发表机构
Indian Institute of Technology Madras（印度马德拉斯理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
AB-PIELM是结合区域分解与超参数联合优化的自适应基物理信息极端学习机，可高效稳定求解PDE，在相关实验中表现优于现有方法，还支持逆问题求解。
AI 中文摘要
物理信息神经求解器因采用全局近似空间和非凸训练，常难以处理多尺度行为与陡峭梯度。我们提出AB-PIELM，一种自适应基物理信息极端学习机，它结合区域分解与空间、光谱超参数的联合优化，同时保留确定性正规方程训练。该方法可自适应子域的数量与位置，以及径向基函数的局部性，能根据问题分配表征能力，无需修改底层求解器。在振荡函数近似和奇异摄动对流扩散方程上的实验表明，该方法在多种刚度工况下可得到精确解，且相比现有神经方法和基于PIELM的方法使用的神经元数量显著更少。学习到的超参数具有可解释性，会在陡峭梯度附近集中分辨率。该框架还支持逆问题，通过贝叶斯优化成功从稀疏带噪观测中恢复扩散系数。这些结果表明，自适应基PIELM公式是梯度训练神经PDE求解器的高效稳定替代方案，且可自然扩展到更广泛的微分方程类别。
英文摘要
Physics-informed neural solvers often struggle with multiscale behavior and sharp gradients due to the use of global approximation spaces and nonconvex training. We introduce AB-PIELM, an adaptive-basis physics-informed extreme learning machine that combines domain decomposition with joint optimisation of spatial and spectral hyperparameters while retaining deterministic normal-equation training. The method adapts both the number and placement of subdomains and the locality of radial basis functions, enabling problem-dependent allocation of representational capacity without modifying the underlying solver. Experiments on oscillatory function approximation and singularly perturbed advection-diffusion equations demonstrate accurate solutions across a wide range of stiffness regimes while using substantially fewer neurons than existing neural and PIELM-based approaches. The learned hyperparameters are interpretable, concentrating resolution near sharp gradients. The framework also supports inverse problems, successfully recovering diffusion coefficients from sparse noisy observations using Bayesian optimisation. These results indicate that adaptive-basis PIELM formulations provide an efficient and stable alternative to gradient-trained neural PDE solvers and naturally extend to broader classes of differential equations.
