Physics-Informed Machine Learning for Two-Phase Moving-Interface and Stefan Problems

TL;DR

This paper introduces a physics-informed neural network framework for accurately solving two-phase Stefan problems with moving interfaces, explicitly tracking interface motion and enforcing temperature discontinuities, outperforming existing neural methods.

Contribution

The work develops a novel neural network approach that explicitly models the moving interface and temperature discontinuities in phase-change problems, enhancing accuracy and robustness.

Findings

Demonstrates superior accuracy over existing neural methods.

Effectively captures unstable interface evolution.

Provides a flexible alternative to traditional numerical methods.

Abstract

The Stefan problem is a classical free-boundary problem that models phase-change processes and poses computational challenges due to its moving interface and nonlinear temperature-phase coupling. In this work, we develop a physics-informed neural network framework for solving two-phase Stefan problems. The proposed method explicitly tracks the interface motion and enforces the discontinuity in the temperature gradient across the interface while maintaining global consistency of the temperature field. Our approach employs two neural networks: one representing the moving interface and the other for the temperature field. The interface network allows rapid categorization of thermal diffusivity in the spatial domain, which is a crucial step for selecting training points for the temperature network. The temperature network's input is augmented with a modified zero-level set function to accurately capture the jump in its normal derivative across the interface. Numerical experiments on two-phase dynamical Stefan problems demonstrate the superior accuracy and effectiveness of our proposed method compared with the ones obtained by other neural network methodology in literature. The results indicate that the proposed framework offers a robust and flexible alternative to traditional numerical methods for solving phase-change problems governed by moving boundaries. In addition, the proposed method can capture an unstable interface evolution associated with the Mullins-Sekerka instability.