```javascript
javascript:(function(){var url=window.location.href;if(url.match(/arxiv\.org\/pdf\//)){window.location.href=url.replace('/pdf/','/abs/');}else if(url.match(/openreview\.net\/pdf\?id=/)){window.location.href=url.replace('/pdf?id=','/forum?id=');}})();
```

Thanks! Yes they should also work for larger nonlinear systems as long as they are not too stiff. And there is also a Python implementation by @pnkraemer.bsky.social: github.com/pnkraemer/pr...

And because methods are only useful if people can actually use them: I wrote ProbNumDiffEq.jl to make all of this accessible. Give it a try!

💻 github.com/nathanaelbos...
📖 nathanaelbosch.github.io/ProbNumDiffE...
▶️ www.youtube.com/watch?v=iH_G...

6/6

There are many more things that I'd love to write about - e.g. robust parameter inference in neuroscience ODEs - but I think my thesis does a better job at explaining everything.

📄 Full thesis: tobias-lib.uni-tuebingen.de/xmlui/handle...

5/6

Two more examples: We can add linear ODEs to the prior to create a probabilistic version of "exponential integrators". onlinear information (e.g. conservation laws) can be included in the likelihood to get more plausible solutions - see gif.

[2] tinyurl.com/2av3e4te
[3] tinyurl.com/bddfkwcu

4/6

It turns out that this framework is quite convenient: You can easily customize each building block - prior, likelihood, inference - to adjust the solver and its properties. For example, by using a time-parallel smoother we obtain a parallel-in-time ODE solver!

[1] www.jmlr.org/papers/v25/2...

3/6

The main trick is to reformulate "solving an ODE" as "Bayesian state estimation" by turning the ODE into a nonlinear observation model. With a suitable prior - a Gauss-Markov process - you can solve the resulting problem with Bayesian filtering to obtain a probabilistic numerical ODE solution.

2/6

🙋