Soft Partition-based KAPI-ELM for Multi-Scale PDEs

TL;DR

This paper introduces a novel soft partition-based kernel method for physics-informed machine learning that effectively solves multiscale and oscillatory PDEs, outperforming existing neural network approaches in accuracy and efficiency.

Contribution

It proposes a deterministic, low-dimensional parameterization with smooth partitions controlling collocation and kernel widths, avoiding Fourier features and hard domain interfaces.

Findings

Matches or exceeds state-of-the-art accuracy on benchmarks

Uses only a single linear solve for complex PDEs

Demonstrates broad potential for multiscale PDE modeling

Abstract

Physics-informed machine learning holds great promise for solving differential equations, yet existing methods struggle with highly oscillatory, multiscale, or singularly perturbed PDEs due to spectral bias, costly backpropagation, and manually tuned kernel or Fourier frequencies. This work introduces a soft partition--based Kernel-Adaptive Physics-Informed Extreme Learning Machine (KAPI-ELM), a deterministic low-dimensional parameterization in which smooth partition lengths jointly control collocation centers and Gaussian kernel widths, enabling continuous coarse-to-fine resolution without Fourier features, random sampling, or hard domain interfaces. A signed-distance-based weighting further stabilizes least-squares learning on irregular geometries. Across eight benchmarks--including oscillatory ODEs, high-frequency Poisson equations, irregular-shaped domains, and stiff singularly perturbed convection-diffusion problems-the proposed method matches or exceeds the accuracy of state-of-the-art Physics-Informed Neural Network (PINN) and Theory of Functional Connections (TFC) variants while using only a single linear solve. Although demonstrated on steady linear PDEs, the results show that soft-partition kernel adaptation provides a fast, architecture-free approach for multiscale PDEs with broad potential for future physics-informed modeling. For reproducibility, the reference codes are available at https://github.com/vikas-dwivedi-2022/soft_kapi