Thread by @pnkraemer

⚡️New research on probabilistic numerical solvers for (partial) differential equations ⚡️

How do I solve nonlinear, time-dependent #pde's with a #probnum algorithm?

A thread. (1/7)

How do you solve a time-dependent PDE efficiently without probabilistic numerics? 🧐 Many will probably answer with the method of lines: discretise the differential equation in space, and use an ODE solver for the rest. ✔️ (2/7)

But this is a pipeline of TWO solvers, not one solver 🤯 What happens if the space-discretisation is bad, but the ODE solver uses high precision, i.e., low absolute and relative tolerances? Computation is wasted, and ODE-solver uncertainty quantification becomes useless 😭 (3/7)

But this need not be the case! If we use specific classes of probabilistic numerical methods, for both the space-discretisation and the ODE solution, the two solver steps can communicate with each other! 📨📨📨 (4/7)

What is the result? Overconfidence is removed for good (see the figure; yellow is desirable; red and blue are bad). The #probnum solver ("PN") is well-calibrated across all dt-dx configurations, but the traditional method of lines solver ("MOL") is not. Not at all, in fact. (5/7)

This work was published at #AISTATS 2022. If you find it interesting: It is too late to drop by the poster session, but you can have a look at the talk instead (link below). Don't worry, it is extremely short. (6/7)

This is joint work with Jonathan Schmidt and @PhilippHennig5.

Paper: arxiv.org/abs/2110.11847
Youtube:
Github: github.com/schmidtjonatha… (7/7)