用概率方法自适应选择物理关键区域，显著提升求解刚性微分方程的精度与速度。
Learning Where the Physics Is: Probabilistic Adaptive Sampling for Stiff PDEs
- 基于高斯混合模型学习物理敏感区域，动态优化径向基函数位置。
- 在ν=10⁻⁴条件下，误差比基准方法低7个数量级，成功捕捉极薄边界层。
- 无需反向传播或贝叶斯调参，保持极限学习机的高速优势，适合快速高精度仿真。
刚性偏微分方程（PDEs）中尖锐梯度的建模仍是科学机器学习的重大挑战。尽管物理信息神经网络（PINNs）受谱偏差和训练慢限制，物理信息极限学习机（PIELMs）虽具快速闭式解优势，却受限于无关物理的随机初始化。本文提出高斯混合模型自适应PIELM（GMM-PIELM），通过概率框架学习“物理位置”分布，实现对PIELM核函数的自适应采样。采用加权期望最大化（EM）算法，自动将径向基函数中心集中于高数值误差区，如激波面与边界层，动态改善隐层条件。该方法避免了昂贵的梯度优化或贝叶斯搜索。我们在一维奇异摄动对流-扩散方程（ν=10⁻⁴）上验证，相较基准RBF-PIELM，L₂误差降低高达7个数量级，成功解析指数级薄边界层，同时保持极限学习机的量级提速优势。
原文摘要 · Abstract (English)
Modeling stiff partial differential equations (PDEs) with sharp gradients remains a significant challenge for scientific machine learning. While Physics-Informed Neural Networks (PINNs) struggle with spectral bias and slow training times, Physics-Informed Extreme Learning Machines (PIELMs) offer a rapid, closed-form linear solution but are fundamentally limited by physics-agnostic, random initialization. We introduce the Gaussian Mixture Model Adaptive PIELM (GMM-PIELM), a probabilistic framework that learns a probability density function representing the ``location of physics'' for adaptively sampling kernels of PIELMs. By employing a weighted Expectation-Maximization (EM) algorithm, GMM-PIELM autonomously concentrates radial basis function centers in regions of high numerical error, such as shock fronts and boundary layers. This approach dynamically improves the conditioning of the hidden layer without the expensive gradient-based optimization(of PINNs) or Bayesian search. We evaluate our methodology on 1D singularly perturbed convection-diffusion equations with diffusion coefficients $ν=10^{-4}$. Our method achieves $L_2$ errors up to $7$ orders of magnitude lower than baseline RBF-PIELMs, successfully resolving exponentially thin boundary layers while retaining the orders-of-magnitude speed advantage of the ELM architecture.
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