Machine Learning of Nonlinear Waves: Data-Driven Methods for Computer-Assisted Discovery of Equations, Symmetries, Conservation Laws, and Integrability

Abstract

The purpose of this article is to provide a perspective -- admittedly, a rather subjective one -- of recent developments at the interface of machine learning/data-driven methods and nonlinear wave studies. We review some recent pillars of the rapidly evolving landscape of scientific machine learning, including deep learning, data-driven equation discovery, {\color{blue} Koopman-based methods,} and operator learning, among others. We then showcase these methods in applications ranging from learning lattice dynamical models and reduced order modeling of effective dynamics to discovery of conservation laws and potential identification of integrability of ODE and PDE models. Our intention is to make clear that these machine learning methods are complementary to the preexisting powerful tools of the nonlinear waves community, and should be integrated into this toolkit to augment and enable mathematical discoveries and computational capabilities in the age of data.