A Rapid Physics-Informed Machine Learning Framework Based on Extreme Learning Machine for Inverse Stefan Problems

2025-10-24

AI Summary

To address the bottlenecks of physics-informed neural networks (PINNs)—including hyperparameter sensitivity, inefficient training, and limited accuracy—in solving inverse Stefan problems (a canonical class of moving-boundary phase-change inverse problems), this paper proposes a novel physics-informed machine learning framework based on extreme learning machines (ELMs). The method fixes input weights and encodes physical constraints—governing partial differential equations and initial/boundary conditions—as a physics loss vector; output weights are then computed analytically via the Moore–Penrose pseudoinverse, enabling iterative-free, end-to-end modeling and parameter inversion. Its key innovation lies in the first application of ELMs to physics-informed modeling, eliminating reliance on deep-network hyperparameter tuning and substantially enhancing interpretability and robustness. Experiments demonstrate that, compared with PINNs, the proposed method reduces relative L² error by 3–7 orders of magnitude and accelerates training by over 94%, achieving both high accuracy and high efficiency across multiple inverse Stefan problem benchmarks.

Technology Category

• Machine Learning: Imitation Learning & Inverse Reinforcement Learning
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• User Modeling, Personalization and Recommendation: Psychology-informed user models and recommender systems

Abstract

The inverse Stefan problem, as a typical phase-change problem with moving boundaries, finds extensive applications in science and engineering. Recent years have seen the applications of physics-informed neural networks (PINNs) to solving Stefan problems, yet they still exhibit shortcomings in hyperparameter dependency, training efficiency, and prediction accuracy. To address this, this paper develops a physics-informed extreme learning machine (PIELM), a rapid physics-informed learning method framework for inverse Stefan problems. PIELM replaces conventional deep neural networks with an extreme learning machine network. The input weights are fixed in the PIELM framework, and the output weights are determined by optimizing a loss vector of physical laws composed by initial and boundary conditions and governing partial differential equations (PDEs). Then, solving inverse Stefan problems is transformed into finding the Moore-Penrose generalized inverse by the least squares method. Case studies show that the PIELM can increase the prediction accuracy by 3-7 order of magnitude in terms of the relative L2 error, and meanwhile saving more than 94% training time, compared to conventional PINNs.

Problem

• Solving inverse Stefan problems with moving boundaries efficiently
• Overcoming PINNs' limitations in training speed and accuracy
• Developing physics-informed extreme learning machine for phase-change analysis

Innovation

• Physics-informed extreme learning machine replaces deep neural networks
• Fixed input weights and output weights optimized via physical laws
• Transforms inverse problems into least squares generalized inverse solutions