PCELM: Perturbation-Correction Extreme Learning Machine for the Stefan problem

TL;DR

The paper introduces PCELM, a neural network framework that effectively solves Stefan problems with moving boundaries by transforming a nonconvex optimization into a convex one, achieving high accuracy.

Contribution

It proposes a novel perturbation-correction approach within extreme learning machines that guarantees convexity and rapid convergence for Stefan problem solutions.

Findings

PCELM overcomes optimization plateaus in Stefan problems.

The correction step improves accuracy by 2-6 orders of magnitude.

Numerical experiments validate the effectiveness across multi-phase and multi-dimensional cases.

Abstract

For Stefan problems, characterized by moving boundaries and discontinuous coefficients due to phase changes, the inherent nonconvexity of the objective functional frequently causes optimization difficulty in randomized neural network approximations; to address this, we propose a Perturbation-Correction Extreme Learning Machine (PCELM) framework, built upon the extreme learning machine framework. This method first establishes a basic approximation during an initialization step by minimizing the original nonconvex residual, typically achieving only moderate accuracy, and then, in a subsequent correction step, determines a correction term by solving a subproblem based on a perturbation expansion around this basic approximation, thereby transforming it into a convex optimization problem for the output coefficients that ensures rapid convergence. We further provide a rigorous a convexity analysis, demonstrating that PCELM method solves a convex sub-problem. Numerical experiments on various Stefan problems, including multi-phase and multi-dimensional Stefan problems, confirm that the proposed PCELM method successfully overcomes optimization plateaus, with the correction step consistently delivering a significant improvement of 2-6 orders of magnitude in the relative L2 accuracy.