Data-Driven Modeling of Nonlinear Traveling Waves

TL;DR

This paper introduces a machine learning framework for identifying and modeling nonlinear traveling wave dynamics in physical systems, using coordinate transformations and sparse regression or neural ODEs.

Contribution

It presents a novel data-driven approach combining coordinate transformation with sparse regression and neural ODEs to model traveling waves when governing equations are unknown.

Findings

Successfully applied to physical systems with traveling wave phenomena

Demonstrated interpretability through sparse regression techniques

Validated effectiveness with examples of wave fronts, pulses, and wavetrains

Abstract

Presented is a data-driven Machine Learning (ML) framework for the identification and modeling of traveling wave spatiotemporal dynamics. The presented framework is based on the steadily-propagating traveling wave ansatz, $u(x,t) = U(\xi=x - ct + a)$. For known evolution equations, this coordinate transformation reduces governing partial differential equations (PDEs) to a set of coupled ordinary differential equations (ODEs) in the traveling wave coordinate $\xi$. Although traveling waves are readily observed in many physical systems, the underlying governing equations may be unknown. For these instances, the traveling wave ODEs can be (i) identified in an interpretable manner through an implementation of sparse regression techniques or (ii) modeled empirically with neural ODEs. Presented are these methods applied to several physical systems that admit traveling waves. Examples include traveling wave fronts, pulses, and wavetrains restricted to one-wave wave propagation in a single spatial dimension.