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

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.