  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.
