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

2026-01-13

AI Summary

This work proposes Kernel-Adaptive Physics-Informed Extreme Learning Machines (KAPI-ELM) to address the limitations of existing physics-informed machine learning methods—such as spectral bias, high computational cost, and reliance on manual hyperparameter tuning—when solving high-frequency oscillatory, multiscale, or singularly perturbed partial differential equations. KAPI-ELM uniquely integrates a soft partitioning mechanism with kernel adaptivity, enabling continuous multiscale modeling without Fourier features, random sampling, or hard domain decomposition by smoothly modulating collocation point distribution and Gaussian kernel width. A signed distance-based weighting scheme further enhances stability on irregular geometries. Requiring only a single linear solve, the method matches or exceeds the accuracy of state-of-the-art PINN and TFC approaches across eight benchmark problems—including high-frequency Poisson and singularly perturbed convection-diffusion equations—while significantly reducing dependence on frequency-aware tuning and complex training procedures.

Technology Category

Machine Learning: Kernel MethodsConstraint Satisfaction and Optimization: Satisfiability Modulo TheoriesComputer Vision: Low Level & Physics-based Vision

Application Category

Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applicationsSemantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMsGraph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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

Problem

• multiscale PDEs
• spectral bias
• singularly perturbed PDEs
• physics-informed machine learning
• highly oscillatory

Innovation

• soft partition
• kernel-adaptive
• physics-informed ELM
• multiscale PDEs
• signed-distance weighting