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

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.

💡 Research Summary

This paper presents a novel Physics-Informed Neural Network (PINN) framework specifically designed to solve the classical two-phase Stefan problem, a free-boundary problem that models phase-change processes like melting and solidification. The core challenge lies in accurately tracking the moving interface between phases while enforcing the discontinuity in the temperature gradient (heat flux) across it, governed by the Stefan condition.

The authors’ key innovation is the use of two coupled neural networks. The first network, denoted as ŝ(y,t), directly models the position of the moving interface. The second network, U(x,y,t, z), models the temperature field. Crucially, the input to this temperature network is augmented with a modified level-set function z = φ_a = |x - ŝ(y,t)|. This architectural choice, inspired by “cusp-capturing PINNs,” enables the network to inherently produce a jump in the normal derivative of the temperature across the interface (where φ_a is non-differentiable), which is essential for satisfying the Stefan condition. The interface network ŝ also serves a critical secondary function: during training, it categorizes each collocation point in the spatial domain as belonging to either the Ω+ or Ω- phase, allowing the correct thermal diffusivity (k+ or k-) to be assigned when evaluating the physics-based loss for the heat equation.

The unified loss function incorporates residuals from the heat equation in each sub-domain, initial and boundary conditions, and three interface conditions: the melting temperature (u=0), the Stefan condition (linking interface velocity to heat flux jump), and the initial interface position. The parameters of both networks are optimized jointly using the Levenberg-Marquardt algorithm. The training process involves iterative updates where, after each parameter adjustment, the domain categorization is refreshed based on the current interface estimate.

Numerical experiments validate the framework’s effectiveness. In a one-dimensional benchmark test, the proposed method achieves significantly higher accuracy (with relative L2 errors on the order of 10^-4) compared to an existing deep learning-based Stefan problem solver. Two-dimensional tests further demonstrate its capability to handle complex scenarios, including problems with heat sources and the capture of nonlinear interface evolution. Most notably, the method successfully simulates the onset and development of the Mullins-Sekerka instability—a complex pattern-forming instability arising from a perturbed initial interface—without needing explicit front-tracking or stabilization techniques common in traditional numerical methods.

In conclusion, this work introduces a robust, mesh-free, and flexible alternative for solving moving-interface problems. By decoupling the interface and temperature representations and ingeniously using a modified level-set function as an input feature, the framework naturally respects the underlying physical discontinuities. The results demonstrate superior accuracy over comparable neural network methods and establish the potential of this PINN-based approach for a wider class of free-boundary and multiphysics problems involving sharp interfaces.