A Predictor Corrector Convex Splitting Method for Stefan Problems Based on Extreme Learning Machines

TL;DR

This paper introduces a novel predictor-corrector convex splitting method using Extreme Learning Machines to efficiently solve Stefan problems by decoupling interface evolution from physical field reconstruction, ensuring stability and high accuracy.

Contribution

It proposes a convex splitting operator splitting approach with ELMs for decoupling Stefan problem components, transforming a non-convex problem into stable convex subproblems.

Findings

Method achieves high accuracy in 1D to 3D Stefan problems.

The iterative operator is shown to be locally contractive.

The approach provides stable and efficient solutions for free boundary problems.

Abstract

Solving Stefan problems via neural networks is inherently challenged by the nonlinear coupling between the solutions and the free boundary, which results in a non-convex optimization problem. To address this, this work proposes an Operator Splitting Method (OSM) based on Extreme Learning Machines (ELM) to decouple the geometric interface evolution from the physical field reconstruction. Within a predictor-corrector framework, the method splits the coupled system into an alternating sequence of two linear and convex subproblems: solving the diffusion equation on fixed subdomains and updating the interface geometry based on the Stefan condition. A key contribution is the formulation of both steps as linear least-squares problems; this transforms the computational strategy from a non-convex gradient-based optimization into a stable fixed-point iteration composed of alternating convex solvers. From a theoretical perspective, the relaxed iterative operator is shown to be locally contractive, and its fixed points are consistent with stationary points of the coupled residual functional. Benchmarks across 1D to 3D domains demonstrate the stability and high accuracy of the method, confirming that the proposed framework provides a highly accurate and efficient numerical solution for free boundary problems.