Adaptive randomized neural networks improve solving complex pdes efficiently
Adaptive Randomized Neural Networks for PDEs: Structure-Aware Feature Redistribution and Residual-Driven Block Enrichment
Machine Learning
Summary
Solving partial differential equations (PDEs), which describe many physical and engineering systems, can be hard when solutions vary in space and scale. The paper introduces a new method that adapts how neural network features are distributed and enriched based on the solution’s structure and errors. This approach helps the neural network focus on important regions and refine its predictions systematically, improving accuracy. The authors also provide a mathematical analysis supporting the method’s error control. Experiments show it works well on challenging localized and nonlinear problems involving space and time.
What this means in practice
- •For computational engineers: Improve numerical simulation accuracy and efficiency when solving PDEs in multiscale and nonlinear problems with limited computational resources.
- •For software developers for physics simulation: Incorporate adaptive randomized neural network techniques to build faster and more accurate PDE solvers for complex dynamic systems.
Authors
You Yang, Fei Wang
Abstract
Randomized neural networks retain an output-linear least-squares solve, but globally sampled random features may fail to capture the spatial and scale structure of PDE solutions. We propose SFRBE--RaNN, a two-stage adaptive trial-space construction. SFR redistributes a fixed feature budget according to regional indicator masses and adapts regional weight scales through low-dimensional residual-based optimization. RBE constructs nonredundant enrichments by screening multiple activation families, removing represented response directions, and selecting the smallest admissible prefix. For solution-dependent operators, candidates are screened with a frozen operator and accepted only after nonlinear residual verification. For fixed linear problems, we establish an exact discrete residual-reduction identity and a conditional solution-error estimate. Building on this identity, we combine safeguarded SFR with RBE capture factors to derive a finite-horizon error bound involving baseline best approximation and empirical consistency, together with a conditional random-search success-probability bound. Numerical experiments on localized, multiscale, and nonlinear space--time problems demonstrate the method's effectiveness.