A Deep Ritz Method for High-Dimensional Steady States of the Cahn--Hilliard Equation

TL;DR

This paper introduces a deep learning framework called the Deep Ritz method to efficiently compute high-dimensional steady states of the Cahn--Hilliard equation, capturing complex phase separation patterns.

Contribution

It develops a novel deep learning approach with augmented Lagrangian and Fourier features to enforce constraints and encode periodicity, enabling fast convergence and discovery of multiple solutions.

Findings

Successfully captures diverse phase separation patterns in 1D, 2D, and 3D.

Achieves faster convergence compared to traditional methods.

Effectively identifies multiple local energy minimizers.

Abstract

The Cahn--Hilliard equation is a fundamental model for describing phase separation phenomena in binary mixtures. Traditional numerical methods, such as finite difference and finite element methods, often incur substantial computational cost, particularly when computing steady-state solutions in high-dimensional settings. To address this challenge, we propose a deep learning-based framework, namely the Deep Ritz method, for computing steady states of the Cahn--Hilliard equation under periodic boundary conditions. An enhanced augmented Lagrangian formulation is incorporated to strictly enforce the mass conservation constraint, while separable Fourier feature mappings are employed to naturally encode periodicity and enhance the representation of nontrivial solution structures. The proposed method exhibits a notable dual capability: it not only achieves fast convergence to steady states but also effectively identifies multiple nontrivial solutions corresponding to different local minimizers of the energy functional. Extensive numerical experiments in one-, two-, and three-dimensional cases demonstrate that the method can successfully capture a rich variety of phase separation patterns, including droplet-type, lamellar, and tubular structures, highlighting its effectiveness and robustness in exploring complex high-dimensional energy landscapes.