Funny things may happen when using NNs to solve PDEs

Date: Thu 17th Sep 2026

Abstract

For many PDEs, classical simulation methods are very costly and therefore ineffective. Such classical methods include for instance finite differences, in which the solution is represented by its values on a grid, and finite elements, in which the solution is represented as a piecewise-polynomial function. 

Neural networks (NNs) provide an alternative approach to numerical simulation of PDEs. In their simplest form, neural networks are parametrized functions, and one way of using NNs in PDE simulation lies in representing candidate solutions directly as such a parametrized function. Numerically `solving the PDE' then reduces to finding the parameter that minimizes some measure of error. 

This approach has been very successful in various different contexts, and many names are associated with such methods (PINNs, VPINNs, Galerkin PINNs, DeepRitz, Energy Nets, ...). At the same time, the state of the theory of such classes of functions is still very much in development. Part of the challenge is that the class of all NN functions is not a nice set: for instance, it is not linear, not convex, not smooth, and not closed. These `interesting' aspects of this class of NN functions cause other `interesting' things to happen when one tries to use NNs as approximation classes in the numerical simulation of PDEs. 

In this talk I will give an example of this coming from the simulation of gradient flows such as the Allen-Cahn equation. This is based on joint work with Olga Mula (Vienna) and Daan Bon and Benjamin Caris (Eindhoven).