Gradient Flow Based Phase-Field Modeling Using Separable Neural Networks

TL;DR

This paper introduces a separable neural network approach with a low-rank tensor decomposition and energy-stable transformation to efficiently solve the Allen-Cahn equation for phase separation, outperforming existing ML methods.

Contribution

It proposes a novel neural network-based method with energy stability guarantees for phase-field modeling, improving accuracy and computational speed.

Findings

Outperforms state-of-the-art ML methods in phase separation tasks.

Achieves an order of magnitude faster computation than finite element methods.

Effectively models sharp interfaces using a bounded neural network approach.

Abstract

The $L^2$ gradient flow of the Ginzburg-Landau free energy functional leads to the Allen Cahn equation that is widely used for modeling phase separation. Machine learning methods for solving the Allen-Cahn equation in its strong form suffer from inaccuracies in collocation techniques, errors in computing higher-order spatial derivatives through automatic differentiation, and the large system size required by the space-time approach. To overcome these limitations, we propose a separable neural network-based approximation of the phase field in a minimizing movement scheme to solve the aforementioned gradient flow problem. At each time step, the separable neural network is used to approximate the phase field in space through a low-rank tensor decomposition thereby accelerating the derivative calculations. The minimizing movement scheme naturally allows for the use of Gauss quadrature technique to compute the functional. A `$tanh$' transformation is applied on the neural network-predicted phase field to strictly bounds the solutions within the values of the two phases. For this transformation, a theoretical guarantee for energy stability of the minimizing movement scheme is established. Our results suggest that bounding the solution through this transformation is the key to effectively model sharp interfaces through separable neural network. The proposed method outperforms the state-of-the-art machine learning methods for phase separation problems and is an order of magnitude faster than the finite element method.