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

• URL: http://arxiv.org/abs/2603.06287v1
• Date: Fri, 06 Mar 2026 13:46:38 GMT
• Title: Learning Where the Physics Is: Probabilistic Adaptive Sampling for Stiff PDEs
• Abstract summary: We introduce a framework that learns a probability density function representing the location of physics'' for adaptively sampling kernels of PIELMs.<n>By employing a weighted Expectation-Maximization (EM) algorithm, GMM-PIELM autonomously concentrates radial basis function centers in regions of high numerical error.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-sa/4.0/
• 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 $ν=10^{-4}$. 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.

Related papers

• Soft Partition-based KAPI-ELM for Multi-Scale PDEs [0.0]
  This work introduces a soft partition-based Kernel-Adaptive Physics-Informed Extreme Learning Machine.<n>A signed-distance-based weighting stabilizes least-squares learning on irregular frequencies.<n>Although demonstrated on steady linear PDEs, the results show that soft-partition kernel adaptation provides a fast, architecture-free approach for multiscale PDEs.
  arXiv  Detail & Related papers  (2026-01-13T16:43:38Z)
• Deep vs. Shallow: Benchmarking Physics-Informed Neural Architectures on the Biharmonic Equation [0.0]
  This paper systematically benchmarks RBF-PIELM, a rapid PINN variant-an extreme learning machine with radial-basis activations-for higher-order PDEs.<n>Our results show up to $(350times)$ faster training than PINNs and over $(10times)$ fewer parameters for comparable solution accuracy.
  arXiv  Detail & Related papers  (2025-10-06T04:54:04Z)
• Kernel-Adaptive PI-ELMs for Forward and Inverse Problems in PDEs with Sharp Gradients [0.0]
  This paper introduces the Kernel Adaptive Physics-Informed Extreme Learning Machine (KAPI-ELM)<n>It is designed to solve both forward and inverse Partial Differential Equation (PDE) problems involving localized sharp gradients.<n>KAPI-ELM achieves state-of-the-art accuracy in both forward and inverse settings.
  arXiv  Detail & Related papers  (2025-07-14T13:03:53Z)
• LaPON: A Lagrange's-mean-value-theorem-inspired operator network for solving PDEs and its application on NSE [8.014720523981385]
  We propose LaPON, an operator network inspired by the Lagrange's mean value theorem.<n>It embeds prior knowledge directly into the neural network architecture instead of the loss function.<n>LaPON provides a scalable and reliable solution for high-fidelity fluid dynamics simulation.
  arXiv  Detail & Related papers  (2025-05-18T10:45:17Z)
• Decentralized Nonconvex Composite Federated Learning with Gradient Tracking and Momentum [78.27945336558987]
  Decentralized server (DFL) eliminates reliance on client-client architecture.<n>Non-smooth regularization is often incorporated into machine learning tasks.<n>We propose a novel novel DNCFL algorithm to solve these problems.
  arXiv  Detail & Related papers  (2025-04-17T08:32:25Z)
• Enabling Automatic Differentiation with Mollified Graph Neural Operators [73.52999622724101]
  We propose the mollified graph neural operator ($m$GNO), the first method to leverage automatic differentiation and compute exact gradients on arbitrary geometries.<n>For a PDE example on regular grids, $m$GNO paired with autograd reduced the L2 relative data error by 20x compared to finite differences.<n>It can also solve PDEs on unstructured point clouds seamlessly, using physics losses only, at resolutions vastly lower than those needed for finite differences to be accurate enough.
  arXiv  Detail & Related papers  (2025-04-11T06:16:30Z)
• Learning Controllable Adaptive Simulation for Multi-resolution Physics [86.8993558124143]
  We introduce Learning controllable Adaptive simulation for Multi-resolution Physics (LAMP) as the first full deep learning-based surrogate model. LAMP consists of a Graph Neural Network (GNN) for learning the forward evolution, and a GNN-based actor-critic for learning the policy of spatial refinement and coarsening. We demonstrate that our LAMP outperforms state-of-the-art deep learning surrogate models, and can adaptively trade-off computation to improve long-term prediction error.
  arXiv  Detail & Related papers  (2023-05-01T23:20:27Z)
• Implicit Stochastic Gradient Descent for Training Physics-informed Neural Networks [51.92362217307946]
  Physics-informed neural networks (PINNs) have effectively been demonstrated in solving forward and inverse differential equation problems. PINNs are trapped in training failures when the target functions to be approximated exhibit high-frequency or multi-scale features. In this paper, we propose to employ implicit gradient descent (ISGD) method to train PINNs for improving the stability of training process.
  arXiv  Detail & Related papers  (2023-03-03T08:17:47Z)
• NeuralStagger: Accelerating Physics-constrained Neural PDE Solver with Spatial-temporal Decomposition [67.46012350241969]
  This paper proposes a general acceleration methodology called NeuralStagger. It decomposing the original learning tasks into several coarser-resolution subtasks. We demonstrate the successful application of NeuralStagger on 2D and 3D fluid dynamics simulations.
  arXiv  Detail & Related papers  (2023-02-20T19:36:52Z)
• Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation [59.45669299295436]
  We propose a Monte Carlo PDE solver for training unsupervised neural solvers.<n>We use the PDEs' probabilistic representation, which regards macroscopic phenomena as ensembles of random particles.<n>Our experiments on convection-diffusion, Allen-Cahn, and Navier-Stokes equations demonstrate significant improvements in accuracy and efficiency.
  arXiv  Detail & Related papers  (2023-02-10T08:05:19Z)
• A hybrid MGA-MSGD ANN training approach for approximate solution of linear elliptic PDEs [0.0]
  We introduce a hybrid "Modified Genetic-Multilevel Gradient Descent" (MGA-MSGD) training algorithm. It considerably improves accuracy and efficiency of solving 3D mechanical problems described, in strong-form, by PDEs via ANNs.
  arXiv  Detail & Related papers  (2020-12-18T10:59:07Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.