Add Jacobian field to StochSystem

[oameye]

In many algorithms one needs the jacobian of the drift field. These are now computed using ForwardDiff. Adding the the field jacobian to StochSystem, makes it possible for the user to hardcode it or compute it symbolically using, e.g., ModeligToolkit.

[reykboerner]

Agreed, integration of ModelingToolkit and the possibility to hardcode the Jacobian of the drift should be included.

[oameye]

A jacobian can be stored in sde.integ.f.jac

[oameye]

Als see JuliaDynamics/DynamicalSystemsBase.jl#228

[Datseris]

probably worth coordinating creating the jacobian of a dynamic rule generically via an interface function. I do it on my own in TangentDynamicalSystem via ForwardDiff. And we have the jacobian function for htis.

How does integ.f.jac do it?

[oameye]

Would be nice that jacobian(sde::CoupledSDEs) works, just as jacobian(sde::CoupledODEs). We just need to make CoupledSDEs <: CoreDynamicalSystem. For some reason CoupledODEs <: CoreDynamicalSystem, but I don't see where that's defined.

integ.f.jac stores the jacobian if you define the system with ModelingToolkit and use jac=true in the ODESystem. In the background it just uses Symbolics.build_function to compile the Symbolics.

[Datseris]

Would be nice that jacobian(sde::CoupledSDEs) works, just as jacobian(sde::CoupledODEs). We just need to make CoupledSDEs <: CoreDynamicalSystem. For some reason CoupledODEs <: CoreDynamicalSystem, but I don't see where that's defined.

You can define jacobian(sde::CoupledSDEs) = jacobian(CoupledODEs(sde)) which to me sounds more correct. Because you refer to the jacobian of the drift, while of course one could ask for the jacobian of the noise term just as well.

integ.f.jac stores the jacobian if you define the system with ModelingToolkit and use jac=true in the ODESystem. In the background it just uses Symbolics.build_function to compile the Symbolics.

Oh, I didn't know that. THat is very useful. DynamicalSystems.jl already supports MTK-made models throughout. So we can add one more option so that if there is a symbolic Jacobian, it uses this one instead of making a forward-diff one. Easy to do via a PR if you are interested!

[oameye]

@Datseris made a PR