Learning Where the Physics Is: Probabilistic Adaptive Sampling for Stiff PDEs

TL;DR

The paper introduces GMM-PIELM, a probabilistic adaptive sampling method that enhances physics-informed neural network solutions for stiff PDEs by dynamically focusing on regions with high error, significantly improving accuracy and efficiency.

Contribution

It proposes a novel GMM-PIELM framework that learns the physics location distribution to adaptively sample kernels, overcoming limitations of traditional physics-agnostic methods.

Findings

Achieves up to 7 orders of magnitude lower $L_2$ error compared to baseline RBF-PIELMs.

Successfully resolves thin boundary layers in stiff PDEs.

Retains the speed advantage of ELM architectures.

Abstract

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 $\nu=10^{-4}$. 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.