Learning of discrete models of variational PDEs from data

TL;DR

This paper introduces a neural network approach to learn discrete Lagrangian models of PDEs from data, ensuring structure preservation, numerical regularity, and the ability to identify simple solutions like traveling waves, even without their explicit presence in training data.

Contribution

It presents a novel structure-preserving neural network framework for learning discrete field theories from data, including regularisation techniques and methods to identify simple solutions.

Findings

Successfully learned discrete models for wave and Schrödinger equations.

Regularisers improve numerical robustness and efficiency of the learned models.

Can identify simple solutions like traveling waves without their explicit presence in data.

Abstract

We show how to learn discrete field theories from observational data of fields on a space-time lattice. For this, we train a neural network model of a discrete Lagrangian density such that the discrete Euler--Lagrange equations are consistent with the given training data. We, thus, obtain a structure-preserving machine learning architecture. Lagrangian densities are not uniquely defined by the solutions of a field theory. We introduce a technique to derive regularisers for the training process which optimise numerical regularity of the discrete field theory. Minimisation of the regularisers guarantees that close to the training data the discrete field theory behaves robust and efficient when used in numerical simulations. Further, we show how to identify structurally simple solutions of the underlying continuous field theory such as travelling waves. This is possible even when travelling waves are not present in the training data. This is compared to data-driven model order reduction based approaches, which struggle to identify suitable latent spaces containing structurally simple solutions when these are not present in the training data. Ideas are demonstrated on examples based on the wave equation and the Schr\"odinger equation.