Adaptive physics-informed solvers improve accuracy and efficiency for complex equations
AB-PIELMs: Adaptive-Basis Physics-Informed Extreme Learning Machines for Residual-Driven Domain Decomposition
Computational Engineering, Finance, and Science
Summary
Solving complex physics equations with traditional neural networks can be tricky when the problems have sharp changes or happen at multiple scales. The authors developed a method called AB-PIELM that breaks the problem into smaller pieces and adjusts how it approximates each piece. This approach uses fewer computational resources while still giving accurate results, even for problems that are usually hard to solve. It can also help find unknown physical parameters from noisy data, making it useful in practical settings.
What this means in practice
- •For computational engineers: Produce accurate numerical solutions for differential equations with sharp gradients using fewer computational resources.
- •For environmental modelers: Estimate unknown physical coefficients in space from sparse and noisy measurements to improve model fidelity.
Authors
Aldis Daniel, Vikas Dwivedi, Balaji Srinivasan
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.