In Chaos (Woodbury, N.Y.)
Presented is a data-driven machine learning framework for modeling traveling wave spatiotemporal dynamics. The presented framework is based on the steadily propagating traveling wave ansatz, u(x,t)=U(ξ=x-ct+a). For known evolution equations, this coordinate transformation reduces governing partial differential equations to a set of coupled ordinary differential equations (ODEs) in the traveling wave coordinate ξ. Although traveling waves are readily observed in many physical systems, the underlying governing equations may be unknown. For these instances, the traveling wave dynamical system can be modeled empirically with neural ODEs. Presented are these ideas applied to several physical systems that admit traveling waves. Examples include traveling wave fronts, pulses, and wavetrains restricted to one-way wave propagation in a single spatial dimension. Last, applicability to real-world physical systems is presented with an exploration of data-driven modeling of rotating detonation waves.