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

Siyuan Lang, Zhiyue Zhang

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.

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

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.
Paper Structure (16 sections, 1 theorem, 72 equations, 20 figures, 1 table, 1 algorithm)

This paper contains 16 sections, 1 theorem, 72 equations, 20 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Methodology
  4. Extreme learning machine
  5. Operator splitting method and alternating iteration
  6. Theoretical Analysis
  7. Local convergence of the relaxed fixed-point iteration
  8. Consistency with the Monolithic Formulation
  9. Numerical experiments
  10. One-dimensional one-phase Stefan problem
  11. One-dimensional two-phase Stefan problem
  12. Two-dimensional two-phase Stefan problem
  13. Stefan problem with thermal resistance
  14. Frank sphere problem
  15. Mullins-Sekerka problem for quasi-stationary Stefan problem
  16. ...and 1 more sections

Key Result

Theorem 1

Assume that the composite operator $\mathcal{M}$ is Fréchet differentiable in an open neighborhood $\mathcal{U} \subset \mathbb{R}^M$ of a fixed point $\mathbf{c}^*$, and that its Jacobian $\mathbf{J}_{\mathcal{M}}(\mathbf{c}^*)$ has bounded spectral radius. Then, there exists a relaxation parameter where $L_\rho < 1$ is the contraction factor. Consequently, the sequence $\{\mathbf{c}^{(k)}\}_{k \

Figures (20)

  • Figure 1: ELM Architecture
  • Figure 2: Schematic diagram of the proposed operator splitting framework.
  • Figure 3: Numerical results for the 1D one-phase Stefan problem. (a) Evolution of the absolute error in the free boundary location $\Gamma(t)$. (b) Heatmap of the pointwise absolute error $|e_u|$ in the space-time domain.
  • Figure 4: Numerical results for the one-dimensional two-phase Stefan problem (Case 1). (a) Absolute error of the predicted free boundary position $\Gamma(t)$ versus time. (b) Spatiotemporal distribution of the pointwise absolute error $|u_{\mathrm{pred}}(x,t) - u_{\mathrm{exact}}(x,t)|$. The dashed line indicates the exact interface trajectory.
  • Figure 5: Numerical results for the one-dimensional two-phase Stefan problem (Case 2). (a) Absolute error of the predicted free boundary position $\Gamma(t)$ versus time. (b) Spatiotemporal distribution of the pointwise absolute error $|u_{\mathrm{pred}}(x,t) - u_{\mathrm{exact}}(x,t)|$. The dashed line indicates the exact interface trajectory.
  • ...and 15 more figures

Theorems & Definitions (4)

  • Remark 1
  • Remark 2
  • Theorem 1: Local Convergence via Relaxation
  • proof