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

TL;DR

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.

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

Figures (13)

• Figure 1: RBF spectral bandwidth induced by partition–adaptive sampling. Left: the proposed sampler generates a multimodal distribution of Gaussian widths $\sigma$, including extremely small kernels. Right: corresponding Fourier magnitudes $\lvert\widehat{\phi}_{\sigma}(\omega)\rvert=\exp(-(\sigma\omega)^2/2)$ for representative widths. Narrow kernels yield very large frequency bandwidths, enabling KAPI--ELM to capture high-frequency and boundary-layer structure without Fourier features.
• Figure 2: KAPI--ELM approximation and exact solution for $u'(x)=\cos(15x)$ on $[-2\pi,2\pi]$.
• Figure 3: KAPI--ELM approximation and exact solution for $u"(x)=\sin(15x)$ with mixed boundary data.
• Figure 4: KAPI--ELM solution and exact multiscale solution for $u'(x)=\omega_{1}\cos(\omega_{1}x)+\omega_{2}\cos(\omega_{2}x)$ with $(\omega_{1},\omega_{2})=(1,15)$.
• Figure 5: 2D Poisson on a unit square (without SDF-weighting): (Left) KAPI--ELM approximation, (Center) exact solution, (Right) pointwise absolute error. The error remains at $10^{-12}$--level across the entire domain, demonstrating accurate recovery of high-frequency two-dimensional structure without Fourier features.