AB-PIELMs: Adaptive-Basis Physics-Informed Extreme Learning Machines for Residual-Driven Domain Decomposition

2026-10-08

AI Summary

This study addresses the solution bottlenecks of physics-informed neural networks in multiscale and sharp-gradient scenarios arising from global approximation and non-convex optimization by proposing an adaptive basis physics-informed extreme learning machine. The method introduces a novel residual-driven adaptive domain decomposition mechanism that dynamically adjusts the number of subdomains and the locality of radial basis functions. By integrating spatial-spectral hyperparameter co-optimization with a Bayesian framework, it achieves deterministic training and problem-dependent capacity allocation without modifying the underlying solver. Experimental results demonstrate that the proposed approach significantly reduces network size while yielding interpretable, high-accuracy solutions for stiff equations, and successfully recovers diffusion coefficients from sparse, noisy data.

Abstract

Physics-informed neural solvers often struggle with multiscale behavior and sharp gradients due to the use of global approximation spaces and nonconvex training. We introduce AB-PIELM, an adaptive-basis physics-informed extreme learning machine that combines domain decomposition with joint optimisation of spatial and spectral hyperparameters while retaining deterministic normal-equation training. The method adapts both the number and placement of subdomains and the locality of radial basis functions, enabling problem-dependent allocation of representational capacity without modifying the underlying solver. Experiments on oscillatory function approximation and singularly perturbed advection-diffusion equations demonstrate accurate solutions across a wide range of stiffness regimes while using substantially fewer neurons than existing neural and PIELM-based approaches. The learned hyperparameters are interpretable, concentrating resolution near sharp gradients. The framework also supports inverse problems, successfully recovering diffusion coefficients from sparse noisy observations using Bayesian optimisation. These results indicate that adaptive-basis PIELM formulations provide an efficient and stable alternative to gradient-trained neural PDE solvers and naturally extend to broader classes of differential equations.

Problem

• physics-informed neural solvers
• multiscale behavior
• sharp gradients
• domain decomposition
• partial differential equations

Innovation

• Physics-Informed Extreme Learning Machine
• Domain Decomposition
• Adaptive Basis
• Radial Basis Functions
• Inverse Problems