Learning Where the Physics Is: Probabilistic Adaptive Sampling for Stiff PDEs
This work provides a significant improvement in accuracy for solving stiff PDEs with sharp gradients, which is a critical problem for engineers and scientists working with complex physical simulations.
This paper addresses the challenge of modeling stiff partial differential equations (PDEs) with sharp gradients, which often cause issues for scientific machine learning methods. The authors introduce GMM-PIELM, a probabilistic framework that adaptively samples kernels by learning the 'location of physics' using a weighted Expectation-Maximization algorithm. This method achieves L2 errors up to 7 orders of magnitude lower than baseline RBF-PIELMs for 1D singularly perturbed convection-diffusion equations, while maintaining the speed advantage of the ELM architecture.
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.