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

[facusapienza21]

Hi @tansongchen ! Sorry for the delay in response.

I will take a look at the documentation for the examples and PR with the same format, thank you so much for the references.

Yes, we wrote this review about differentiable programming for differential equations. We are not planning to include examples of Taylor AD here, although I will include a sentence with a reference to Taylor AD in the next iteration of the manuscript. This PR is not related to the review, rather I recently learned about Taylor AD and I was playing with the library for some other examples.

Very excited to see where this projects move and happy to be in touch in the future about for possible collaborations!