Black and Gray Box Learning of Amplitude Equations: Application to Phase Field Systems

TL;DR

This paper develops a data-driven method to learn surrogate amplitude equations for phase field systems, improving accuracy over analytical models and introducing gray box PDEs with data-driven corrections.

Contribution

It introduces a novel approach for learning amplitude equations from data, including gray box models that combine analytical and data-driven components for phase field interface dynamics.

Findings

Data-driven models outperform analytical equations beyond their validity range.

Gray box PDEs with data-driven corrections improve modeling accuracy.

Effective surrogate models are demonstrated on phase field interface dynamics.

Abstract

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.