Black and gray box learning of amplitude equations: Application to phase field systems.

Abstract

We present a data-driven approach to learning surrogate models for amplitude equationsand illustrate its application to interfacial dynamics of phase field systems. In particular, we demonstrate learning effective partial differential equationsdescribing the evolution of phase field interfaces from full phase field data. We illustrate this on a model phase field system, where analytical approximate equationsfor the dynamics of the phase field interface (a higher-order eikonal equationand its approximation, the Kardar-Parisi-Zhang equation) are known. For this system, we discuss data-driven approaches for the identification of equationsthat 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 equationsoutperform them. In these regimes, going beyond black box identification, we explore different approaches to learning data-driven corrections to the analytically approximate models, leading to effective gray box partial differential equations.