A Thermo-Kinematic Operator Splitting Framework for Stefan Problems Based on Extreme Learning Machines

Solving Stefan problems via neural networks is challenging because of the nonlinear coupling between the solution and the free boundary, which generally leads to a non-convex optimization problem when they are parameterized and optimized simultaneously. To address this difficulty, we propose a thermo-kinematic operator splitting framework based on Extreme Learning Machines (ELMs) to decouple the solution reconstruction from the free boundary evolution. Both the solution and the complete free-boundary trajectory are represented over the full space-time domain. At each outer iteration, the solution is computed on the space-time subdomains determined by the current free-boundary trajectory, after which the interface geometry is updated through the corresponding free-boundary conditions using the computed heat fluxes. For the models considered here, each split subproblem is linear in the unknown coefficients when the required geometric quantities are held fixed. Both steps are therefore formulated as convex linear least-squares problems. The proposed framework thus avoids direct minimization of the monolithic non-convex residual functional by isolating the nonlinear solution free-boundary coupling in a reduced relaxed fixed-point iteration for the complete free-boundary trajectory. For a fixed discretization, the analysis establishes local linear convergence of the free-boundary trajectory iteration under the stated assumptions and with suitable relaxation. Numerical experiments in one, two, and three spatial dimensions verify the accuracy of the proposed framework on benchmark problems and illustrate its applicability to physically motivated Stefan problems.