  We present a data-driven approach to learning surrogate models for amplitude
equations, and illustrate its application to interfacial dynamics of phase
field systems. In particular, we demonstrate learning effective partial
differential equations describing the evolution of phase field interfaces from
full phase field data. We illustrate this on a model phase field system, where
analytical approximate equations for the dynamics of the phase field interface
(a higher order eikonal equation and its approximation, the Kardar-Parisi-Zhang
(KPZ) equation) are known. For this system, we discuss data-driven approaches
for the identification of equations that accurately describe the front
interface dynamics. When the analytical approximate models mentioned above
become inaccurate, as we move beyond the region of validity of the underlying
assumptions, the data-driven equations outperform them. In these regimes, going
beyond black-box identification, we explore different approaches to learn
data-driven corrections to the analytically approximate models, leading to
effective gray box partial differential equations.
