Learning discrete Lagrangians for variational PDEs from data and detection of travelling waves

TL;DR

This paper introduces a neural network-based approach to learn discrete Lagrangian models of variational PDEs from data and develops a method to detect traveling wave solutions within these learned models.

Contribution

It presents a novel method to learn discrete Lagrangians directly from data and introduces a technique to identify traveling wave solutions in the learned models.

Findings

Successful learning of discrete Lagrangians from data

Effective regularisation for numerical stability

Method for detecting traveling waves in learned models

Abstract

The article shows how to learn models of dynamical systems from data which are governed by an unknown variational PDE. Rather than employing reduction techniques, we learn a discrete field theory governed by a discrete Lagrangian density $L_d$ that is modelled as a neural network. Careful regularisation of the loss function for training $L_d$ is necessary to obtain a field theory that is suitable for numerical computations: we derive a regularisation term which optimises the solvability of the discrete Euler--Lagrange equations. Secondly, we develop a method to find solutions to machine learned discrete field theories which constitute travelling waves of the underlying continuous PDE.