Adding support for ODE differentiation and interface for DifferentialEquations.jl

[facusapienza21]

Hi!

I had been playing around with using TaylorDiff.jl for higher-order differentiation of numerical solutions of ODEs. I have a simple implementation of the method introduced in Appendix D in Taylor-Mode Automatic Differentiation for
Higher-Order Derivatives in JAX together with some simple examples of ODEs. I think this quite add some value to the library, since it is particularly important to compute higher-order derivatives of solutions of differential equations for some very specific applications.

I would be happy to make a PR with this, probably into a new module ode.jl. Would this be useful for the library? I would love to contribute :)

[tansongchen]

Hi,

Sure! This would be very helpful, and really appreciate your pull request.

In addition to adding code to src/, you can also optionally add tutorial-like examples in documentations. Doc tests ensure that examples are always run when the code gets updated, which is better than the current separate examples/ dir.

I will start to move the examples to documentations, but in the mean time feel free to start your work!

[facusapienza21]

That is great! Yes, I was planning to also provide with an example we can include in the documentation.

I may need some feedback on specific TaylorDiff or SciML generic styles, which I am happy to provide. All this to say that I am sure there will be changes to make to my original PR, which I am looking for ;)

[tansongchen]

Sure! Don't worry too much about styles at the beginning, we can definitely refine after examples are up and running.

[tansongchen]

Hi, I just moved an example from examples/ folder to documentation, which could serve as a reference for your example.

In addition, I also cleaned up a script https://github.com/JuliaDiff/TaylorDiff.jl/blob/main/examples/integration.jl which I initially wrote in the last year to demonstrate how to use TaylorDiff.jl to compute jet coefficients for Taylor integration. I didn't go further to make it fully functional, but you can use it as a starting point to implement your solver!

[tansongchen]

Doc example: https://juliadiff.org/TaylorDiff.jl/dev/examples/halley/

[tansongchen]

Chris mentioned to me today that you are working on a review paper that will use higher-order autodiff to show several things. Let me know if there's anything specific I can help. In addition, I will also prepare a paper on the implementation and applications of TaylorDiff.jl early next year, and by then we can also collaborate