Discovering Governing Equations in Discrete Systems Using PINNs

TL;DR

This paper demonstrates how Physics-Informed Neural Networks (PINNs) can be adapted to identify governing equations in high-dimensional discrete nonlinear dynamical systems, including Hamiltonian and dissipative cases.

Contribution

It introduces a novel adaptation of PINNs for inverse problems in lattice nonlinear systems, extending their application beyond continuous systems.

Findings

Successfully identified equations in diverse nonlinear systems

Demonstrated applicability to both Hamiltonian and dissipative models

Discussed limitations and potential of PINNs in discrete systems

Abstract

Sparse identification of nonlinear dynamical systems is a topic of continuously increasing significance in the dynamical systems community. Here we explore it at the level of lattice nonlinear dynamical systems of many degrees of freedom. We illustrate the ability of a suitable adaptation of Physics-Informed Neural Networks (PINNs) to solve the inverse problem of parameter identification in such discrete, high-dimensional systems inspired by physical applications. The methodology is illustrated in a diverse array of examples including real-field ones ($\phi^4$ and sine-Gordon), as well as complex-field (discrete nonlinear Schr{\"o}dinger equation) and going beyond Hamiltonian to dissipative cases (the discrete complex Ginzburg-Landau equation). Both the successes, as well as some limitations of the method are discussed along the way.