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

2025-12-15

AI Summary

The Stefan problem—a canonical moving-boundary phase-change model—poses significant challenges for numerical simulation due to strong nonlinear coupling between interfacial dynamics and the temperature field, leading to high computational cost in conventional methods and insufficient accuracy/stability in existing physics-informed neural networks (PINNs). This work proposes a dual-network cooperative PINN framework: one network explicitly learns the interface trajectory, while the other solves the temperature field using a modified zero-level-set function, enabling precise resolution of gradient discontinuities across the phase boundary and ensuring global physical consistency. Notably, our method is the first PINN-based approach to reproduce the Mullins–Sekerka instability. Coupled with an interface-driven adaptive sampling strategy, the framework achieves state-of-the-art performance on dynamic two-phase Stefan problems—demonstrating superior accuracy, robustness, and generalizability over prior neural-network methods—and establishes a viable alternative paradigm to traditional numerical simulation for moving-boundary problems.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Multiagent Systems: Adversarial Agents
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Social Networks and Social Media: Computational social science

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.

Problem

• Develops a physics-informed neural network for two-phase Stefan problems
• Explicitly tracks interface motion and enforces temperature gradient discontinuity
• Captures unstable interface evolution like Mullins-Sekerka instability

Innovation

• Physics-informed neural network for Stefan problems
• Two neural networks track interface and temperature
• Modified zero-level set captures derivative jumps