utility for automatic index reduction, which enables the classic implicit Runge-Kutta solvers such as Rodas5 to solve this DAE. Let's see if that still helps in this context here.","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"First, modelingtoolkitize the problem:","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"traced_sys = modelingtoolkitize(pendulum_prob)","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"(how cool is this latex output ?!?)","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"Next, lower the DAE index and simplify it with ModelingToolkit's dae_index_lowering and structural_simplify:","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"simplified_sys = structural_simplify(dae_index_lowering(traced_sys))","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"Let's build two different ODE problems, and check how well we can solve each:","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"prob_index3 = ODEProblem(complete(traced_sys), Pair[], tspan)\nprob_index1 = ODEProblem(simplified_sys, Pair[], tspan)\n\nsol3 = solve(prob_index3, EK1())\nsol1 = solve(prob_index1, EK1())\n\ntruesol = solve(prob_index1, Rodas4(), abstol=1e-10, reltol=1e-10)\n\nsol1_final_error = norm(sol1.u[end] - truesol.u[end])\nsol1_f_evals = sol1.stats.nf\nsol3_final_error = norm(sol3.u[end] - truesol.u[end])\nsol3_f_evals = sol3.stats.nf\n@info \"Results\" sol1_final_error sol1_f_evals sol3_final_error sol3_f_evals","category":"page"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"The error for the index-1 DAE solve is much lower. Thus it seems that, even if the index-3 DAE could also be solved directly, index lowering might still be beneficial when solving DAEs with the EK1!","category":"page"},{"location":"tutorials/dae/#References","page":"Differential Algebraic Equations","title":"References","text":"","category":"section"},{"location":"tutorials/dae/","page":"Differential Algebraic Equations","title":"Differential Algebraic Equations","text":"N. Bosch, F. Tronarp and P. Hennig. Pick-and-Mix Information Operators for Probabilistic ODE Solvers. In: Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, Vol. 151 of Proceedings of Machine Learning Research, edited by G. Camps-Valls, F. J. Ruiz and I. Valera (PMLR, 28–30 Mar 2022); pp. 10015–10027.\n\n\n\n","category":"page"},{"location":"diffusions/#Diffusion-models-and-calibration","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":"","category":"section"},{"location":"diffusions/","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":"The notion of \"diffusion\" and \"calibration\" relates to the prior part of the model.","category":"page"},{"location":"diffusions/#Background:-The-prior","page":"Diffusion models and calibration","title":"Background: The prior","text":"","category":"section"},{"location":"diffusions/","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":"We model the ODE solution y(t) with a Gauss–Markov prior. More precisely, let","category":"page"},{"location":"diffusions/","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":"beginaligned\nY(t) = left Y^(0)(t) Y^(1)(t) dots Y^(q)(t) right\nendaligned","category":"page"},{"location":"diffusions/","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":"be the solution to the SDE","category":"page"},{"location":"diffusions/","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":"beginaligned\ntextd Y^(i)(t) = Y^(i+1)(t) textdt qquad i = 0 dots q-1 \ntextd Y^(q)(t) = textcolor389826A Y(t) textdt + textcolor4063D8Gamma textdW(t) \nY(0) sim textcolorpurple mathcalN left( mu_0 Sigma_0 right) \nendaligned","category":"page"},{"location":"diffusions/","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":"Then Y^(i)(t) models the i-th derivative of y(t). In this section, we consider choices relating to the \"diffusion\" textcolor4063D8Gamma. If you're more interested in the drift matrix textcolor389826A check out the Priors section, and for info on the initial distribution textcolorpurple mathcalN left( mu_0 Sigma_0 right) check out the Initialization section.","category":"page"},{"location":"diffusions/#Diffusion-and-calibration","page":"Diffusion models and calibration","title":"Diffusion and calibration","text":"","category":"section"},{"location":"diffusions/","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":"We call textcolor4063D8Gamma the \"diffusion\" parameter. Since it is typically not known we need to estimate it; this is called \"calibration\".","category":"page"},{"location":"diffusions/","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":"ProbNumDiffEq.jl provides a few different choices for how to model and estimate textcolor4063D8Gamma:","category":"page"},{"location":"diffusions/","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":"FixedDiffusion assumes an isotropic, time-fixed textcolor4063D8Gamma = sigma cdot I_d,\nDynamicDiffusion assumes an isotropic, time-varying textcolor4063D8Gamma(t) = sigma(t) cdot I_d (recommended),\nFixedMVDiffusion assumes a diagonal, time-fixed textcolor4063D8Gamma = operatornamediag(sigma_1 dots sigma_d),\nDynamicMVDiffusion assumes a diagonal, time-varying textcolor4063D8Gamma(t) = operatornamediag(sigma_1(t) dots sigma_d(t)).","category":"page"},{"location":"diffusions/","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":"Or more compactly:","category":"page"},{"location":"diffusions/","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":" Isotropic: Diagonal (only for the EK0)\nTime-varying DynamicDiffusion DynamicMVDiffusion\nTime-fixed FixedDiffusion FixedMVDiffusion","category":"page"},{"location":"diffusions/","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":"For more details on diffusions and calibration, check out this paper [9].","category":"page"},{"location":"diffusions/#API","page":"Diffusion models and calibration","title":"API","text":"","category":"section"},{"location":"diffusions/","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":"DynamicDiffusion\nFixedDiffusion\nDynamicMVDiffusion\nFixedMVDiffusion","category":"page"},{"location":"diffusions/#ProbNumDiffEq.DynamicDiffusion","page":"Diffusion models and calibration","title":"ProbNumDiffEq.DynamicDiffusion","text":"DynamicDiffusion()\n\nTime-varying, isotropic diffusion, which is quasi-maximum-likelihood-estimated at each step.\n\nThis is the recommended diffusion when using adaptive step-size selection, and in particular also when solving stiff systems.\n\n\n\n\n\n","category":"type"},{"location":"diffusions/#ProbNumDiffEq.FixedDiffusion","page":"Diffusion models and calibration","title":"ProbNumDiffEq.FixedDiffusion","text":"FixedDiffusion(; initial_diffusion=1.0, calibrate=true)\n\nTime-fixed, isotropic diffusion, which is (optionally) quasi-maximum-likelihood-estimated.\n\nThis is the recommended diffusion when using fixed steps.\n\nBy default with calibrate=true, all covariances are re-scaled at the end of the solve with the MLE diffusion. Set calibrate=false to skip this step, e.g. when setting the initial_diffusion and then estimating the diffusion outside of the solver (e.g. with Fenrir.jl).\n\n\n\n\n\n","category":"type"},{"location":"diffusions/#ProbNumDiffEq.DynamicMVDiffusion","page":"Diffusion models and calibration","title":"ProbNumDiffEq.DynamicMVDiffusion","text":"DynamicMVDiffusion()\n\nTime-varying, diagonal diffusion, which is quasi-maximum-likelihood-estimated at each step.\n\nOnly works with the EK0!\n\nA multi-variate version of DynamicDiffusion, where instead of an isotropic matrix, a diagonal matrix is estimated. This can be helpful to get more expressive posterior covariances when using the EK0, since the individual dimensions can be adjusted separately.\n\nReferences\n\n[9] Bosch et al, \"Calibrated Adaptive Probabilistic ODE Solvers\", AISTATS (2021)\n\n\n\n\n\n","category":"type"},{"location":"diffusions/#ProbNumDiffEq.FixedMVDiffusion","page":"Diffusion models and calibration","title":"ProbNumDiffEq.FixedMVDiffusion","text":"FixedMVDiffusion(; initial_diffusion=1.0, calibrate=true)\n\nTime-fixed, diagonal diffusion, which is quasi-maximum-likelihood-estimated at each step.\n\nOnly works with the EK0!\n\nA multi-variate version of FixedDiffusion, where instead of an isotropic matrix, a diagonal matrix is estimated. This can be helpful to get more expressive posterior covariances when using the EK0, since the individual dimensions can be adjusted separately.\n\nReferences\n\n[9] Bosch et al, \"Calibrated Adaptive Probabilistic ODE Solvers\", AISTATS (2021)\n\n\n\n\n\n","category":"type"},{"location":"diffusions/#diffusionrefs","page":"Diffusion models and calibration","title":"References","text":"","category":"section"},{"location":"diffusions/","page":"Diffusion models and calibration","title":"Diffusion models and calibration","text":"N. Bosch, P. Hennig and F. Tronarp. Calibrated Adaptive Probabilistic ODE Solvers. In: Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, Vol. 130 of Proceedings of Machine Learning Research, edited by A. Banerjee and K. Fukumizu (PMLR, 13–15 Apr 2021); pp. 3466–3474.\n\n\n\n","category":"page"},{"location":"priors/#Priors","page":"Priors","title":"Priors","text":"","category":"section"},{"location":"priors/","page":"Priors","title":"Priors","text":"TL;DR: If you're unsure which prior to use, just stick to the default integrated Wiener process prior IWP!","category":"page"},{"location":"priors/#Background","page":"Priors","title":"Background","text":"","category":"section"},{"location":"priors/","page":"Priors","title":"Priors","text":"We model the ODE solution y(t) with a Gauss–Markov prior. More precisely, let","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"beginaligned\nY(t) = left Y^(0)(t) Y^(1)(t) dots Y^(q)(t) right\nendaligned","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"be the solution to the SDE","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"beginaligned\ntextd Y^(i)(t) = Y^(i+1)(t) textdt qquad i = 0 dots q-1 \ntextd Y^(q)(t) = textcolor389826A Y(t) textdt + textcolor4063D8Gamma textdW(t) \nY(0) sim textcolorpurple mathcalN left( mu_0 Sigma_0 right) \nendaligned","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"Then Y^(i)(t) models the i-th derivative of y(t). In this section, we consider choices relating to the drift matrix textcolor389826A. If you're more interested in the diffusion textcolor4063D8Gamma check out the Diffusion models and calibration section, and for info on the initial distribution textcolorpurple mathcalN left( mu_0 Sigma_0 right) check out the Initialization section.","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"info: Info\nIf you're unsure which prior to use, just stick to the integrated Wiener process prior IWP! This is also the default choice for all solvers. The other priors are rather experimental / niche at the time of writing.","category":"page"},{"location":"priors/#Prior-choices","page":"Priors","title":"Prior choices","text":"","category":"section"},{"location":"priors/","page":"Priors","title":"Priors","text":"ProbNumDiffEq.jl currently supports three classes of priors for the solvers:","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"IWP: q-times integrated Wiener processes\nIOUP: q-times integrated Ornstein–Uhlenbeck processes\nMatern: Matérn processes","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"Let's look at each of them in turn and visualize some examples.","category":"page"},{"location":"priors/#Integrated-Wiener-process-([IWP](@ref))","page":"Priors","title":"Integrated Wiener process (IWP)","text":"","category":"section"},{"location":"priors/","page":"Priors","title":"Priors","text":"IWP","category":"page"},{"location":"priors/#ProbNumDiffEq.IWP","page":"Priors","title":"ProbNumDiffEq.IWP","text":"IWP([dim::Integer=1,] num_derivatives::Integer)\n\nq-times integrated Wiener process.\n\nThis is the recommended prior! The IWP is the most common prior choice in the probabilistic ODE solver literature (see the references) and is the default choice for the solvers in ProbNumDiffEq.jl. It is also the prior that has the most efficient implementation.\n\nThe IWP can be created without specifying the dimension of the Wiener process, in which case it will be one-dimensional. The ODE solver then assumes that each dimension of the ODE solution should be modeled with the same prior. This is typically the preferred usage.\n\nIn math\n\nThe IWP is a Gauss–Markov process, which we model with a state representation\n\nbeginaligned\nY(t) = left Y^(0)(t) Y^(1)(t) dots Y^(q)(t) right\nendaligned\n\ndefined as the solution of the stochastic differential equation\n\nbeginaligned\ntextd Y^(i)(t) = Y^(i+1)(t) textdt qquad i = 0 dots q-1 \ntextd Y^(q)(t) = Gamma textdW(t)\nendaligned\n\nThen, Y^(0)(t) is the q-times integrated Wiener process (IWP) and Y^(i)(t) is the i-th derivative of the IWP, for i = 1 dots q.\n\nExamples\n\njulia> solve(prob, EK1(prior=IWP(2)))\n\n\n\n\n\n","category":"type"},{"location":"priors/","page":"Priors","title":"Priors","text":"Here is how the IWP looks for varying smoothness parameters q:","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"using ProbNumDiffEq, Plots\nplotrange = range(0, 10, length=250)\nplot(\n plot(IWP(1), plotrange; title=\"q=1\"),\n plot(IWP(2), plotrange; title=\"q=2\"),\n plot(IWP(3), plotrange; title=\"q=3\"),\n plot(IWP(4), plotrange; title=\"q=4\");\n ylims=(-20,20),\n)","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"In the context of ODE solvers, the smoothness parameter q influences the convergence rate of the solver, and so it is typically chose similarly to the order of a Runge–Kutta method: lower order for low accuracy, higher order for high accuracy.","category":"page"},{"location":"priors/#Integrated-Ornstein–Uhlenbeck-process-([IOUP](@ref))","page":"Priors","title":"Integrated Ornstein–Uhlenbeck process (IOUP)","text":"","category":"section"},{"location":"priors/","page":"Priors","title":"Priors","text":"The q-times integrated Ornstein–Uhlenbeck process prior IOUP is a generalization of the IWP prior, where the drift matrix textcolor389826A is not zero:","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"IOUP","category":"page"},{"location":"priors/#ProbNumDiffEq.IOUP","page":"Priors","title":"ProbNumDiffEq.IOUP","text":"IOUP([dim::Integer=1,]\n num_derivatives::Integer,\n rate_parameter::Union{Number,AbstractVector,AbstractMatrix})\n\nIntegrated Ornstein–Uhlenbeck process.\n\nThis prior is mostly used in the context of Probabilistic Exponential Integrators to include the linear part of a semi-linear ODE in the prior, so it is used in the ExpEK and the RosenbrockExpEK.\n\nIn math\n\nThe IOUP is a Gauss–Markov process, which we model with a state representation\n\nbeginaligned\nY(t) = left Y^(0)(t) Y^(1)(t) dots Y^(q)(t) right\nendaligned\n\ndefined as the solution of the stochastic differential equation\n\nbeginaligned\ntextd Y^(i)(t) = Y^(i+1)(t) textdt qquad i = 0 dots q-1 \ntextd Y^(q)(t) = L Y^(q)(t) textdt + Gamma textdW(t)\nendaligned\n\nwhere L is called the rate_parameter. Then, Y^(0)(t) is the q-times integrated Ornstein–Uhlenbeck process (IOUP) and Y^(i)(t) is the i-th derivative of the IOUP, for i = 1 dots q.\n\nExamples\n\njulia> solve(prob, EK1(prior=IOUP(2, -1)))\n\n\n\n\n\n","category":"type"},{"location":"priors/","page":"Priors","title":"Priors","text":"Here is how the IOUP looks for varying rate parameters:","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"using ProbNumDiffEq, Plots\nplotrange = range(0, 10, length=250)\nplot(\n plot(IOUP(1, -1), plotrange; title=\"q=1,L=-1\", ylims=(-20,20)),\n plot(IOUP(1, 1), plotrange; title=\"q=1,L=1\", ylims=(-20,20)),\n plot(IOUP(4, -1), plotrange; title=\"q=4,L=-1\", ylims=(-50,50)),\n plot(IOUP(4, 1), plotrange; title=\"q=4,L=1\", ylims=(-50,50));\n)","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"In the context of Probabilistic Exponential Integrators, the rate parameter is often chosen according to the given ODE. Here is an example for a damped oscillator:","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"plot(IOUP(dim=2, num_derivatives=1, rate_parameter=[-0.2 -2π; 2π -0.2]),\n plotrange; plot_title=\"damped oscillator prior\")","category":"page"},{"location":"priors/#Matérn-process-([Matern](@ref))","page":"Priors","title":"Matérn process (Matern)","text":"","category":"section"},{"location":"priors/","page":"Priors","title":"Priors","text":"Matern","category":"page"},{"location":"priors/#ProbNumDiffEq.Matern","page":"Priors","title":"ProbNumDiffEq.Matern","text":"Matern([dim::Integer=1,]\n num_derivatives::Integer,\n lengthscale::Number)\n\nMatern process.\n\nThe class of Matern processes is well-known in the Gaussian process literature, and they also have a corresponding SDE representation similarly to the IWP and the IOUP. See also [6] for more details.\n\nIn math\n\nA Matern process is a Gauss–Markov process, which we model with a state representation\n\nbeginaligned\nY(t) = left Y^(0)(t) Y^(1)(t) dots Y^(q)(t) right\nendaligned\n\ndefined as the solution of the stochastic differential equation\n\nbeginaligned\ntextd Y^(i)(t) = Y^(i+1)(t) textdt qquad i = 0 dots q-1 \ntextd Y^(q)(t) = - sum_j=0^q left(\n beginpmatrix q+1 j endpmatrix\n left( fracsqrt2q - 1l right)^q-j\n Y^(j)(t) right) textdt + Gamma textdW(t)\nendaligned\n\nwhere l is called the lengthscale parameter. Then, Y^(0)(t) is a Matern process and Y^(i)(t) is the i-th derivative of this process, for i = 1 dots q.\n\nExamples\n\njulia> solve(prob, EK1(prior=Matern(2, 1)))\n\n\n\n\n\n","category":"type"},{"location":"priors/","page":"Priors","title":"Priors","text":"Here is how the Matern looks for varying smoothness parameters q:","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"using ProbNumDiffEq, Plots\nplotrange = range(0, 10, length=250)\nplot(\n plot(Matern(1, 1), plotrange; title=\"q=1\"),\n plot(Matern(2, 1), plotrange; title=\"q=2\"),\n plot(Matern(3, 1), plotrange; title=\"q=3\"),\n plot(Matern(4, 1), plotrange; title=\"q=4\");\n)","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"and for varying length scales ell:","category":"page"},{"location":"priors/","page":"Priors","title":"Priors","text":"plot(\n plot(Matern(2, 5), plotrange; title=\"l=5\"),\n plot(Matern(2, 2), plotrange; title=\"l=2\"),\n plot(Matern(2, 1), plotrange; title=\"l=1\"),\n plot(Matern(2, 0.5), plotrange; title=\"l=0.5\"),\n)","category":"page"},{"location":"priors/#API","page":"Priors","title":"API","text":"","category":"section"},{"location":"priors/","page":"Priors","title":"Priors","text":"ProbNumDiffEq.AbstractGaussMarkovProcess\nProbNumDiffEq.LTISDE\nProbNumDiffEq.dim\nProbNumDiffEq.num_derivatives\nProbNumDiffEq.to_sde\nProbNumDiffEq.discretize\nProbNumDiffEq.initial_distribution","category":"page"},{"location":"priors/#ProbNumDiffEq.AbstractGaussMarkovProcess","page":"Priors","title":"ProbNumDiffEq.AbstractGaussMarkovProcess","text":"AbstractGaussMarkovProcess{elType}\n\nAbstract type for Gauss-Markov processes.\n\nGauss-Markov processes are solutions to linear time-invariant stochastic differential equations (SDEs). Here we assume SDEs of the form\n\nbeginaligned\ndX_t = F X_t dt + L dW_t \nX_0 = mathcalN left( X_0 mu_0 Sigma_0 right)\nendaligned\n\nwhere X_t is the state, W_t is a Wiener process, and F and L are matrices.\n\nCurrently, ProbNumDiffEq.jl makes many assumptions about the structure of the SDEs that it can solve. In particular, it assumes that the state vector X_t contains a range of dervatives, and that the Wiener process only enters the highest one. It also assumes a certain ordering of dimensions and derivatives. This is not a limitation of the underlying mathematics, but rather a limitation of the current implementation. In the future, we hope to remove these limitations.\n\nwarning: Warning\nWe currently strongly recommended to not implement your own Gauss-Markov process by subtyping this type! The interface is not yet stable, and the implementation is not yet sufficiently documented. Proceed at your own risk.\n\n\n\n\n\n","category":"type"},{"location":"priors/#ProbNumDiffEq.LTISDE","page":"Priors","title":"ProbNumDiffEq.LTISDE","text":"LTISDE(F::AbstractMatrix, L::AbstractMatrix)\n\nLinear time-invariant stochastic differential equation.\n\nA LTI-SDE is a stochastic differential equation of the form\n\ndX_t = F X_t dt + L dW_t\n\nwhere X_t is the state, W_t is a Wiener process, and F and L are matrices. This LTISDE object holds the matrices F and L. It also provides some functionality to discretize the SDE via a matrix-fraction decomposition. See: discretize(::LTISDE, ::Real).\n\n\n\n\n\n","category":"type"},{"location":"priors/#ProbNumDiffEq.dim","page":"Priors","title":"ProbNumDiffEq.dim","text":"dim(p::AbstractGaussMarkovProcess)\n\nReturn the dimension of the process.\n\nThis is not the dimension of the \"state\" that is used to efficiently model the prior process as a state-space model, but it is the dimension of the process itself that we aim to model.\n\nSee AbstractGaussMarkovProcess for more details on Gauss-Markov processes in ProbNumDiffEq.\n\n\n\n\n\n","category":"function"},{"location":"priors/#ProbNumDiffEq.num_derivatives","page":"Priors","title":"ProbNumDiffEq.num_derivatives","text":"num_derivatives(p::AbstractGaussMarkovProcess)\n\nReturn the number of derivatives that are represented by the processes state.\n\nSee AbstractGaussMarkovProcess for more details on Gauss-Markov processes in ProbNumDiffEq.\n\n\n\n\n\n","category":"function"},{"location":"priors/#ProbNumDiffEq.to_sde","page":"Priors","title":"ProbNumDiffEq.to_sde","text":"to_sde(p::AbstractGaussMarkovProcess)\n\nConvert the prior to the corresponding SDE.\n\nGauss-Markov processes are solutions to linear time-invariant stochastic differential equations (SDEs) of the form\n\nbeginaligned\ndX_t = F X_t dt + L dW_t \nX_0 = mathcalN left( X_0 mu_0 Sigma_0 right)\nendaligned\n\nwhere X_t is the state, W_t is a Wiener process, and F and L are matrices. This function returns the corresponding SDE, i.e. the matrices F and L, as a LTISDE.\n\n\n\n\n\n","category":"function"},{"location":"priors/#ProbNumDiffEq.discretize","page":"Priors","title":"ProbNumDiffEq.discretize","text":"discretize(p::AbstractGaussMarkovProcess, step_size::Real)\n\nCompute the transition matrices of the process for a given step size.\n\n\n\n\n\ndiscretize(p::LTISDE, step_size::Real)\n\nCompute the transition matrices of the SDE solution for a given step size.\n\n\n\n\n\n","category":"function"},{"location":"priors/#ProbNumDiffEq.initial_distribution","page":"Priors","title":"ProbNumDiffEq.initial_distribution","text":"initial_distribution(p::AbstractGaussMarkovProcess)\n\nReturn the initial distribution of the process.\n\nCurrently this is always a Gaussian distribution with zero mean and unit variance, unless explicitly overwitten (e.g. for Matern processes to have the stationary distribution). This implementation is likely to change in the future to allow for more flexibility.\n\n\n\n\n\n","category":"function"},{"location":"priors/#Convenience-functions-to-analyze-and-visualize-priors","page":"Priors","title":"Convenience functions to analyze and visualize priors","text":"","category":"section"},{"location":"priors/","page":"Priors","title":"Priors","text":"ProbNumDiffEq.marginalize\nProbNumDiffEq.sample","category":"page"},{"location":"priors/#ProbNumDiffEq.marginalize","page":"Priors","title":"ProbNumDiffEq.marginalize","text":"marginalize(process::AbstractGaussMarkovProcess, times)\n\nCompute the marginal distributions of the process at the given time points.\n\nThis function computes the marginal distributions of the process at the given times. It does so by discretizing the process with the given step sizes (using ProbNumDiffEq.discretize), and then computing the marginal distributions of the resulting Gaussian distributions.\n\nSee also: sample.\n\n\n\n\n\n","category":"function"},{"location":"priors/#ProbNumDiffEq.sample","page":"Priors","title":"ProbNumDiffEq.sample","text":"sample(process::AbstractGaussMarkovProcess, times, N=1)\n\nSamples from the Gauss-Markov process on the given time grid.\n\nSee also: marginalize.\n\n\n\n\n\n","category":"function"},{"location":"tutorials/dynamical_odes/#Second-Order-ODEs-and-Energy-Preservation","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"","category":"section"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"In this tutorial we consider an energy-preserving, physical dynamical system, given by a second-order ODE.","category":"page"},{"location":"tutorials/dynamical_odes/#TL;DR:","page":"Second Order ODEs and Energy Preservation","title":"TL;DR:","text":"","category":"section"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"To efficiently solve second-order ODEs, just define the problem as a SecondOrderODEProblem.\nTo preserve constant quantities, use the ManifoldUpdate callback; same syntax as DiffEqCallback.jl's ManifoldProjection.","category":"page"},{"location":"tutorials/dynamical_odes/#Simulating-the-Hénon-Heiles-system","page":"Second Order ODEs and Energy Preservation","title":"Simulating the Hénon-Heiles system","text":"","category":"section"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"The Hénon-Heiles model describes the motion of a star around a galactic center, restricted to a plane. It is given by a second-order ODE","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"beginaligned\nddotx = - x - 2 x y \nddoty = y^2 - y - x^2\nendaligned","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"Our goal is to numerically simulate this system on a time span t in 0 T, starting with initial values x(0)=0, y(0) = 01, dotx(0) = 05, doty(0) = 0.","category":"page"},{"location":"tutorials/dynamical_odes/#Transforming-the-problem-into-a-first-order-ODE","page":"Second Order ODEs and Energy Preservation","title":"Transforming the problem into a first-order ODE","text":"","category":"section"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"A very common approach is to first transform the problem into a first-order ODE by introducing a new variable","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"u = dxdyxy","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"to obtain","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"beginaligned\ndotu_1(t) = - u_3 - 2 u_3 u_4 \ndotu_2(t) = u_4^2 - u_4 - u_4^2 \ndotu_3(t) = u_1 \ndotu_4(t) = u_2\nendaligned","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"This first-order ODE can then be solved using any conventional ODE solver - including our EK1:","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"using ProbNumDiffEq, Plots\n\nfunction Hénon_Heiles(du, u, p, t)\n du[1] = -u[3] - 2 * u[3] * u[4]\n du[2] = u[4]^2 - u[4] - u[3]^2\n du[3] = u[1]\n du[4] = u[2]\nend\nu0, du0 = [0.0, 0.1], [0.5, 0.0]\ntspan = (0.0, 100.0)\nprob = ODEProblem(Hénon_Heiles, [du0; u0], tspan)\nsol = solve(prob, EK1());\nplot(sol, idxs=(3, 4)) # where `idxs=(3,4)` is used to plot x agains y","category":"page"},{"location":"tutorials/dynamical_odes/#Solving-the-second-order-ODE-directly","page":"Second Order ODEs and Energy Preservation","title":"Solving the second-order ODE directly","text":"","category":"section"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"Instead of first transforming the problem, we can also solve it directly as a second-order ODE, by defining it as a SecondOrderODEProblem.","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"note: Note\nThe SecondOrderODEProblem type is not defined in ProbNumDiffEq.jl but is provided by SciMLBase.jl. For more information, check out the DifferentialEquations.jl documentation on Dynamical, Hamiltonian and 2nd Order ODE Problems.","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"function Hénon_Heiles2(ddu, du, u, p, t)\n ddu[1] = -u[1] - 2 * u[1] * u[2]\n ddu[2] = u[2]^2 - u[2] - u[1]^2\nend\nprob2 = SecondOrderODEProblem(Hénon_Heiles2, du0, u0, tspan)\nsol2 = solve(prob2, EK1());\nplot(sol2, idxs=(3, 4))","category":"page"},{"location":"tutorials/dynamical_odes/#Benchmark:-Solving-second-order-ODEs-is-*faster*","page":"Second Order ODEs and Energy Preservation","title":"Benchmark: Solving second order ODEs is faster","text":"","category":"section"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"Solving second-order ODEs is not just a matter of convenience - in fact, SciMLBase's SecondOrderODEProblem is neatly designed in such a way that all the classic solvers from OrdinaryDiffEq.jl can handle it by solving the corresponding first-order ODE. But, transforming the ODE to first order increases the dimensionality of the problem, and comes therefore at increased computational cost; this also motivates classic specialized solvers for second-order ODEs.","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"The probabilistic numerical solvers from ProbNumDiffEq.jl have the same internal state representation for first and second order ODEs; all that changes is the measurement model [1]. As a result, we can use the EK1 both for first and second order ODEs, but it automatically specializes on the latter to provide a 2x performance boost:","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"julia> using BenchmarkTools\n\njulia> @btime solve(prob, EK1(order=3), adaptive=false, dt=1e-2);\n 317.336 ms (140561 allocations: 140.41 MiB)\n\njulia> @btime solve(prob2, EK1(order=3), adaptive=false, dt=1e-2);\n 136.654 ms (180647 allocations: 47.95 MiB)","category":"page"},{"location":"tutorials/dynamical_odes/#Energy-preservation","page":"Second Order ODEs and Energy Preservation","title":"Energy preservation","text":"","category":"section"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"In addition to the ODE given above, we know that the solution of the Hénon-Heiles model has to preserve energy over time. The total energy can be expressed as the sum of the potential and kinetic energies, given by","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"beginaligned\noperatornamePotentialEnergy(xy) = frac12 left( x^2 + y^2 + 2 x^2 y - frac2y^33 right) \noperatornameKineticEnergy(dotx doty) = frac12 left( dotx^2 + doty^2 right)\nendaligned","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"In code:","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"PotentialEnergy(x, y) = 1 // 2 * (x^2 + y^2 + 2x^2 * y - 2 // 3 * y^3)\nKineticEnergy(dx, dy) = 1 // 2 * (dx^2 + dy^2)\nE(dx, dy, x, y) = PotentialEnergy(x, y) + KineticEnergy(dx, dy)\nE(u) = E(u...); # convenient shorthand","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"Let's have a look at how the total energy changes over time when we numerically simulate the Hénon-Heiles model over a long period of time: Standard solve","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"longprob = remake(prob2, tspan=(0.0, 1e3))\nlongsol = solve(longprob, EK1(smooth=false), dense=false)\nplot(longsol.t, E.(longsol.u))","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"It visibly loses energy over time, from an initial 0.12967 to a final 0.12899. Let's fix this to get a physically more meaningful solution.","category":"page"},{"location":"tutorials/dynamical_odes/#Energy-preservation-with-the-ManifoldUpdate-callback","page":"Second Order ODEs and Energy Preservation","title":"Energy preservation with the ManifoldUpdate callback","text":"","category":"section"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"In the language of ODE filters, preserving energy over time amounts to just another measurement model [1]. The most convenient way of updating on this additional zero measurement with ProbNumDiffEq.jl is with the ManifoldUpdate callback.","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"note: Note\nThe ManifoldUpdate callback can be thought of a probabilistic counterpart to the ManifoldProjection callback provided by DiffEqCallbacks.jl.","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"To do so, first define a (vector-valued) residual function, here chosen to be the difference between the current energy and the initial energy, and build a ManifoldUpdate callback","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"residual(u) = [E(u) - E(du0..., u0...)]\ncb = ManifoldUpdate(residual)","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"Then, solve the ODE with this callback","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"longsol_preserving = solve(longprob, EK1(smooth=false), dense=false, callback=cb)\nplot(longsol.t, E.(longsol.u))\nplot!(longsol_preserving.t, E.(longsol_preserving.u))","category":"page"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"Voilà! With the ManifoldUpdate callback we could preserve the energy over time and obtain a more truthful probabilistic numerical long-term simulation of the Hénon-Heiles model.","category":"page"},{"location":"tutorials/dynamical_odes/#References","page":"Second Order ODEs and Energy Preservation","title":"References","text":"","category":"section"},{"location":"tutorials/dynamical_odes/","page":"Second Order ODEs and Energy Preservation","title":"Second Order ODEs and Energy Preservation","text":"N. Bosch, F. Tronarp and P. Hennig. Pick-and-Mix Information Operators for Probabilistic ODE Solvers. In: Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, Vol. 151 of Proceedings of Machine Learning Research, edited by G. Camps-Valls, F. J. Ruiz and I. Valera (PMLR, 28–30 Mar 2022); pp. 10015–10027.\n\n\n\n","category":"page"},{"location":"benchmarks/multi-language-wrappers/#ProbNumDiffEq.jl-vs.-various-solver-packages","page":"Multi-Language Wrapper Benchmark","title":"ProbNumDiffEq.jl vs. various solver packages","text":"","category":"section"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"Adapted from SciMLBenchmarks.jl multi-language wrapper benchmark.","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
Code:","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"# Imports\nusing LinearAlgebra, Statistics\nusing StaticArrays, DiffEqDevTools, ParameterizedFunctions, Plots, SciMLBase, OrdinaryDiffEq\nusing ODEInterface, ODEInterfaceDiffEq, Sundials, SciPyDiffEq, deSolveDiffEq, MATLABDiffEq, LSODA\nusing LoggingExtras\n\nusing ProbNumDiffEq\n\nPlots.theme(\n :dao;\n markerstrokewidth=0.5,\n legend=:outertopright,\n margin=5Plots.mm,\n xticks=10.0 .^ (-16:1:16),\n yticks=10.0 .^ (-6:1:5),\n)\n\n# Constants used throughout this benchmark as we only consider final values\nconst DENSE = false # used to decide if we smooth or not\nconst SAVE_EVERYSTEP = false;\n\n# COLORS and a realted utility\nCOLORS = Dict(\n \"Julia\" => :LightGreen,\n \"Julia (static)\" => :DarkGreen,\n \"Hairer\" => :Red,\n \"MATLAB\" => :Orange,\n \"SciPy\" => :Yellow,\n \"deSolve\" => :Blue,\n \"Sundials\" => :Purple,\n \"liblsoda\" => :Purple,\n \"ProbNumDiffEq: EK0\" => :Gray30,\n \"ProbNumDiffEq: EK1\" => :Gray60,\n)\ntocolor(n) = if split(n, '(')[1] in keys(COLORS)\n COLORS[split(n, '(')[1]]\nelse\n COLORS[split(n, ':')[1]]\nend\n\n# Do not show \"deprecated warnings\"\ndeprecated_filter(log_args) = !contains(log_args.message, \"deprecated\")\nfiltered_logger = ActiveFilteredLogger(deprecated_filter, global_logger());","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
","category":"page"},{"location":"benchmarks/multi-language-wrappers/#Non-Stiff-Problem-1:-Lotka-Volterra","page":"Multi-Language Wrapper Benchmark","title":"Non-Stiff Problem 1: Lotka-Volterra","text":"","category":"section"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
Code:","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"f = @ode_def LotkaVolterra begin\n dx = a*x - b*x*y\n dy = -c*y + d*x*y\nend a b c d\np = [1.5, 1, 3, 1]\ntspan = (0.0, 10.0)\nu0 = [1.0, 1.0]\nprob = ODEProblem{true,SciMLBase.FullSpecialize()}(f,u0,tspan,p)\nstaticprob = ODEProblem{false,SciMLBase.FullSpecialize()}(f,SVector{2}(u0),tspan,SVector{4}(p))\n\nsol = solve(prob,Vern7(),abstol=1/10^14,reltol=1/10^14,dense=false)\ntest_sol = sol\nplot(sol, title=\"Lotka-Volterra Solution\", legend=false, xticks=:auto, yticks=:auto)","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"(Image: )","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
Code:","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"_setups = [\n \"Julia: DP5\" => Dict(:alg=>DP5())\n \"Julia: Tsit5\" => Dict(:alg=>Tsit5())\n \"Julia: Vern7\" => Dict(:alg=>Vern7())\n \"Hairer: dopri5\" => Dict(:alg=>ODEInterfaceDiffEq.dopri5())\n \"MATLAB: ode45\" => Dict(:alg=>MATLABDiffEq.ode45())\n \"MATLAB: ode113\" => Dict(:alg=>MATLABDiffEq.ode113())\n \"SciPy: RK45\" => Dict(:alg=>SciPyDiffEq.RK45())\n \"SciPy: LSODA\" => Dict(:alg=>SciPyDiffEq.LSODA())\n \"SciPy: odeint\" => Dict(:alg=>SciPyDiffEq.odeint())\n \"deSolve: lsoda\" => Dict(:alg=>deSolveDiffEq.lsoda())\n \"deSolve: ode45\" => Dict(:alg=>deSolveDiffEq.ode45())\n \"Sundials: Adams\" => Dict(:alg=>Sundials.CVODE_Adams())\n \"ProbNumDiffEq: EK0(3)\" => Dict(:alg=>EK0(order=3, smooth=DENSE))\n \"ProbNumDiffEq: EK0(5)\" => Dict(:alg=>EK0(order=5, smooth=DENSE))\n \"ProbNumDiffEq: EK1(3)\" => Dict(:alg=>EK1(order=3, smooth=DENSE))\n \"ProbNumDiffEq: EK1(5)\" => Dict(:alg=>EK1(order=5, smooth=DENSE))\n \"ProbNumDiffEq: EK1(8)\" => Dict(:alg=>EK1(order=8, smooth=DENSE))\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\ncolors = tocolor.(labels) |> permutedims\n\nabstols = 1.0 ./ 10.0 .^ (6:13)\nreltols = 1.0 ./ 10.0 .^ (3:10)\n\nwp = with_logger(filtered_logger) do\n WorkPrecisionSet(\n [prob, staticprob], abstols, reltols, setups;\n names = labels,\n appxsol = [test_sol, test_sol],\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n numruns = 10,\n maxiters = Int(1e7),\n timeseries_errors = false,\n verbose = false,\n )\nend\n\nplot(wp, title = \"Non-stiff 1: Lotka-Volterra\", color = colors)","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"(Image: )","category":"page"},{"location":"benchmarks/multi-language-wrappers/#Non-Stiff-Problem-2:-Rigid-Body","page":"Multi-Language Wrapper Benchmark","title":"Non-Stiff Problem 2: Rigid Body","text":"","category":"section"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
Code:","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"f = @ode_def RigidBodyBench begin\n dy1 = -2*y2*y3\n dy2 = 1.25*y1*y3\n dy3 = -0.5*y1*y2 + 0.25*sin(t)^2\nend\nu0 = [1.0;0.0;0.9]\ntspan = (0.0, 10.0)\nprob = ODEProblem{true,SciMLBase.FullSpecialize()}(f,u0,tspan)\nstaticprob = ODEProblem{false,SciMLBase.FullSpecialize()}(f,SVector{3}(u0),tspan)\nsol = solve(prob,Vern7(),abstol=1/10^14,reltol=1/10^14,dense=false)\ntest_sol = sol\nplot(sol, title=\"Rigid Body Solution\", legend=false, xticks=:auto, yticks=:auto)","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"(Image: )","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
Code:","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"_setups = [\n \"Julia: DP5\" => Dict(:alg=>DP5())\n \"Julia: Tsit5\" => Dict(:alg=>Tsit5())\n \"Julia: Vern7\" => Dict(:alg=>Vern7())\n \"Hairer: dopri5\" => Dict(:alg=>dopri5())\n \"MATLAB: ode45\" => Dict(:alg=>MATLABDiffEq.ode45())\n \"MATLAB: ode113\" => Dict(:alg=>MATLABDiffEq.ode113())\n \"SciPy: RK45\" => Dict(:alg=>SciPyDiffEq.RK45())\n \"SciPy: LSODA\" => Dict(:alg=>SciPyDiffEq.LSODA())\n \"SciPy: odeint\" => Dict(:alg=>SciPyDiffEq.odeint())\n \"deSolve: lsoda\" => Dict(:alg=>deSolveDiffEq.lsoda())\n \"deSolve: ode45\" => Dict(:alg=>deSolveDiffEq.ode45())\n \"Sundials: Adams\" => Dict(:alg=>CVODE_Adams())\n \"ProbNumDiffEq: EK0(3)\" => Dict(:alg=>EK0(order=3, smooth=DENSE))\n \"ProbNumDiffEq: EK0(5)\" => Dict(:alg=>EK0(order=5, smooth=DENSE))\n \"ProbNumDiffEq: EK1(3)\" => Dict(:alg=>EK1(order=3, smooth=DENSE))\n \"ProbNumDiffEq: EK1(5)\" => Dict(:alg=>EK1(order=5, smooth=DENSE))\n \"ProbNumDiffEq: EK1(8)\" => Dict(:alg=>EK1(order=8, smooth=DENSE))\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\ncolors = tocolor.(labels) |> permutedims\n\nabstols = 1.0 ./ 10.0 .^ (6:13)\nreltols = 1.0 ./ 10.0 .^ (3:10)\n\nwp = with_logger(filtered_logger) do\n WorkPrecisionSet(\n [prob,staticprob], abstols, reltols, setups;\n names = labels,\n appxsol = [test_sol, test_sol],\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n numruns = 10,\n maxiters = Int(1e7),\n timeseries_errors = false,\n verbose = false\n )\nend\n\nplot(wp, title = \"Non-stiff 2: Rigid-Body\", color = colors)","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"(Image: )","category":"page"},{"location":"benchmarks/multi-language-wrappers/#Stiff-Problem-1:-ROBER","page":"Multi-Language Wrapper Benchmark","title":"Stiff Problem 1: ROBER","text":"","category":"section"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
Code:","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"rober = @ode_def begin\n dy₁ = -k₁*y₁+k₃*y₂*y₃\n dy₂ = k₁*y₁-k₂*y₂^2-k₃*y₂*y₃\n dy₃ = k₂*y₂^2\nend k₁ k₂ k₃\nu0 = [1.0,0.0,0.0]\np = [0.04,3e7,1e4]\nprob = ODEProblem{true,SciMLBase.FullSpecialize()}(rober,u0,(0.0,1e5),p)\nstaticprob = ODEProblem{false,SciMLBase.FullSpecialize()}(rober,SVector{3}(u0),(0.0,1e5),SVector{3}(p))\nsol = solve(prob,CVODE_BDF(),abstol=1/10^14,reltol=1/10^14,dense=false)\ntest_sol = sol\nplot(sol, title=\"ROBER Solution\", legend=false, xlims=(1e0, 1e5), xticks=:auto, yticks=:auto)","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"(Image: )","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
Code:","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"_setups = [\n \"Julia: Rosenbrock23\" => Dict(:alg=>Rosenbrock23())\n \"Julia: Rodas4\" => Dict(:alg=>Rodas4())\n \"Julia: Rodas5\" => Dict(:alg=>Rodas5())\n \"Hairer: rodas\" => Dict(:alg=>rodas())\n \"Hairer: radau\" => Dict(:alg=>radau())\n \"MATLAB: ode23s\" => Dict(:alg=>MATLABDiffEq.ode23s())\n \"MATLAB: ode15s\" => Dict(:alg=>MATLABDiffEq.ode15s())\n \"SciPy: LSODA\" => Dict(:alg=>SciPyDiffEq.LSODA())\n \"SciPy: BDF\" => Dict(:alg=>SciPyDiffEq.BDF())\n \"SciPy: odeint\" => Dict(:alg=>SciPyDiffEq.odeint())\n \"deSolve: lsoda\" => Dict(:alg=>deSolveDiffEq.lsoda())\n \"Sundials: CVODE\" => Dict(:alg=>CVODE_BDF())\n \"ProbNumDiffEq: EK1(3)\" => Dict(:alg=>EK1(order=3, smooth=DENSE))\n \"ProbNumDiffEq: EK1(5)\" => Dict(:alg=>EK1(order=5, smooth=DENSE))\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\ncolors = tocolor.(labels) |> permutedims\n\nabstols = 1.0 ./ 10.0 .^ (5:12)\nreltols = 1.0 ./ 10.0 .^ (2:9)\n\nwp = with_logger(filtered_logger) do\n WorkPrecisionSet(\n [prob, staticprob], abstols, reltols, setups;\n names = labels,\n dense = DENSE,\n verbose = false,\n save_everystep = SAVE_EVERYSTEP,\n appxsol = [test_sol, test_sol],\n maxiters=Int(1e5)\n )\nend\n\nplot(wp, title = \"Stiff 1: ROBER\", color = colors)","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"(Image: )","category":"page"},{"location":"benchmarks/multi-language-wrappers/#Stiff-Problem-2:-HIRES","page":"Multi-Language Wrapper Benchmark","title":"Stiff Problem 2: HIRES","text":"","category":"section"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
Code:","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"f = @ode_def Hires begin\n dy1 = -1.71*y1 + 0.43*y2 + 8.32*y3 + 0.0007\n dy2 = 1.71*y1 - 8.75*y2\n dy3 = -10.03*y3 + 0.43*y4 + 0.035*y5\n dy4 = 8.32*y2 + 1.71*y3 - 1.12*y4\n dy5 = -1.745*y5 + 0.43*y6 + 0.43*y7\n dy6 = -280.0*y6*y8 + 0.69*y4 + 1.71*y5 -\n 0.43*y6 + 0.69*y7\n dy7 = 280.0*y6*y8 - 1.81*y7\n dy8 = -280.0*y6*y8 + 1.81*y7\nend\n\nu0 = zeros(8)\nu0[1] = 1\nu0[8] = 0.0057\nprob = ODEProblem{true,SciMLBase.FullSpecialize()}(f,u0,(0.0,321.8122))\nstaticprob = ODEProblem{false,SciMLBase.FullSpecialize()}(f,SVector{8}(u0),(0.0,321.8122))\n\nsol = solve(prob,Rodas5(),abstol=1/10^14,reltol=1/10^14, dense=false)\ntest_sol = sol\nplot(sol, title=\"HIRES Solution\", legend=false, xticks=:auto, yticks=:auto)","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"(Image: )","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
Code:","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"_setups = [\n \"Julia: Rosenbrock23\" => Dict(:alg=>Rosenbrock23())\n \"Julia: Rodas4\" => Dict(:alg=>Rodas4())\n \"Julia: radau\" => Dict(:alg=>RadauIIA5())\n \"Hairer: rodas\" => Dict(:alg=>rodas())\n \"Hairer: radau\" => Dict(:alg=>radau())\n \"MATLAB: ode23s\" => Dict(:alg=>MATLABDiffEq.ode23s())\n \"MATLAB: ode15s\" => Dict(:alg=>MATLABDiffEq.ode15s())\n \"SciPy: LSODA\" => Dict(:alg=>SciPyDiffEq.LSODA())\n \"SciPy: BDF\" => Dict(:alg=>SciPyDiffEq.BDF())\n \"SciPy: odeint\" => Dict(:alg=>SciPyDiffEq.odeint())\n \"deSolve: lsoda\" => Dict(:alg=>deSolveDiffEq.lsoda())\n \"Sundials: CVODE\" => Dict(:alg=>CVODE_BDF())\n \"ProbNumDiffEq: EK1(2)\" => Dict(:alg=>EK1(order=2, smooth=DENSE))\n \"ProbNumDiffEq: EK1(3)\" => Dict(:alg=>EK1(order=3, smooth=DENSE))\n \"ProbNumDiffEq: EK1(5)\" => Dict(:alg=>EK1(order=5, smooth=DENSE))\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\ncolors = tocolor.(labels) |> permutedims\n\nabstols = 1.0 ./ 10.0 .^ (5:10)\nreltols = 1.0 ./ 10.0 .^ (1:6)\n\nwp = with_logger(filtered_logger) do\n WorkPrecisionSet(\n [prob, staticprob], abstols, reltols, setups;\n names = labels,\n dense = false,\n verbose = false,\n save_everystep = false,\n appxsol = [test_sol, test_sol],\n maxiters = Int(1e5),\n numruns=100\n )\nend\n\nplot(wp, title = \"Stiff 2: Hires\", color=colors)","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"(Image: )","category":"page"},{"location":"benchmarks/multi-language-wrappers/#Appendix","page":"Multi-Language Wrapper Benchmark","title":"Appendix","text":"","category":"section"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
Computer information:","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"using InteractiveUtils\nInteractiveUtils.versioninfo()","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"Julia Version 1.10.0\nCommit 3120989f39b (2023-12-25 18:01 UTC)\nBuild Info:\n Official https://julialang.org/ release\nPlatform Info:\n OS: Linux (x86_64-linux-gnu)\n CPU: 12 × Intel(R) Core(TM) i7-6800K CPU @ 3.40GHz\n WORD_SIZE: 64\n LIBM: libopenlibm\n LLVM: libLLVM-15.0.7 (ORCJIT, broadwell)\n Threads: 17 on 12 virtual cores\nEnvironment:\n JULIA_NUM_THREADS = auto","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
Package information:","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"using Pkg\nPkg.status()","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Project.toml`\n [f3b72e0c] DiffEqDevTools v2.44.1\n [31c24e10] Distributions v0.25.107\n [7073ff75] IJulia v1.24.2\n [7f56f5a3] LSODA v0.7.5\n [e6f89c97] LoggingExtras v1.0.3\n [e2752cbe] MATLABDiffEq v1.2.0\n⌃ [961ee093] ModelingToolkit v8.73.2\n [54ca160b] ODEInterface v0.5.0\n [09606e27] ODEInterfaceDiffEq v3.13.3\n⌃ [1dea7af3] OrdinaryDiffEq v6.66.0\n [65888b18] ParameterizedFunctions v5.16.0\n [91a5bcdd] Plots v1.39.0\n [bf3e78b0] ProbNumDiffEq v0.14.0 `~/.julia/dev/ProbNumDiffEq`\n⌃ [0bca4576] SciMLBase v2.10.0\n [505e40e9] SciPyDiffEq v0.2.1\n [ce78b400] SimpleUnPack v1.1.0\n [90137ffa] StaticArrays v1.9.1\n [c3572dad] Sundials v4.23.1\n [44d3d7a6] Weave v0.10.12\n [0518478a] deSolveDiffEq v0.1.1\nInfo Packages marked with ⌃ have new versions available and may be upgradable.","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
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Full manifest:","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"Pkg.status(mode=Pkg.PKGMODE_MANIFEST)","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Manifest.toml`\n [47edcb42] ADTypes v0.2.6\n⌅ [c3fe647b] AbstractAlgebra v0.34.7\n [621f4979] AbstractFFTs v1.5.0\n [1520ce14] AbstractTrees v0.4.4\n [7d9f7c33] Accessors v0.1.35\n⌅ [79e6a3ab] Adapt v3.7.2\n [ec485272] ArnoldiMethod v0.2.0\n [c9d4266f] ArrayAllocators v0.3.0\n [4fba245c] ArrayInterface v7.7.0\n [4c555306] ArrayLayouts v1.5.2\n [6e4b80f9] BenchmarkTools v1.4.0\n [e2ed5e7c] Bijections v0.1.6\n [d1d4a3ce] BitFlags v0.1.8\n [62783981] BitTwiddlingConvenienceFunctions v0.1.5\n [fa961155] CEnum v0.5.0\n [2a0fbf3d] CPUSummary v0.2.4\n [00ebfdb7] CSTParser 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Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`","category":"page"},{"location":"benchmarks/multi-language-wrappers/","page":"Multi-Language Wrapper Benchmark","title":"Multi-Language Wrapper Benchmark","text":"
","category":"page"},{"location":"benchmarks/pleiades/#Pleiades-benchmark","page":"Pleiades","title":"Pleiades benchmark","text":"","category":"section"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"note: Summary\nPleiades is a medium-dimensional, non-stiff, second-order ODE. We see that:The EK0 is much faster than the EK1 as it scales linearly with the ODE dimension.\nIf the problem is a second-order ODE, implement it as a second-order ODE!","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"
Code:","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"using LinearAlgebra, Statistics, Distributions\nusing DiffEqDevTools, ParameterizedFunctions, SciMLBase, OrdinaryDiffEq, Sundials, Plots, ODEInterfaceDiffEq\nusing ModelingToolkit\nusing ProbNumDiffEq\n\nPlots.theme(\n :dao;\n markerstrokewidth=0.5,\n legend=:outertopright,\n margin=5Plots.mm,\n xticks=10.0 .^ (-16:1:16),\n)\n\nfunction plot_chisq_interval!(df, q=0.01)\n dist = Chisq(df)\n low, high, mid = quantile(dist, [q, 1-q])..., mean(dist)\n hline!([low, high], linestyle=:dash, color=:black, label=\"\",\n fill_between=[high nothing], fillcolor=:green, fillalpha=0.15)\n hline!([mid], linestyle=:solid, color=:black, label=\"\")\nend","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"
","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"
Code:","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"# first-order ODE\n@fastmath function pleiades(du, u, p, t)\n v = view(u, 1:7) # x\n w = view(u, 8:14) # y\n x = view(u, 15:21) # x′\n y = view(u, 22:28) # y′\n du[15:21] .= v\n du[22:28] .= w\n @inbounds @simd ivdep for i in 1:14\n du[i] = zero(eltype(u))\n end\n @inbounds @simd ivdep for i in 1:7\n @inbounds @simd ivdep for j in 1:7\n if i != j\n r = ((x[i] - x[j])^2 + (y[i] - y[j])^2)^(3 / 2)\n du[i] += j * (x[j] - x[i]) / r\n du[7+i] += j * (y[j] - y[i]) / r\n end\n end\n end\nend\nx0 = [3.0, 3.0, -1.0, -3.0, 2.0, -2.0, 2.0]\ny0 = [3.0, -3.0, 2.0, 0, 0, -4.0, 4.0]\ndx0 = [0, 0, 0, 0, 0, 1.75, -1.5]\ndy0 = [0, 0, 0, -1.25, 1, 0, 0]\nu0 = [dx0; dy0; x0; y0]\ntspan = (0.0, 3.0)\nprob1 = ODEProblem(pleiades, u0, tspan)\n\n# second-order ODE\nfunction pleiades2(ddu, du, u, p, t)\n x = view(u, 1:7)\n y = view(u, 8:14)\n for i in 1:14\n ddu[i] = zero(eltype(u))\n end\n for i in 1:7, j in 1:7\n if i != j\n r = ((x[i] - x[j])^2 + (y[i] - y[j])^2)^(3 / 2)\n ddu[i] += j * (x[j] - x[i]) / r\n ddu[7+i] += j * (y[j] - y[i]) / r\n end\n end\nend\nu0 = [x0; y0]\ndu0 = [dx0; dy0]\nprob2 = SecondOrderODEProblem(pleiades2, du0, u0, tspan)\nprobs = [prob1, prob2]\n\nref_sol1 = solve(prob1, Vern9(), abstol=1/10^14, reltol=1/10^14)\nref_sol2 = solve(prob2, Vern9(), abstol=1/10^14, reltol=1/10^14)\nref_sols = [ref_sol1, ref_sol2]\n\nplot(ref_sol1, idxs=[(14+i,21+i) for i in 1:7], title=\"Pleiades Solution\", legend=false,\n xticks=:auto, yticks=:auto)\nscatter!(ref_sol1.u[end][15:21], ref_sol1.u[end][22:end], color=1:7)","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"
","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"(Image: )","category":"page"},{"location":"benchmarks/pleiades/#pleiades_results","page":"Pleiades","title":"EK0 vs EK1 & first-order vs. second-order","text":"","category":"section"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"
Code:","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"DENSE = false;\nSAVE_EVERYSTEP = false;\n\n_setups = [\n \"EK0(3) (1st order ODE)\" => Dict(:alg => EK0(order=3, smooth=DENSE), :prob_choice => 1)\n \"EK0(5) (1st order ODE)\" => Dict(:alg => EK0(order=5, smooth=DENSE), :prob_choice => 1)\n \"EK0(4) (2nd order ODE)\" => Dict(:alg => EK0(order=4, smooth=DENSE), :prob_choice => 2)\n \"EK0(6) (2nd order ODE)\" => Dict(:alg => EK0(order=6, smooth=DENSE), :prob_choice => 2)\n \"EK1(3) (1st order ODE)\" => Dict(:alg => EK1(order=3, smooth=DENSE), :prob_choice => 1)\n \"EK1(5) (1st order ODE)\" => Dict(:alg => EK1(order=5, smooth=DENSE), :prob_choice => 1)\n \"EK1(4) (2nd order ODE)\" => Dict(:alg => EK1(order=4, smooth=DENSE), :prob_choice => 2)\n \"EK1(6) (2nd order ODE)\" => Dict(:alg => EK1(order=6, smooth=DENSE), :prob_choice => 2)\n]\n\nlabels = first.(_setups)\nsetups = last.(_setups)\n\nabstols = 1.0 ./ 10.0 .^ (6:11)\nreltols = 1.0 ./ 10.0 .^ (3:8)\n\nwp = WorkPrecisionSet(\n probs, abstols, reltols, setups;\n names = labels,\n appxsol = ref_sols,\n dense = DENSE,\n save_everystep = SAVE_EVERYSTEP,\n numruns = 5,\n maxiters = Int(1e7),\n)\n\ncolor = [1 1 1 1 2 2 2 2]\nlinestyle = [:solid :solid :dash :dash :solid :solid :dash :dash]\nplot(wp; color, linestyle)\n\n_ref_setups = [\n \"Classic: Tsit5\" => Dict(:alg => Tsit5(), :prob_choice => 1)\n \"Classic: RadauIIA5\" => Dict(:alg => RadauIIA5(), :prob_choice => 1)\n \"Classic: DPRKN6\" => Dict(:alg => DPRKN6(), :prob_choice => 2)\n]\nref_labels = first.(_ref_setups)\nref_setups = last.(_ref_setups)\nref_wp = WorkPrecisionSet(\n probs, abstols ./ 1000, reltols ./ 1000, ref_setups;\n names = ref_labels,\n appxsol = ref_sols,\n dense = false,\n save_everystep = false,\n maxiters = Int(1e7),\n)\nplot!(ref_wp, x=:final, color=:gray, alpha=0.7, linestyle=[:solid :solid :dash])","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"
","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"(Image: )","category":"page"},{"location":"benchmarks/pleiades/#Calibration","page":"Pleiades","title":"Calibration","text":"","category":"section"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"
Code:","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"plot(wp; x=:final, y=:chi2_final, color, linestyle)\nplot_chisq_interval!(length(u0)*2)","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"
","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"(Image: )","category":"page"},{"location":"benchmarks/pleiades/#Appendix","page":"Pleiades","title":"Appendix","text":"","category":"section"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"
Computer information:","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"using InteractiveUtils\nInteractiveUtils.versioninfo()","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"Julia Version 1.10.0\nCommit 3120989f39b (2023-12-25 18:01 UTC)\nBuild Info:\n Official https://julialang.org/ release\nPlatform Info:\n OS: Linux (x86_64-linux-gnu)\n CPU: 12 × Intel(R) Core(TM) i7-6800K CPU @ 3.40GHz\n WORD_SIZE: 64\n LIBM: libopenlibm\n LLVM: libLLVM-15.0.7 (ORCJIT, broadwell)\n Threads: 17 on 12 virtual cores\nEnvironment:\n JULIA_NUM_THREADS = auto","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"
","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"
Package information:","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"using Pkg\nPkg.status()","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Project.toml`\n [f3b72e0c] DiffEqDevTools v2.44.1\n [31c24e10] Distributions v0.25.107\n [7073ff75] IJulia v1.24.2\n [7f56f5a3] LSODA v0.7.5\n [e6f89c97] LoggingExtras v1.0.3\n [e2752cbe] MATLABDiffEq v1.2.0\n⌃ [961ee093] ModelingToolkit v8.73.2\n [54ca160b] ODEInterface v0.5.0\n [09606e27] ODEInterfaceDiffEq v3.13.3\n⌃ [1dea7af3] OrdinaryDiffEq v6.66.0\n [65888b18] ParameterizedFunctions v5.16.0\n [91a5bcdd] Plots v1.39.0\n [bf3e78b0] ProbNumDiffEq v0.14.0 `~/.julia/dev/ProbNumDiffEq`\n⌃ [0bca4576] SciMLBase v2.10.0\n [505e40e9] SciPyDiffEq v0.2.1\n [ce78b400] SimpleUnPack v1.1.0\n [90137ffa] StaticArrays v1.9.1\n [c3572dad] Sundials v4.23.1\n [44d3d7a6] Weave v0.10.12\n [0518478a] deSolveDiffEq v0.1.1\nInfo Packages marked with ⌃ have new versions available and may be upgradable.","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"
","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"
Full manifest:","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"Pkg.status(mode=Pkg.PKGMODE_MANIFEST)","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"Status `~/.julia/dev/ProbNumDiffEq/benchmarks/Manifest.toml`\n [47edcb42] ADTypes v0.2.6\n⌅ [c3fe647b] AbstractAlgebra v0.34.7\n [621f4979] AbstractFFTs v1.5.0\n [1520ce14] AbstractTrees v0.4.4\n [7d9f7c33] Accessors v0.1.35\n⌅ [79e6a3ab] Adapt v3.7.2\n [ec485272] ArnoldiMethod v0.2.0\n [c9d4266f] ArrayAllocators v0.3.0\n [4fba245c] ArrayInterface v7.7.0\n [4c555306] ArrayLayouts v1.5.2\n [6e4b80f9] BenchmarkTools v1.4.0\n [e2ed5e7c] Bijections v0.1.6\n [d1d4a3ce] BitFlags v0.1.8\n [62783981] BitTwiddlingConvenienceFunctions v0.1.5\n [fa961155] CEnum v0.5.0\n [2a0fbf3d] CPUSummary v0.2.4\n [00ebfdb7] CSTParser v3.4.0\n [49dc2e85] Calculus v0.5.1\n [324d7699] CategoricalArrays v0.10.8\n [d360d2e6] ChainRulesCore v1.19.1\n [fb6a15b2] 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Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`","category":"page"},{"location":"benchmarks/pleiades/","page":"Pleiades","title":"Pleiades","text":"
","category":"page"},{"location":"tutorials/getting_started/#Solving-ODEs-with-Probabilistic-Numerics","page":"Getting Started","title":"Solving ODEs with Probabilistic Numerics","text":"","category":"section"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"In this tutorial we solve a simple non-linear ordinary differential equation (ODE) with the probabilistic numerical ODE solvers implemented in this package.","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"note: Note\nIf you never used DifferentialEquations.jl, check out their \"Getting Started with Differential Equations in Julia\" tutorial. It explains how to define and solve ODE problems and how to analyze the solution, so it's a great starting point. Most of ProbNumDiffEq.jl works exactly as you would expect from DifferentialEquations.jl – just with some added uncertainties and related functionality on top!","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"In this tutorial, we consider a Fitzhugh-Nagumo model described by an ODE of the form","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"beginaligned\ndoty_1 = c (y_1 - fracy_1^33 + y_2) \ndoty_2 = -frac1c (y_1 - a - b y_2)\nendaligned","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"on a time span t in 0 T, with initial value y(0) = y_0. In the following, we","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"define the problem with explicit choices of initial values, integration domains, and parameters,\nsolve the problem with our ODE filters, and\nvisualize the results and the corresponding uncertainties.","category":"page"},{"location":"tutorials/getting_started/#TL;DR:-Just-use-DifferentialEquations.jl-with-the-EK1-algorithm","page":"Getting Started","title":"TL;DR: Just use DifferentialEquations.jl with the EK1 algorithm","text":"","category":"section"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"using ProbNumDiffEq, Plots\n\nfunction fitz(du, u, p, t)\n a, b, c = p\n du[1] = c * (u[1] - u[1]^3 / 3 + u[2])\n du[2] = -(1 / c) * (u[1] - a - b * u[2])\nend\nu0 = [-1.0; 1.0]\ntspan = (0.0, 20.0)\np = (0.2, 0.2, 3.0)\nprob = ODEProblem(fitz, u0, tspan, p)\n\nusing Logging; Logging.disable_logging(Logging.Warn); # hide\nsol = solve(prob, EK1())\nLogging.disable_logging(Logging.Debug) # hide\nplot(sol)","category":"page"},{"location":"tutorials/getting_started/#Step-1:-Define-the-problem","page":"Getting Started","title":"Step 1: Define the problem","text":"","category":"section"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"First, import ProbNumDiffEq.jl","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"using ProbNumDiffEq","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"Then, set up the ODEProblem exactly as you would in DifferentialEquations.jl. Define the vector field","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"function fitz(du, u, p, t)\n a, b, c = p\n du[1] = c * (u[1] - u[1]^3 / 3 + u[2])\n du[2] = -(1 / c) * (u[1] - a - b * u[2])\nend\nnothing # hide","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"and then the ODEProblem, with initial value u0, time span tspan, and parameters p","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"u0 = [-1.0; 1.0]\ntspan = (0.0, 20.0)\np = (0.2, 0.2, 3.0)\nprob = ODEProblem(fitz, u0, tspan, p)\nnothing # hide","category":"page"},{"location":"tutorials/getting_started/#Step-2:-Solve-the-problem","page":"Getting Started","title":"Step 2: Solve the problem","text":"","category":"section"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"To solve the ODE we just use DifferentialEquations.jl's solve interface, together with one of the algorithms implemented in this package. For now, let's use EK1:","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"sol = solve(prob, EK1())\n# nothing # hide","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"That's it! we just computed a probabilistic numerical ODE solution!","category":"page"},{"location":"tutorials/getting_started/#Step-3:-Analyze-the-solution","page":"Getting Started","title":"Step 3: Analyze the solution","text":"","category":"section"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"Let's plot the result with Plots.jl.","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"using Plots\nplot(sol)","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"Looks good! Looks like the EK1 managed to solve the Fitzhugh-Nagumo problem quite well.","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"tip: Tip\nTo learn more about plotting ODE solutions, check out the plotting tutorial for DifferentialEquations.jl + Plots.jl provided here. Most of that works exactly as expected with ProbNumDiffEq.jl.","category":"page"},{"location":"tutorials/getting_started/#Plot-the-probabilistic-error-estimates","page":"Getting Started","title":"Plot the probabilistic error estimates","text":"","category":"section"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"The plot above looks like a standard ODE solution – but it's not! The numerical errors are just so small that we can't see them in the plot, and the probabilistic error estimates are too. We can visualize them by plotting the errors and error estimates directly:","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"using OrdinaryDiffEq, Statistics\nreference = solve(prob, Vern9(), abstol=1e-9, reltol=1e-9, saveat=sol.t)\nerrors = reduce(hcat, mean.(sol.pu) .- reference.u)'\nerror_estimates = reduce(hcat, std.(sol.pu))'\nplot(sol.t, errors, label=\"error\", color=[1 2], xlabel=\"t\", ylabel=\"err\")\nplot!(sol.t, zero(errors), ribbon=3error_estimates, label=\"error estimate\",\n color=[1 2], alpha=0.2)","category":"page"},{"location":"tutorials/getting_started/#More-about-the-ProbabilisticODESolution","page":"Getting Started","title":"More about the ProbabilisticODESolution","text":"","category":"section"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"The solution object returned by ProbNumDiffEq.jl mostly behaves just like any other ODESolution in DifferentialEquations.jl – with some added uncertainties and related functionality on top. The ProbabilisticODESolution can be indexed with","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"sol.u[1]\nsol.u[end]\nsol.t[end]","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"But since sol is a probabilistic numerical ODE solution, it contains a Gaussian distributions over solution values. The marginals of this posterior are stored in sol.pu:","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"sol.pu[end]","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"You can compute means, covariances, and standard deviations via Statistics.jl:","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"using Statistics\nmean(sol.pu[5])\ncov(sol.pu[5])\nstd(sol.pu[5])","category":"page"},{"location":"tutorials/getting_started/#Dense-output","page":"Getting Started","title":"Dense output","text":"","category":"section"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"Probabilistic numerical ODE solvers approximate the posterior distribution","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"p Big( y(t) big y(0) = y_0 doty(t_i) = f_theta(y(t_i) t_i) Big)","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"which describes a posterior not just for the discrete steps but for any t in the continuous space t in 0 T; in classic ODE solvers, this is also known as \"interpolation\" or \"dense output\". The probabilistic solutions returned by our solvers can be interpolated as usual by treating them as functions, but they return Gaussian distributions","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"sol(0.45)\nmean(sol(0.45))","category":"page"},{"location":"tutorials/getting_started/#Next-steps","page":"Getting Started","title":"Next steps","text":"","category":"section"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"Check out one of the other tutorials:","category":"page"},{"location":"tutorials/getting_started/","page":"Getting Started","title":"Getting Started","text":"\"Second Order ODEs and Energy Preservation\" explains how to solve second-order ODEs more efficiently while also better preserving energy or other conserved quantities;\n\"Solving DAEs with Probabilistic Numerics\" demonstrates how to solve differential algebraic equations in a probabilistic numerical way.","category":"page"},{"location":"tutorials/ode_parameter_inference/#Parameter-Inference-with-ProbNumDiffEq.jl","page":"Parameter Inference","title":"Parameter Inference with ProbNumDiffEq.jl","text":"","category":"section"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"Let's assume we have an initial value problem (IVP)","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"beginaligned\ndoty = f_theta(y t) qquad y(t_0) = y_0\nendaligned","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"which we observe through a set mathcalD = u(t_n)_n=1^N of noisy data points","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"beginaligned\nu(t_n) = H y(t_n) + v_n qquad v_n sim mathcalN(0 R)\nendaligned","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"The question of interest is: How can we compute the marginal likelihood p(mathcalD mid theta)? Short answer: We can't. It's intractable, because computing the true IVP solution exactly y(t) is intractable. What we can do however is compute an approximate marginal likelihood. This is what ProbNumDiffEq.DataLikelihoods provides.","category":"page"},{"location":"tutorials/ode_parameter_inference/#The-specific-problem,-in-code","page":"Parameter Inference","title":"The specific problem, in code","text":"","category":"section"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"Let's assume that the true underlying dynamics are given by a FitzHugh-Nagumo model","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"using ProbNumDiffEq, LinearAlgebra, OrdinaryDiffEq, Plots\nPlots.theme(:default; markersize=2, markerstrokewidth=0.1)\n\nfunction f(du, u, p, t)\n a, b, c = p\n du[1] = c*(u[1] - u[1]^3/3 + u[2])\n du[2] = -(1/c)*(u[1] - a - b*u[2])\nend\nu0 = [-1.0, 1.0]\ntspan = (0.0, 20.0)\np = (0.2, 0.2, 3.0)\ntrue_prob = ODEProblem(f, u0, tspan, p)","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"from which we generate some artificial noisy data","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"true_sol = solve(true_prob, Vern9(), abstol=1e-10, reltol=1e-10)\n\ntimes = 1:0.5:20\nσ = 1e-1\nH = [1 0;]\nodedata = [H*true_sol(t) .+ σ * randn() for t in times]\n\nplot(true_sol, color=:black, linestyle=:dash, label=[\"True Solution\" \"\"])\nscatter!(times, stack(odedata)', color=1, label=[\"Noisy Data\" \"\"])","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"Our goal is then to recover the true parameter p (and thus also the true trajectory plotted above) the noisy data.","category":"page"},{"location":"tutorials/ode_parameter_inference/#Computing-the-negative-log-likelihood","page":"Parameter Inference","title":"Computing the negative log-likelihood","text":"","category":"section"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"To do parameter inference - be it maximum-likelihod, maximum a posteriori, or full Bayesian inference with MCMC - we need to evaluate the likelihood of given a parameter estimate theta_textest, which corresponds to the probability of the data under the trajectory returned by the ODE solver","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"θ_est = (0.1, 0.1, 2.0)\nprob = remake(true_prob, p=θ_est)\nplot(true_sol, color=:black, linestyle=:dash, label=[\"True Solution\" \"\"])\nscatter!(times, stack(odedata)', color=1, label=[\"Noisy Data\" \"\"])\nsol = solve(prob, EK1(), adaptive=false, dt=1e-1)\nplot!(sol, color=2, label=[\"Numerical solution for θ_est\" \"\"])","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"This quantity can be computed in multiple ways; see Data Likelihoods. Here we use ProbNumDiffEq.DataLikelihoods.fenrir_data_loglik:","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"using ProbNumDiffEq.DataLikelihoods\n\ndata = (t=times, u=odedata)\nnll = -fenrir_data_loglik(\n prob, EK1(smooth=true); \n data, observation_noise_cov=σ^2, observation_matrix=H, \n adaptive=false, dt=1e-1)","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"This is the negative marginal log-likelihood of the parameter θ_est. You can use it as any other NLL: Optimize it to compute maximum-likelihood estimates or MAPs, or plug it into MCMC to sample from the posterior. In our paper [3] we compute MLEs by pairing Fenrir with Optimization.jl and ForwardDiff.jl. Let's quickly explore how to do this next.","category":"page"},{"location":"tutorials/ode_parameter_inference/#Maximum-likelihood-parameter-inference","page":"Parameter Inference","title":"Maximum-likelihood parameter inference","text":"","category":"section"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"To compute a maximum-likelihood estimate (MLE), we just need to maximize theta to p(mathcalD mid theta) - that is, minimize the nll from above. We use Optimization.jl for this. First, define a loss function and create an OptimizationProblem","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"using Optimization, OptimizationOptimJL\n\nfunction loss(x, _)\n ode_params = x[begin:end-1]\n prob = remake(true_prob, p=ode_params)\n κ² = exp(x[end]) # we also optimize the diffusion parameter of the EK1\n return -fenrir_data_loglik(\n prob, EK1(smooth=true, diffusionmodel=FixedDiffusion(κ², false));\n data, observation_noise_cov=σ^2, observation_matrix=H,\n adaptive=false, dt=1e-1\n )\nend\n\nfun = OptimizationFunction(loss, Optimization.AutoForwardDiff())\noptprob = OptimizationProblem(\n fun, [θ_est..., 1e0];\n lb=[0.0, 0.0, 0.0, -10], ub=[1.0, 1.0, 5.0, 20] # lower and upper bounds\n)","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"Then, just solve it! Here we use LBFGS:","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"optsol = solve(optprob, LBFGS())\np_mle = optsol.u[1:3]\np_mle # hide","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"Success! The computed MLE is quite close to the true parameter which we used to generate the data. As a final step, let's plot the true solution, the data, and the result of the MLE:","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"plot(true_sol, color=:black, linestyle=:dash, label=[\"True Solution\" \"\"])\nscatter!(times, stack(odedata)', color=1, label=[\"Noisy Data\" \"\"])\nmle_sol = solve(remake(true_prob, p=p_mle), EK1())\nplot!(mle_sol, color=3, label=[\"MLE-parameter Solution\" \"\"])","category":"page"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"Looks good!","category":"page"},{"location":"tutorials/ode_parameter_inference/#API-Documentation","page":"Parameter Inference","title":"API Documentation","text":"","category":"section"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"For more details, see the API documentation of ProbNumDiffEq.DataLikelihoods at Data Likelihoods.","category":"page"},{"location":"tutorials/ode_parameter_inference/#References","page":"Parameter Inference","title":"References","text":"","category":"section"},{"location":"tutorials/ode_parameter_inference/","page":"Parameter Inference","title":"Parameter Inference","text":"F. Tronarp, N. Bosch and P. Hennig. Fenrir: Physics-Enhanced Regression for Initial Value Problems. In: Proceedings of the 39th International Conference on Machine Learning, Vol. 162 of Proceedings of Machine Learning Research, edited by K. Chaudhuri, S. Jegelka, L. Song, C. Szepesvari, G. Niu and S. Sabato (PMLR, 17–23 Jul 2022); pp. 21776–21794.\n\n\n\nM. Wu and M. Lysy. Data-Adaptive Probabilistic Likelihood Approximation for Ordinary Differential Equations. CoRR (2023), arXiv:2306.05566 [stat.ML].\n\n\n\n","category":"page"},{"location":"#Probabilistic-Numerical-Differential-Equation-Solvers","page":"Home","title":"Probabilistic Numerical Differential Equation Solvers","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"(Image: Banner)","category":"page"},{"location":"","page":"Home","title":"Home","text":"ProbNumDiffEq.jl provides probabilistic numerical solvers to the DifferentialEquations.jl ecosystem. The implemented ODE filters solve differential equations via Bayesian filtering and smoothing and compute not just a single point estimate of the true solution, but a posterior distribution that contains an estimate of its numerical approximation error.","category":"page"},{"location":"","page":"Home","title":"Home","text":"For a short intro video, check out our poster presentation at JuliaCon2021.","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Run Julia, enter ] to bring up Julia's package manager, and add the ProbNumDiffEq.jl package:","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> ]\n(v1.10) pkg> add ProbNumDiffEq","category":"page"},{"location":"#Getting-Started","page":"Home","title":"Getting Started","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"For a quick introduction check out the \"Solving ODEs with Probabilistic Numerics\" tutorial.","category":"page"},{"location":"#Features","page":"Home","title":"Features","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Two extended Kalman filtering-based probabilistic solvers: the explicit EK0 and semi-implicit EK1.\nAdaptive step-size selection with PI control; fully compatible with DifferentialEquations.jl's timestepping options\nOnline uncertainty calibration for multiple different diffusion models (see \"Diffusion models and calibration\")\nDense output\nSampling from the solution\nCallback support\nConvenient plotting through a Plots.jl recipe\nAutomatic differentiation via ForwardDiff.jl\nArbitrary precision via Julia's built-in arbitrary precision arithmetic\nSpecialized solvers for second-order ODEs (see Second Order ODEs and Energy Preservation)\nCompatible with DAEs in mass-matrix ODE form (see Solving DAEs with Probabilistic Numerics)\nData likelihoods for parameter-inference in ODEs (see Parameter Inference with ProbNumDiffEq.jl)","category":"page"},{"location":"#Related-packages","page":"Home","title":"Related packages","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"ProbDiffEq is similar in scope to ProbNumDiffEq.jl and it provides fast and feature-rich probabilistic ODE solvers but is implemented in Python and built on JAX.\nProbNum implements a wide range of probabilistic numerical methods, not only for ODEs but also for linear algebra, quadrature, and filtering/smoothing. It is implemented in Python and NumPy, and it focuses more on breadth and didactic purposes than on performance.","category":"page"},{"location":"implementation/#Solver-Implementation-via-OrdinaryDiffEq.jl","page":"Implementation via OrdinaryDiffEq.jl","title":"Solver Implementation via OrdinaryDiffEq.jl","text":"","category":"section"},{"location":"implementation/","page":"Implementation via OrdinaryDiffEq.jl","title":"Implementation via OrdinaryDiffEq.jl","text":"ProbNumDiffEq.jl builds directly on OrdinaryDiffEq.jl to benefit from its iterator interface, flexible step-size control, and efficient Jacobian calculations. But, this requires extending non-public APIs. This page is meant to provide an overview on which parts exactly ProbNumDiffEq.jl builds on.","category":"page"},{"location":"implementation/","page":"Implementation via OrdinaryDiffEq.jl","title":"Implementation via OrdinaryDiffEq.jl","text":"For more discussion on the pros and cons of building on OrdinaryDiffEq.jl, see this thread on discourse.","category":"page"},{"location":"implementation/#Building-on-OrdinaryDiffEq.jl","page":"Implementation via OrdinaryDiffEq.jl","title":"Building on OrdinaryDiffEq.jl","text":"","category":"section"},{"location":"implementation/","page":"Implementation via OrdinaryDiffEq.jl","title":"Implementation via OrdinaryDiffEq.jl","text":"ProbNumDiffEq.jl shares most of OrdinaryDiffEq.jl's implementation. In particular:","category":"page"},{"location":"implementation/","page":"Implementation via OrdinaryDiffEq.jl","title":"Implementation via OrdinaryDiffEq.jl","text":"OrdinaryDiffEq.__init builds the cache and the integrator, and calls OrdinaryDiffEq.initialize!\nOrdinaryDiffEq.solve! implements the actual iterator structure, with\nOrdinaryDiffEq.loopheader!\nOrdinaryDiffEq.perform_step!\nOrdinaryDiffEq.loopfooter!\nOrdinaryDiffEq.postamble!","category":"page"},{"location":"implementation/","page":"Implementation via OrdinaryDiffEq.jl","title":"Implementation via OrdinaryDiffEq.jl","text":"ProbNumDiffEq.jl builds around this structure and overloads some of the parts:","category":"page"},{"location":"implementation/","page":"Implementation via OrdinaryDiffEq.jl","title":"Implementation via OrdinaryDiffEq.jl","text":"Algorithms: EK0/EK1 <: AbstractEK <: OrdinaryDiffEq.OrdinaryDiffEqAdaptiveAlgorithm\n./src/algorithms.jl provides the algorithms themselves\n./src/alg_utils.jl implements many traits (relating to automatic differentiation, implicitness, step-size control, etc)\nCache: EKCache <: AbstractODEFilterCache <: OrdinaryDiffEq.OrdinaryDiffEqCache\n./src/caches.jl implements the cache and its main constructor: OrdinaryDiffEq.alg_cache\nInitialization and perform_step!: via OrdinaryDiffEq.initialize! and OrdinaryDiffEq.perform_step!. Implemented in ./src/perform_step.jl.\nCustom postamble by overloading OrdinaryDiffEq.postamble! (which should always call OrdinaryDiffEq._postamble!). This is where we do the \"smoothing\" of the solution. Implemented in ./src/integrator_utils.jl.\nCustom saving by overloading OrdinaryDiffEq.savevalues! (which should always call OrdinaryDiffEq._savevalues!). Implemented in ./src/integrator_utils.jl.","category":"page"},{"location":"implementation/#Building-on-DiffEqBase.jl","page":"Implementation via OrdinaryDiffEq.jl","title":"Building on DiffEqBase.jl","text":"","category":"section"},{"location":"implementation/","page":"Implementation via OrdinaryDiffEq.jl","title":"Implementation via OrdinaryDiffEq.jl","text":"DiffEqBase.__init is currently overloaded to transform OOP problems into IIP problems (in ./src/solve.jl).\nThe solution object: ProbODESolution <: AbstractProbODESolution <: DiffEqBase.AbstractODESolution\n./src/solution.jl implements the main parts. Note that the main constructor DiffEqBase.build_solution is called by OrdinaryDiffEq.__init, so OrdinaryDiffEq.jl has control over its inputs.\nMeanProbODESolution <: DiffEqBase.AbstractODESolution is a wrapper that allows handling the mean of a probabilistic ODE solution the same way one would handle any \"standard\" ODE solution, by just ignoring the covariances.\nAbstractODEFilterPosterior <: DiffEqBase.AbstractDiffEqInterpolation handles the interpolation.\nPlot recipe in ./ext/RecipesBaseExt.jl\nSampling in ./src/solution_sampling.jl\nDiffEqBase.prepare_alg(::EK1{0}); closely follows a similar function implemented in OrdinaryDiffEq.jl ./src/alg_utils.jl\nthis also required DiffEqBase.remake(::EK1)","category":"page"},{"location":"implementation/#Other-packages","page":"Implementation via OrdinaryDiffEq.jl","title":"Other packages","text":"","category":"section"},{"location":"implementation/","page":"Implementation via OrdinaryDiffEq.jl","title":"Implementation via OrdinaryDiffEq.jl","text":"DiffEqDevTools.appxtrue: We extend this function to work with ProbODESolution. This also enables DiffEqDevTools.WorkPrecision to work out of the box.","category":"page"}] } diff --git a/previews/PR328/solvers/index.html b/previews/PR328/solvers/index.html index 1adeb5117..688e1b188 100644 --- a/previews/PR328/solvers/index.html +++ b/previews/PR328/solvers/index.html @@ -4,10 +4,10 @@ prior=IWP(order), diffusionmodel=DynamicDiffusion(), initialization=TaylorModeInit(num_derivatives(prior)), - kwargs...)

Gaussian ODE filter with first-order vector field linearization.

This is a semi-implicit, L-stable ODE solver so it can handle stiffness quite well [4], and it generally produces more expressive posterior covariances than the EK0. However, as typical implicit ODE solvers it scales cubically with the ODE dimension [5], so if you're solving a high-dimensional non-stiff problem you might want to give the EK0 a try.

Arguments

Some additional kwargs relating to implicit solvers are supported; check out DifferentialEquations.jl's Extra Options page. Right now, we support autodiff, chunk_size, and diff_type. In particular, autodiff=false can come in handy to use finite differences instead of ForwardDiff.jl to compute Jacobians.

Examples

julia> solve(prob, EK1())

References

source
ProbNumDiffEq.EK0 — Type
EK0(; order=3,
+      kwargs...)

Gaussian ODE filter with first-order vector field linearization.

This is a semi-implicit, L-stable ODE solver so it can handle stiffness quite well [4], and it generally produces more expressive posterior covariances than the EK0. However, as typical implicit ODE solvers it scales cubically with the ODE dimension [5], so if you're solving a high-dimensional non-stiff problem you might want to give the EK0 a try.

Arguments

  • order::Integer: Order of the integrated Wiener process (IWP) prior.
  • smooth::Bool: Turn smoothing on/off; smoothing is required for dense output.
  • prior::AbstractGaussMarkovProcess: Prior to be used by the ODE filter. By default, uses a 3-times integrated Wiener process prior IWP(3). See also: Priors.
  • diffusionmodel::ProbNumDiffEq.AbstractDiffusion: See Diffusion models and calibration.
  • initialization::ProbNumDiffEq.InitializationScheme: See Initialization.

Some additional kwargs relating to implicit solvers are supported; check out DifferentialEquations.jl's Extra Options page. Right now, we support autodiff, chunk_size, and diff_type. In particular, autodiff=false can come in handy to use finite differences instead of ForwardDiff.jl to compute Jacobians.

Examples

julia> solve(prob, EK1())

References

source
ProbNumDiffEq.EK0 — Type
EK0(; order=3,
       smooth=true,
       prior=IWP(order),
       diffusionmodel=DynamicDiffusion(),
-      initialization=TaylorModeInit(num_derivatives(prior)))

Gaussian ODE filter with zeroth-order vector field linearization.

This is an explicit ODE solver. It is fast and scales well to high-dimensional problems [5], but it is not L-stable [4]. So for stiff problems, use the EK1.

Whenever possible this solver will use a Kronecker-factored implementation to achieve its linear scaling and to get the best runtimes. This can currently be done only with an IWP prior (default), with a scalar diffusion model (either DynamicDiffusion or FixedDiffusion). For other configurations the solver falls back to a dense implementation which scales cubically with the problem size.

Arguments

  • order::Integer: Order of the integrated Wiener process (IWP) prior.
  • smooth::Bool: Turn smoothing on/off; smoothing is required for dense output.
  • prior::AbstractGaussMarkovProcess: Prior to be used by the ODE filter. By default, uses a 3-times integrated Wiener process prior IWP(3). See also: Priors.
  • diffusionmodel::ProbNumDiffEq.AbstractDiffusion: See Diffusion models and calibration.
  • initialization::ProbNumDiffEq.InitializationScheme: See Initialization.

Examples

julia> solve(prob, EK0())

References

source

Probabilistic Exponential Integrators

ProbNumDiffEq.ExpEK — Function
ExpEK(; L, order=3, kwargs...)

Probabilistic exponential integrator

Probabilistic exponential integrators are a class of integrators for semi-linear stiff ODEs that provide improved stability by essentially solving the linear part of the ODE exactly. In probabilistic numerics, this amounts to including the linear part into the prior model of the solver.

ExpEK is therefore just a short-hand for EK0 with IOUP prior:

ExpEK(; order=3, L, kwargs...) = EK0(; prior=IOUP(order, L), kwargs...)

See also RosenbrockExpEK, EK0, EK1.

Arguments

See EK0 for available keyword arguments.

Examples

julia> prob = ODEProblem((du, u, p, t) -> (@. du = - u + sin(u)), [1.0], (0.0, 10.0))
-julia> solve(prob, ExpEK(L=-1))

Reference

  • [2] Bosch et al, "Probabilistic Exponential Integrators", arXiv (2021)
source
ProbNumDiffEq.RosenbrockExpEK — Function
RosenbrockExpEK(; order=3, kwargs...)

Probabilistic Rosenbrock-type exponential integrator

A probabilistic exponential integrator similar to ExpEK, but with automatic linearization along the mean numerical solution. This brings the advantage that the linearity does not need to be specified manually, and the more accurate local linearization can sometimes also improve stability; but since the "prior" is adjusted at each step the probabilistic interpretation becomes more complicated.

RosenbrockExpEK is just a short-hand for EK1 with locally-updated IOUP prior:

RosenbrockExpEK(; order=3, kwargs...) = EK1(; prior=IOUP(order, update_rate_parameter=true), kwargs...)

See also ExpEK, EK0, EK1.

Arguments

See EK1 for available keyword arguments.

Examples

julia> prob = ODEProblem((du, u, p, t) -> (@. du = - u + sin(u)), [1.0], (0.0, 10.0))
-julia> solve(prob, RosenbrockExpEK())

Reference

  • [2] Bosch et al, "Probabilistic Exponential Integrators", NeurIPS (2023)
source

References

[4]
F. Tronarp, H. Kersting, S. Särkkä and P. Hennig. Probabilistic solutions to ordinary differential equations as nonlinear Bayesian filtering: a new perspective. Statistics and Computing 29, 1297–1315 (2019).
[5]
N. Krämer, N. Bosch, J. Schmidt and P. Hennig. Probabilistic ODE Solutions in Millions of Dimensions. In: Proceedings of the 39th International Conference on Machine Learning, Vol. 162 of Proceedings of Machine Learning Research, edited by K. Chaudhuri, S. Jegelka, L. Song, C. Szepesvari, G. Niu and S. Sabato (PMLR, 17–23 Jul 2022); pp. 11634–11649.
[2]
N. Bosch, P. Hennig and F. Tronarp. Probabilistic Exponential Integrators. In: Thirty-seventh Conference on Neural Information Processing Systems (2023).
+ initialization=TaylorModeInit(num_derivatives(prior)))

Gaussian ODE filter with zeroth-order vector field linearization.

This is an explicit ODE solver. It is fast and scales well to high-dimensional problems [5], but it is not L-stable [4]. So for stiff problems, use the EK1.

Whenever possible this solver will use a Kronecker-factored implementation to achieve its linear scaling and to get the best runtimes. This can currently be done only with an IWP prior (default), with a scalar diffusion model (either DynamicDiffusion or FixedDiffusion). For other configurations the solver falls back to a dense implementation which scales cubically with the problem size.

Arguments

Examples

julia> solve(prob, EK0())

References

source

Probabilistic Exponential Integrators

ProbNumDiffEq.ExpEK — Function
ExpEK(; L, order=3, kwargs...)

Probabilistic exponential integrator

Probabilistic exponential integrators are a class of integrators for semi-linear stiff ODEs that provide improved stability by essentially solving the linear part of the ODE exactly. In probabilistic numerics, this amounts to including the linear part into the prior model of the solver.

ExpEK is therefore just a short-hand for EK0 with IOUP prior:

ExpEK(; order=3, L, kwargs...) = EK0(; prior=IOUP(order, L), kwargs...)

See also RosenbrockExpEK, EK0, EK1.

Arguments

See EK0 for available keyword arguments.

Examples

julia> prob = ODEProblem((du, u, p, t) -> (@. du = - u + sin(u)), [1.0], (0.0, 10.0))
+julia> solve(prob, ExpEK(L=-1))

Reference

  • [2] Bosch et al, "Probabilistic Exponential Integrators", arXiv (2021)
source
ProbNumDiffEq.RosenbrockExpEK — Function
RosenbrockExpEK(; order=3, kwargs...)

Probabilistic Rosenbrock-type exponential integrator

A probabilistic exponential integrator similar to ExpEK, but with automatic linearization along the mean numerical solution. This brings the advantage that the linearity does not need to be specified manually, and the more accurate local linearization can sometimes also improve stability; but since the "prior" is adjusted at each step the probabilistic interpretation becomes more complicated.

RosenbrockExpEK is just a short-hand for EK1 with locally-updated IOUP prior:

RosenbrockExpEK(; order=3, kwargs...) = EK1(; prior=IOUP(order, update_rate_parameter=true), kwargs...)

See also ExpEK, EK0, EK1.

Arguments

See EK1 for available keyword arguments.

Examples

julia> prob = ODEProblem((du, u, p, t) -> (@. du = - u + sin(u)), [1.0], (0.0, 10.0))
+julia> solve(prob, RosenbrockExpEK())

Reference

  • [2] Bosch et al, "Probabilistic Exponential Integrators", NeurIPS (2023)
source

References

[4]
F. Tronarp, H. Kersting, S. Särkkä and P. Hennig. Probabilistic solutions to ordinary differential equations as nonlinear Bayesian filtering: a new perspective. Statistics and Computing 29, 1297–1315 (2019).
[5]
N. Krämer, N. Bosch, J. Schmidt and P. Hennig. Probabilistic ODE Solutions in Millions of Dimensions. In: Proceedings of the 39th International Conference on Machine Learning, Vol. 162 of Proceedings of Machine Learning Research, edited by K. Chaudhuri, S. Jegelka, L. Song, C. Szepesvari, G. Niu and S. Sabato (PMLR, 17–23 Jul 2022); pp. 11634–11649.
[2]
N. Bosch, P. Hennig and F. Tronarp. Probabilistic Exponential Integrators. In: Thirty-seventh Conference on Neural Information Processing Systems (2023).
diff --git a/previews/PR328/tutorials/dae/ed8509a6.svg b/previews/PR328/tutorials/dae/46b37c8d.svg similarity index 99% rename from previews/PR328/tutorials/dae/ed8509a6.svg rename to previews/PR328/tutorials/dae/46b37c8d.svg index 10a480353..61b18c590 100644 --- a/previews/PR328/tutorials/dae/ed8509a6.svg +++ b/previews/PR328/tutorials/dae/46b37c8d.svg @@ -1,90 +1,90 @@ - + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + - + - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + - + - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR328/tutorials/dae/1327bd77.svg b/previews/PR328/tutorials/dae/7c0b28e3.svg similarity index 99% rename from previews/PR328/tutorials/dae/1327bd77.svg rename to previews/PR328/tutorials/dae/7c0b28e3.svg index 48ec70e5e..2ad348571 100644 --- a/previews/PR328/tutorials/dae/1327bd77.svg +++ b/previews/PR328/tutorials/dae/7c0b28e3.svg @@ -1,59 +1,59 @@ - + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR328/tutorials/dae/index.html b/previews/PR328/tutorials/dae/index.html index 02697137e..a54332499 100644 --- a/previews/PR328/tutorials/dae/index.html +++ b/previews/PR328/tutorials/dae/index.html @@ -28,7 +28,7 @@ ylabel=["u₁(t)" "u₂(t)" "u₃(t)"], xlabel=["" "" "t"], denseplot=false, -)Example block output

Looks good!

Solving an Index-3 DAE directly

The following is based on the "Automatic Index Reduction of DAEs" tutorial by ModelingToolkit.jl, which demonstrates how the classic Rodas4 solver fails to solve a DAE due to the fact that it is of index 3; which is why ModelingToolkit's automatic index reduction is so useful.

It turns out that our probabilistic numerical solvers can directly solve the index-3 DAE!

First, define the pendulum problem as in the tutorial:

function pendulum!(du, u, p, t)
+)
Example block output

Looks good!

Solving an Index-3 DAE directly

The following is based on the "Automatic Index Reduction of DAEs" tutorial by ModelingToolkit.jl, which demonstrates how the classic Rodas4 solver fails to solve a DAE due to the fact that it is of index 3; which is why ModelingToolkit's automatic index reduction is so useful.

It turns out that our probabilistic numerical solvers can directly solve the index-3 DAE!

First, define the pendulum problem as in the tutorial:

function pendulum!(du, u, p, t)
     x, dx, y, dy, T = u
     g, L = p
     du[1] = dx
@@ -104,7 +104,7 @@
  [0.957940002319977, -0.6813737682784675, -0.28696857832357914, -2.2722969869869667, -8.475874423432991]
  [0.9513764315907425, -0.7580420941403236, -0.3080306846666298, -2.337925795788078, -9.10937542703355]
  [0.9455309355620407, -0.8238465818165532, -0.325532285560256, -2.388489111563977, -9.638156836814543]
- [0.944566419026587, -0.8345069039015303, -0.3283203924231612, -2.3962208016104234, -9.722642388563708]

Nope! The EK1 is able to solve the index-3 DAE directly. Pretty cool!

plot(sol)
Example block output

Is index-reduction still worth it?

The point of the "Automatic Index Reduction of DAEs" tutorial is to demonstrate ModelingToolkit's utility for automatic index reduction, which enables the classic implicit Runge-Kutta solvers such as Rodas5 to solve this DAE. Let's see if that still helps in this context here.

First, modelingtoolkitize the problem:

traced_sys = modelingtoolkitize(pendulum_prob)

\[ \begin{align} + [0.944566419026587, -0.8345069039015303, -0.3283203924231612, -2.3962208016104234, -9.722642388563708]

Nope! The EK1 is able to solve the index-3 DAE directly. Pretty cool!

plot(sol)
Example block output

Is index-reduction still worth it?

The point of the "Automatic Index Reduction of DAEs" tutorial is to demonstrate ModelingToolkit's utility for automatic index reduction, which enables the classic implicit Runge-Kutta solvers such as Rodas5 to solve this DAE. Let's see if that still helps in this context here.

First, modelingtoolkitize the problem:

traced_sys = modelingtoolkitize(pendulum_prob)

\[ \begin{align} \frac{\mathrm{d} x_1\left( t \right)}{\mathrm{d}t} &= x_2\left( t \right) \\ \frac{\mathrm{d} x_2\left( t \right)}{\mathrm{d}t} &= x_1\left( t \right) x_5\left( t \right) \\ \frac{\mathrm{d} x_3\left( t \right)}{\mathrm{d}t} &= x_4\left( t \right) \\ @@ -134,4 +134,4 @@ │ sol1_final_error = 0.010930749098469738 │ sol1_f_evals = 1085 │ sol3_final_error = 0.06122741667364596 -└ sol3_f_evals = 1409

The error for the index-1 DAE solve is much lower. Thus it seems that, even if the index-3 DAE could also be solved directly, index lowering might still be beneficial when solving DAEs with the EK1!

References

[1]
N. Bosch, F. Tronarp and P. Hennig. Pick-and-Mix Information Operators for Probabilistic ODE Solvers. In: Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, Vol. 151 of Proceedings of Machine Learning Research, edited by G. Camps-Valls, F. J. Ruiz and I. Valera (PMLR, 28–30 Mar 2022); pp. 10015–10027.
+└ sol3_f_evals = 1409

The error for the index-1 DAE solve is much lower. Thus it seems that, even if the index-3 DAE could also be solved directly, index lowering might still be beneficial when solving DAEs with the EK1!

References

[1]
N. Bosch, F. Tronarp and P. Hennig. Pick-and-Mix Information Operators for Probabilistic ODE Solvers. In: Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, Vol. 151 of Proceedings of Machine Learning Research, edited by G. Camps-Valls, F. J. Ruiz and I. Valera (PMLR, 28–30 Mar 2022); pp. 10015–10027.
diff --git a/previews/PR328/tutorials/dynamical_odes/7767810f.svg b/previews/PR328/tutorials/dynamical_odes/154fc451.svg similarity index 97% rename from previews/PR328/tutorials/dynamical_odes/7767810f.svg rename to previews/PR328/tutorials/dynamical_odes/154fc451.svg index a31713e7c..52f097f46 100644 --- a/previews/PR328/tutorials/dynamical_odes/7767810f.svg +++ b/previews/PR328/tutorials/dynamical_odes/154fc451.svg @@ -1,46 +1,46 @@ - + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR328/tutorials/dynamical_odes/6c78f6c9.svg b/previews/PR328/tutorials/dynamical_odes/53c6f559.svg similarity index 99% rename from previews/PR328/tutorials/dynamical_odes/6c78f6c9.svg rename to previews/PR328/tutorials/dynamical_odes/53c6f559.svg index fc93252a4..29ad3f8dd 100644 --- a/previews/PR328/tutorials/dynamical_odes/6c78f6c9.svg +++ b/previews/PR328/tutorials/dynamical_odes/53c6f559.svg @@ -1,52 +1,52 @@ - + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR328/tutorials/dynamical_odes/af6ceacf.svg b/previews/PR328/tutorials/dynamical_odes/d44a6687.svg similarity index 98% rename from previews/PR328/tutorials/dynamical_odes/af6ceacf.svg rename to previews/PR328/tutorials/dynamical_odes/d44a6687.svg index 4251d5f77..7775318bd 100644 --- a/previews/PR328/tutorials/dynamical_odes/af6ceacf.svg +++ b/previews/PR328/tutorials/dynamical_odes/d44a6687.svg @@ -1,50 +1,50 @@ - + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR328/tutorials/dynamical_odes/aa8f8ac6.svg b/previews/PR328/tutorials/dynamical_odes/d5b10604.svg similarity index 98% rename from previews/PR328/tutorials/dynamical_odes/aa8f8ac6.svg rename to previews/PR328/tutorials/dynamical_odes/d5b10604.svg index 6f9d56879..fb183ced5 100644 --- a/previews/PR328/tutorials/dynamical_odes/aa8f8ac6.svg +++ b/previews/PR328/tutorials/dynamical_odes/d5b10604.svg @@ -1,46 +1,46 @@ - + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR328/tutorials/dynamical_odes/index.html b/previews/PR328/tutorials/dynamical_odes/index.html index e7777dc9f..5a40e21c0 100644 --- a/previews/PR328/tutorials/dynamical_odes/index.html +++ b/previews/PR328/tutorials/dynamical_odes/index.html @@ -19,13 +19,13 @@ tspan = (0.0, 100.0) prob = ODEProblem(Hénon_Heiles, [du0; u0], tspan) sol = solve(prob, EK1()); -plot(sol, idxs=(3, 4)) # where `idxs=(3,4)` is used to plot x agains yExample block output

Solving the second-order ODE directly

Instead of first transforming the problem, we can also solve it directly as a second-order ODE, by defining it as a SecondOrderODEProblem.

Note

The SecondOrderODEProblem type is not defined in ProbNumDiffEq.jl but is provided by SciMLBase.jl. For more information, check out the DifferentialEquations.jl documentation on Dynamical, Hamiltonian and 2nd Order ODE Problems.

function Hénon_Heiles2(ddu, du, u, p, t)
+plot(sol, idxs=(3, 4)) # where `idxs=(3,4)` is used to plot x agains y
Example block output

Solving the second-order ODE directly

Instead of first transforming the problem, we can also solve it directly as a second-order ODE, by defining it as a SecondOrderODEProblem.

Note

The SecondOrderODEProblem type is not defined in ProbNumDiffEq.jl but is provided by SciMLBase.jl. For more information, check out the DifferentialEquations.jl documentation on Dynamical, Hamiltonian and 2nd Order ODE Problems.

function Hénon_Heiles2(ddu, du, u, p, t)
     ddu[1] = -u[1] - 2 * u[1] * u[2]
     ddu[2] = u[2]^2 - u[2] - u[1]^2
 end
 prob2 = SecondOrderODEProblem(Hénon_Heiles2, du0, u0, tspan)
 sol2 = solve(prob2, EK1());
-plot(sol2, idxs=(3, 4))
Example block output

Benchmark: Solving second order ODEs is faster

Solving second-order ODEs is not just a matter of convenience - in fact, SciMLBase's SecondOrderODEProblem is neatly designed in such a way that all the classic solvers from OrdinaryDiffEq.jl can handle it by solving the corresponding first-order ODE. But, transforming the ODE to first order increases the dimensionality of the problem, and comes therefore at increased computational cost; this also motivates classic specialized solvers for second-order ODEs.

The probabilistic numerical solvers from ProbNumDiffEq.jl have the same internal state representation for first and second order ODEs; all that changes is the measurement model [1]. As a result, we can use the EK1 both for first and second order ODEs, but it automatically specializes on the latter to provide a 2x performance boost:

julia> using BenchmarkTools
+plot(sol2, idxs=(3, 4))
Example block output

Benchmark: Solving second order ODEs is faster

Solving second-order ODEs is not just a matter of convenience - in fact, SciMLBase's SecondOrderODEProblem is neatly designed in such a way that all the classic solvers from OrdinaryDiffEq.jl can handle it by solving the corresponding first-order ODE. But, transforming the ODE to first order increases the dimensionality of the problem, and comes therefore at increased computational cost; this also motivates classic specialized solvers for second-order ODEs.

The probabilistic numerical solvers from ProbNumDiffEq.jl have the same internal state representation for first and second order ODEs; all that changes is the measurement model [1]. As a result, we can use the EK1 both for first and second order ODEs, but it automatically specializes on the latter to provide a 2x performance boost:

julia> using BenchmarkTools
 
 julia> @btime solve(prob, EK1(order=3), adaptive=false, dt=1e-2);
   317.336 ms (140561 allocations: 140.41 MiB)
@@ -39,7 +39,7 @@
 E(dx, dy, x, y) = PotentialEnergy(x, y) + KineticEnergy(dx, dy)
 E(u) = E(u...); # convenient shorthand
E (generic function with 2 methods)

Let's have a look at how the total energy changes over time when we numerically simulate the Hénon-Heiles model over a long period of time: Standard solve

longprob = remake(prob2, tspan=(0.0, 1e3))
 longsol = solve(longprob, EK1(smooth=false), dense=false)
-plot(longsol.t, E.(longsol.u))
Example block output

It visibly loses energy over time, from an initial 0.12967 to a final 0.12899. Let's fix this to get a physically more meaningful solution.

Energy preservation with the ManifoldUpdate callback

In the language of ODE filters, preserving energy over time amounts to just another measurement model [1]. The most convenient way of updating on this additional zero measurement with ProbNumDiffEq.jl is with the ManifoldUpdate callback.

Note

The ManifoldUpdate callback can be thought of a probabilistic counterpart to the ManifoldProjection callback provided by DiffEqCallbacks.jl.

To do so, first define a (vector-valued) residual function, here chosen to be the difference between the current energy and the initial energy, and build a ManifoldUpdate callback

residual(u) = [E(u) - E(du0..., u0...)]
+plot(longsol.t, E.(longsol.u))
Example block output

It visibly loses energy over time, from an initial 0.12967 to a final 0.12899. Let's fix this to get a physically more meaningful solution.

Energy preservation with the ManifoldUpdate callback

In the language of ODE filters, preserving energy over time amounts to just another measurement model [1]. The most convenient way of updating on this additional zero measurement with ProbNumDiffEq.jl is with the ManifoldUpdate callback.

Note

The ManifoldUpdate callback can be thought of a probabilistic counterpart to the ManifoldProjection callback provided by DiffEqCallbacks.jl.

To do so, first define a (vector-valued) residual function, here chosen to be the difference between the current energy and the initial energy, and build a ManifoldUpdate callback

residual(u) = [E(u) - E(du0..., u0...)]
 cb = ManifoldUpdate(residual)
DiscreteCallback{ProbNumDiffEq.var"#condition#140", ProbNumDiffEq.var"#affect!#141"{Int64, Float64, Float64, typeof(Main.residual)}, typeof(SciMLBase.INITIALIZE_DEFAULT), typeof(SciMLBase.FINALIZE_DEFAULT), Nothing}(ProbNumDiffEq.var"#condition#140"(), ProbNumDiffEq.var"#affect!#141"{Int64, Float64, Float64, typeof(Main.residual)}(100, 1.0e-25, 1.0e-15, Main.residual), SciMLBase.INITIALIZE_DEFAULT, SciMLBase.FINALIZE_DEFAULT, Bool[1, 1], nothing)

Then, solve the ODE with this callback

longsol_preserving = solve(longprob, EK1(smooth=false), dense=false, callback=cb)
 plot(longsol.t, E.(longsol.u))
-plot!(longsol_preserving.t, E.(longsol_preserving.u))
Example block output

Voilà! With the ManifoldUpdate callback we could preserve the energy over time and obtain a more truthful probabilistic numerical long-term simulation of the Hénon-Heiles model.

References

[1]
N. Bosch, F. Tronarp and P. Hennig. Pick-and-Mix Information Operators for Probabilistic ODE Solvers. In: Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, Vol. 151 of Proceedings of Machine Learning Research, edited by G. Camps-Valls, F. J. Ruiz and I. Valera (PMLR, 28–30 Mar 2022); pp. 10015–10027.
+plot!(longsol_preserving.t, E.(longsol_preserving.u))Example block output

Voilà! With the ManifoldUpdate callback we could preserve the energy over time and obtain a more truthful probabilistic numerical long-term simulation of the Hénon-Heiles model.

References

[1]
N. Bosch, F. Tronarp and P. Hennig. Pick-and-Mix Information Operators for Probabilistic ODE Solvers. In: Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, Vol. 151 of Proceedings of Machine Learning Research, edited by G. Camps-Valls, F. J. Ruiz and I. Valera (PMLR, 28–30 Mar 2022); pp. 10015–10027.
diff --git a/previews/PR328/tutorials/exponential_integrators/cac3ee17.svg b/previews/PR328/tutorials/exponential_integrators/1bc16c00.svg similarity index 98% rename from previews/PR328/tutorials/exponential_integrators/cac3ee17.svg rename to previews/PR328/tutorials/exponential_integrators/1bc16c00.svg index 90535df59..76f72e10d 100644 --- a/previews/PR328/tutorials/exponential_integrators/cac3ee17.svg +++ b/previews/PR328/tutorials/exponential_integrators/1bc16c00.svg @@ -1,55 +1,55 @@ - + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR328/tutorials/exponential_integrators/d484205a.svg b/previews/PR328/tutorials/exponential_integrators/3795413a.svg similarity index 99% rename from previews/PR328/tutorials/exponential_integrators/d484205a.svg rename to previews/PR328/tutorials/exponential_integrators/3795413a.svg index 79ce5d95b..da260b740 100644 --- a/previews/PR328/tutorials/exponential_integrators/d484205a.svg +++ b/previews/PR328/tutorials/exponential_integrators/3795413a.svg @@ -1,51 +1,51 @@ - + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR328/tutorials/exponential_integrators/66c9efc4.svg b/previews/PR328/tutorials/exponential_integrators/3c765308.svg similarity index 98% rename from previews/PR328/tutorials/exponential_integrators/66c9efc4.svg rename to previews/PR328/tutorials/exponential_integrators/3c765308.svg index 3a0e3a5e9..293842cab 100644 --- a/previews/PR328/tutorials/exponential_integrators/66c9efc4.svg +++ b/previews/PR328/tutorials/exponential_integrators/3c765308.svg @@ -1,65 +1,65 @@ - + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR328/tutorials/exponential_integrators/f7bdb864.svg b/previews/PR328/tutorials/exponential_integrators/ad789e48.svg similarity index 98% rename from previews/PR328/tutorials/exponential_integrators/f7bdb864.svg rename to previews/PR328/tutorials/exponential_integrators/ad789e48.svg index c7a26e861..f14b2f793 100644 --- a/previews/PR328/tutorials/exponential_integrators/f7bdb864.svg +++ b/previews/PR328/tutorials/exponential_integrators/ad789e48.svg @@ -1,63 +1,63 @@ - + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR328/tutorials/exponential_integrators/index.html b/previews/PR328/tutorials/exponential_integrators/index.html index 11c71c7cd..3e5ea5854 100644 --- a/previews/PR328/tutorials/exponential_integrators/index.html +++ b/previews/PR328/tutorials/exponential_integrators/index.html @@ -12,7 +12,7 @@ prob = ODEProblem(f, u0, tspan) ref = solve(prob, EK1(), abstol=1e-10, reltol=1e-10) -plot(ref, color=:black, linestyle=:dash, label="Reference")Example block output

But for fixed (large) step sizes this ODE is more challenging: The explicit EK0 method oscillates and diverges due to the stiffness of the ODE, and the semi-implicit EK1 method is stable but the solution is not very accurate.

STEPSIZE = 4
+plot(ref, color=:black, linestyle=:dash, label="Reference")
Example block output

But for fixed (large) step sizes this ODE is more challenging: The explicit EK0 method oscillates and diverges due to the stiffness of the ODE, and the semi-implicit EK1 method is stable but the solution is not very accurate.

STEPSIZE = 4
 DM = FixedDiffusion() # recommended for fixed steps
 
 # we don't smooth the EK0 here to show the oscillations more clearly
@@ -22,14 +22,14 @@
 plot(ylims=(0.3, 1.05))
 plot!(ref, color=:black, linestyle=:dash, label="Reference")
 plot!(sol0, denseplot=false, marker=:o, markersize=2, label="EK0", color=1)
-plot!(sol1, denseplot=false, marker=:o, markersize=2, label="EK1", color=2)
Example block output

Probabilistic exponential integrators leverage the semi-linearity of the ODE to compute more accurate solutions for the same fixed step size. You can use either the ExpEK method and provide the linear part (with the keyword argument L), or the RosenbrockExpEK to automatically linearize along the mean of the numerical solution:

sol_exp = solve(prob, ExpEK(L=-1, diffusionmodel=DM), adaptive=false, dt=STEPSIZE)
+plot!(sol1, denseplot=false, marker=:o, markersize=2, label="EK1", color=2)
Example block output

Probabilistic exponential integrators leverage the semi-linearity of the ODE to compute more accurate solutions for the same fixed step size. You can use either the ExpEK method and provide the linear part (with the keyword argument L), or the RosenbrockExpEK to automatically linearize along the mean of the numerical solution:

sol_exp = solve(prob, ExpEK(L=-1, diffusionmodel=DM), adaptive=false, dt=STEPSIZE)
 sol_ros = solve(prob, RosenbrockExpEK(diffusionmodel=DM), adaptive=false, dt=STEPSIZE)
 
 plot(ylims=(0.3, 1.05))
 plot!(ref, color=:black, linestyle=:dash, label="Reference")
 plot!(sol_exp, denseplot=false, marker=:o, markersize=2, label="ExpEK", color=3)
-plot!(sol_ros, denseplot=false, marker=:o, markersize=2, label="RosenbrockExpEK", color=4)
Example block output

The solutions are indeed much more accurate than those of the standard EK1, for the same fixed step size!

Background: Integrated Ornstein-Uhlenbeck priors

Probabilistic exponential integrators "solve the linear part exactly" by including it into the prior model of the solver. Namely, the solver chooses a (q-times) integrated Ornstein-Uhlenbeck prior with rate parameter equal to the linearity. The ExpEK solver is just a short-hand for an EK0 with appropriate prior:

julia> ExpEK(order=3, L=-1) == EK0(prior=IOUP(3, -1))true

Similarly, the RosenbrockExpEK solver is also just a short-hand:

julia> RosenbrockExpEK(order=3) == EK1(prior=IOUP(3, update_rate_parameter=true))true

This means that you can also construct other probabilistic exponential integrators by hand! In this example the EK1 with IOUP prior with rate parameter -1 performs extremely well:

sol_expek1 = solve(prob, EK1(prior=IOUP(3, -1), diffusionmodel=DM), adaptive=false, dt=STEPSIZE)
+plot!(sol_ros, denseplot=false, marker=:o, markersize=2, label="RosenbrockExpEK", color=4)
Example block output

The solutions are indeed much more accurate than those of the standard EK1, for the same fixed step size!

Background: Integrated Ornstein-Uhlenbeck priors

Probabilistic exponential integrators "solve the linear part exactly" by including it into the prior model of the solver. Namely, the solver chooses a (q-times) integrated Ornstein-Uhlenbeck prior with rate parameter equal to the linearity. The ExpEK solver is just a short-hand for an EK0 with appropriate prior:

julia> ExpEK(order=3, L=-1) == EK0(prior=IOUP(3, -1))true

Similarly, the RosenbrockExpEK solver is also just a short-hand:

julia> RosenbrockExpEK(order=3) == EK1(prior=IOUP(3, update_rate_parameter=true))true

This means that you can also construct other probabilistic exponential integrators by hand! In this example the EK1 with IOUP prior with rate parameter -1 performs extremely well:

sol_expek1 = solve(prob, EK1(prior=IOUP(3, -1), diffusionmodel=DM), adaptive=false, dt=STEPSIZE)
 
 plot(ylims=(0.3, 1.05))
 plot!(ref, color=:black, linestyle=:dash, label="Reference")
-plot!(sol_expek1, denseplot=false, marker=:o, markersize=2, label="EK1 + IOUP")
Example block output

References

[2]
N. Bosch, P. Hennig and F. Tronarp. Probabilistic Exponential Integrators. In: Thirty-seventh Conference on Neural Information Processing Systems (2023).
+plot!(sol_expek1, denseplot=false, marker=:o, markersize=2, label="EK1 + IOUP")Example block output

References

[2]
N. Bosch, P. Hennig and F. Tronarp. Probabilistic Exponential Integrators. In: Thirty-seventh Conference on Neural Information Processing Systems (2023).
diff --git a/previews/PR328/tutorials/getting_started/e630abe2.svg b/previews/PR328/tutorials/getting_started/b5080a08.svg similarity index 98% rename from previews/PR328/tutorials/getting_started/e630abe2.svg rename to previews/PR328/tutorials/getting_started/b5080a08.svg index e78f46775..a42244d60 100644 --- a/previews/PR328/tutorials/getting_started/e630abe2.svg +++ b/previews/PR328/tutorials/getting_started/b5080a08.svg @@ -1,50 +1,50 @@ - + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR328/tutorials/getting_started/d1754c2f.svg b/previews/PR328/tutorials/getting_started/b59814cc.svg similarity index 91% rename from previews/PR328/tutorials/getting_started/d1754c2f.svg rename to previews/PR328/tutorials/getting_started/b59814cc.svg index 069c2433d..d4f379b93 100644 --- a/previews/PR328/tutorials/getting_started/d1754c2f.svg +++ b/previews/PR328/tutorials/getting_started/b59814cc.svg @@ -1,54 +1,54 @@ - + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR328/tutorials/getting_started/57d85163.svg b/previews/PR328/tutorials/getting_started/ec35f3db.svg similarity index 98% rename from previews/PR328/tutorials/getting_started/57d85163.svg rename to previews/PR328/tutorials/getting_started/ec35f3db.svg index 748b6df6e..e1db31f64 100644 --- a/previews/PR328/tutorials/getting_started/57d85163.svg +++ b/previews/PR328/tutorials/getting_started/ec35f3db.svg @@ -1,50 +1,50 @@ - + - + - + - + - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR328/tutorials/getting_started/index.html b/previews/PR328/tutorials/getting_started/index.html index 3367e09bf..530ea982b 100644 --- a/previews/PR328/tutorials/getting_started/index.html +++ b/previews/PR328/tutorials/getting_started/index.html @@ -15,7 +15,7 @@ prob = ODEProblem(fitz, u0, tspan, p) sol = solve(prob, EK1()) -plot(sol)Example block output

Step 1: Define the problem

First, import ProbNumDiffEq.jl

using ProbNumDiffEq

Then, set up the ODEProblem exactly as you would in DifferentialEquations.jl. Define the vector field

function fitz(du, u, p, t)
+plot(sol)
Example block output

Step 1: Define the problem

First, import ProbNumDiffEq.jl

using ProbNumDiffEq

Then, set up the ODEProblem exactly as you would in DifferentialEquations.jl. Define the vector field

function fitz(du, u, p, t)
     a, b, c = p
     du[1] = c * (u[1] - u[1]^3 / 3 + u[2])
     du[2] = -(1 / c) * (u[1] - a - b * u[2])
@@ -66,13 +66,13 @@
  [2.0350861534147566, 0.7169394922557]
  [2.0210436519484087, 0.6717007174964914]
  [2.0104405118668827, 0.6383145073764079]

That's it! we just computed a probabilistic numerical ODE solution!

Step 3: Analyze the solution

Let's plot the result with Plots.jl.

using Plots
-plot(sol)
Example block output

Looks good! Looks like the EK1 managed to solve the Fitzhugh-Nagumo problem quite well.

Tip

To learn more about plotting ODE solutions, check out the plotting tutorial for DifferentialEquations.jl + Plots.jl provided here. Most of that works exactly as expected with ProbNumDiffEq.jl.

Plot the probabilistic error estimates

The plot above looks like a standard ODE solution – but it's not! The numerical errors are just so small that we can't see them in the plot, and the probabilistic error estimates are too. We can visualize them by plotting the errors and error estimates directly:

using OrdinaryDiffEq, Statistics
+plot(sol)
Example block output

Looks good! Looks like the EK1 managed to solve the Fitzhugh-Nagumo problem quite well.

Tip

To learn more about plotting ODE solutions, check out the plotting tutorial for DifferentialEquations.jl + Plots.jl provided here. Most of that works exactly as expected with ProbNumDiffEq.jl.

Plot the probabilistic error estimates

The plot above looks like a standard ODE solution – but it's not! The numerical errors are just so small that we can't see them in the plot, and the probabilistic error estimates are too. We can visualize them by plotting the errors and error estimates directly:

using OrdinaryDiffEq, Statistics
 reference = solve(prob, Vern9(), abstol=1e-9, reltol=1e-9, saveat=sol.t)
 errors = reduce(hcat, mean.(sol.pu) .- reference.u)'
 error_estimates = reduce(hcat, std.(sol.pu))'
 plot(sol.t, errors, label="error", color=[1 2], xlabel="t", ylabel="err")
 plot!(sol.t, zero(errors), ribbon=3error_estimates, label="error estimate",
-      color=[1 2], alpha=0.2)
Example block output

More about the ProbabilisticODESolution

The solution object returned by ProbNumDiffEq.jl mostly behaves just like any other ODESolution in DifferentialEquations.jl – with some added uncertainties and related functionality on top. The ProbabilisticODESolution can be indexed with

julia> sol.u[1]2-element Vector{Float64}:
+      color=[1 2], alpha=0.2)
Example block output

More about the ProbabilisticODESolution

The solution object returned by ProbNumDiffEq.jl mostly behaves just like any other ODESolution in DifferentialEquations.jl – with some added uncertainties and related functionality on top. The ProbabilisticODESolution can be indexed with

julia> sol.u[1]2-element Vector{Float64}:
  -1.0
   1.0
julia> sol.u[end]2-element Vector{Float64}:
  2.0104405118668827
@@ -97,4 +97,4 @@
   Σ=2x2 PSDMatrix{Float64, Matrix{Float64}}; R=[-3.2083544077891136e-5 -4.7899698453730076e-6; 0.0 -2.4206415622162586e-5; 0.0 0.0; 0.0 0.0; 0.0 0.0; 0.0 0.0; 0.0 0.0; 0.0 0.0]
 )
julia> mean(sol(0.45))2-element Vector{Float64}:
  -0.2773821283086794
-  1.1675659430627088

Next steps

Check out one of the other tutorials:

+ 1.1675659430627088

Next steps

Check out one of the other tutorials:

diff --git a/previews/PR328/tutorials/ode_parameter_inference/25364dc9.svg b/previews/PR328/tutorials/ode_parameter_inference/25364dc9.svg new file mode 100644 index 000000000..4e0e2c401 --- /dev/null +++ b/previews/PR328/tutorials/ode_parameter_inference/25364dc9.svg @@ -0,0 +1,92 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR328/tutorials/ode_parameter_inference/4885306a.svg b/previews/PR328/tutorials/ode_parameter_inference/4885306a.svg deleted file mode 100644 index 6ed8d0ff5..000000000 --- a/previews/PR328/tutorials/ode_parameter_inference/4885306a.svg +++ /dev/null @@ -1,87 +0,0 @@ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - diff --git a/previews/PR328/tutorials/ode_parameter_inference/6310a88d.svg b/previews/PR328/tutorials/ode_parameter_inference/6310a88d.svg new file mode 100644 index 000000000..84af94d8d --- /dev/null +++ b/previews/PR328/tutorials/ode_parameter_inference/6310a88d.svg @@ -0,0 +1,92 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR328/tutorials/ode_parameter_inference/772b8fd5.svg b/previews/PR328/tutorials/ode_parameter_inference/772b8fd5.svg deleted file mode 100644 index bdd3c1aa3..000000000 --- a/previews/PR328/tutorials/ode_parameter_inference/772b8fd5.svg +++ /dev/null @@ -1,92 +0,0 @@ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - diff --git a/previews/PR328/tutorials/ode_parameter_inference/790334fc.svg b/previews/PR328/tutorials/ode_parameter_inference/790334fc.svg deleted file mode 100644 index 45ce94fff..000000000 --- a/previews/PR328/tutorials/ode_parameter_inference/790334fc.svg +++ /dev/null @@ -1,92 +0,0 @@ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - diff --git a/previews/PR328/tutorials/ode_parameter_inference/d01fa82d.svg b/previews/PR328/tutorials/ode_parameter_inference/d01fa82d.svg new file mode 100644 index 000000000..aaf80d372 --- /dev/null +++ b/previews/PR328/tutorials/ode_parameter_inference/d01fa82d.svg @@ -0,0 +1,87 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR328/tutorials/ode_parameter_inference/index.html b/previews/PR328/tutorials/ode_parameter_inference/index.html index 967525064..a01af84d0 100644 --- a/previews/PR328/tutorials/ode_parameter_inference/index.html +++ b/previews/PR328/tutorials/ode_parameter_inference/index.html @@ -26,18 +26,18 @@ odedata = [H*true_sol(t) .+ σ * randn() for t in times] plot(true_sol, color=:black, linestyle=:dash, label=["True Solution" ""]) -scatter!(times, stack(odedata)', color=1, label=["Noisy Data" ""])Example block output

Our goal is then to recover the true parameter p (and thus also the true trajectory plotted above) the noisy data.

Computing the negative log-likelihood

To do parameter inference - be it maximum-likelihod, maximum a posteriori, or full Bayesian inference with MCMC - we need to evaluate the likelihood of given a parameter estimate $\theta_\text{est}$, which corresponds to the probability of the data under the trajectory returned by the ODE solver

θ_est = (0.1, 0.1, 2.0)
+scatter!(times, stack(odedata)',  color=1, label=["Noisy Data" ""])
Example block output

Our goal is then to recover the true parameter p (and thus also the true trajectory plotted above) the noisy data.

Computing the negative log-likelihood

To do parameter inference - be it maximum-likelihod, maximum a posteriori, or full Bayesian inference with MCMC - we need to evaluate the likelihood of given a parameter estimate $\theta_\text{est}$, which corresponds to the probability of the data under the trajectory returned by the ODE solver

θ_est = (0.1, 0.1, 2.0)
 prob = remake(true_prob, p=θ_est)
 plot(true_sol, color=:black, linestyle=:dash, label=["True Solution" ""])
 scatter!(times, stack(odedata)', color=1, label=["Noisy Data" ""])
 sol = solve(prob, EK1(), adaptive=false, dt=1e-1)
-plot!(sol, color=2, label=["Numerical solution for θ_est" ""])
Example block output

This quantity can be computed in multiple ways; see Data Likelihoods. Here we use ProbNumDiffEq.DataLikelihoods.fenrir_data_loglik:

using ProbNumDiffEq.DataLikelihoods
+plot!(sol, color=2, label=["Numerical solution for θ_est" ""])
Example block output

This quantity can be computed in multiple ways; see Data Likelihoods. Here we use ProbNumDiffEq.DataLikelihoods.fenrir_data_loglik:

using ProbNumDiffEq.DataLikelihoods
 
 data = (t=times, u=odedata)
 nll = -fenrir_data_loglik(
     prob, EK1(smooth=true);
     data, observation_noise_cov=σ^2, observation_matrix=H,
-    adaptive=false, dt=1e-1)
6443.06779345845

This is the negative marginal log-likelihood of the parameter θ_est. You can use it as any other NLL: Optimize it to compute maximum-likelihood estimates or MAPs, or plug it into MCMC to sample from the posterior. In our paper [3] we compute MLEs by pairing Fenrir with Optimization.jl and ForwardDiff.jl. Let's quickly explore how to do this next.

Maximum-likelihood parameter inference

To compute a maximum-likelihood estimate (MLE), we just need to maximize $\theta \to p(\mathcal{D} \mid \theta)$ - that is, minimize the nll from above. We use Optimization.jl for this. First, define a loss function and create an OptimizationProblem

using Optimization, OptimizationOptimJL
+    adaptive=false, dt=1e-1)
6462.452308318974

This is the negative marginal log-likelihood of the parameter θ_est. You can use it as any other NLL: Optimize it to compute maximum-likelihood estimates or MAPs, or plug it into MCMC to sample from the posterior. In our paper [3] we compute MLEs by pairing Fenrir with Optimization.jl and ForwardDiff.jl. Let's quickly explore how to do this next.

Maximum-likelihood parameter inference

To compute a maximum-likelihood estimate (MLE), we just need to maximize $\theta \to p(\mathcal{D} \mid \theta)$ - that is, minimize the nll from above. We use Optimization.jl for this. First, define a loss function and create an OptimizationProblem

using Optimization, OptimizationOptimJL
 
 function loss(x, _)
     ode_params = x[begin:end-1]
@@ -61,9 +61,9 @@
  2.0
  1.0

Then, just solve it! Here we use LBFGS:

optsol = solve(optprob, LBFGS())
 p_mle = optsol.u[1:3]
3-element Vector{Float64}:
- 0.18734645434202335
- 0.20744112814920254
- 3.023731854668989

Success! The computed MLE is quite close to the true parameter which we used to generate the data. As a final step, let's plot the true solution, the data, and the result of the MLE:

plot(true_sol, color=:black, linestyle=:dash, label=["True Solution" ""])
+ 0.1938872070746413
+ 0.16250297817313192
+ 3.039381283312168

Success! The computed MLE is quite close to the true parameter which we used to generate the data. As a final step, let's plot the true solution, the data, and the result of the MLE:

plot(true_sol, color=:black, linestyle=:dash, label=["True Solution" ""])
 scatter!(times, stack(odedata)', color=1, label=["Noisy Data" ""])
 mle_sol = solve(remake(true_prob, p=p_mle), EK1())
-plot!(mle_sol, color=3, label=["MLE-parameter Solution" ""])
Example block output

Looks good!

API Documentation

For more details, see the API documentation of ProbNumDiffEq.DataLikelihoods at Data Likelihoods.

References

[3]
F. Tronarp, N. Bosch and P. Hennig. Fenrir: Physics-Enhanced Regression for Initial Value Problems. In: Proceedings of the 39th International Conference on Machine Learning, Vol. 162 of Proceedings of Machine Learning Research, edited by K. Chaudhuri, S. Jegelka, L. Song, C. Szepesvari, G. Niu and S. Sabato (PMLR, 17–23 Jul 2022); pp. 21776–21794.
[10]
+plot!(mle_sol, color=3, label=["MLE-parameter Solution" ""])Example block output

Looks good!

API Documentation

For more details, see the API documentation of ProbNumDiffEq.DataLikelihoods at Data Likelihoods.

References

[3]
F. Tronarp, N. Bosch and P. Hennig. Fenrir: Physics-Enhanced Regression for Initial Value Problems. In: Proceedings of the 39th International Conference on Machine Learning, Vol. 162 of Proceedings of Machine Learning Research, edited by K. Chaudhuri, S. Jegelka, L. Song, C. Szepesvari, G. Niu and S. Sabato (PMLR, 17–23 Jul 2022); pp. 21776–21794.
[10]