用极限学习机加速求解相变边界反问题，精度提升百倍以上。
A Rapid Physics-Informed Machine Learning Framework Based on Extreme Learning Machine for Inverse Stefan Problems
- 用极限学习机替代深度网络，输入权固定，仅优化输出权。
- 相对L2误差降低3-7个数量级，训练时间节省超94%。
- 适合需要快速高精度求解相变问题的工程与科研场景。
逆Stefan问题作为具有移动边界的典型相变问题，在科学与工程中应用广泛。近年来，物理信息神经网络（PINNs）被用于求解Stefan问题，但仍存在超参数依赖性强、训练效率低和预测精度不足等问题。为此，本文提出一种基于极限学习机的物理信息学习框架（PIELM），用于求解逆Stefan问题。该框架用极限学习机替代传统深度神经网络，输入权重固定，输出权重通过最小二乘法求解由初始条件、边界条件及控制偏微分方程（PDEs）组成的物理损失向量的Moore-Penrose广义逆。案例研究表明，相较于传统PINNs，PIELM在相对L2误差上可提升3至7个数量级，同时节省超过94%的训练时间。
原文摘要 · Abstract (English)
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.
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