AB-PIELMS: ADAPTIVE-BASIS PHYSICS-INFORMED EXTREME LEARNING MACHINES FOR RESIDUAL-DRIVEN DOMAIN DECOMPOSITION

Aldis Daniel, Vikas Dwivedi, Balaji Srinivasan

Keywords: Physics-Informed Learning, Extreme Learning Machines, Domain decomposition, PDE solvers, Advection Diffusion, Inverse problems, Parameter estimation

TL;DR: We propose an adaptive, domain-decomposed physics-informed extreme learning machine that automatically adjusts its representation to localized solution features, solving forward and inverse PDE problems accurately with deterministic training.

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.

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Submission Number: 70