Adaptive Randomized Neural Networks for PDEs: Structure-Aware Feature Redistribution and Residual-Driven Block Enrichment
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.
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BibTeX
@misc{yang2026adaptive,
title = {{Adaptive Randomized Neural Networks for PDEs: Structure-Aware Feature Redistribution and Residual-Driven Block Enrichment}},
author = {You Yang and Fei Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21077},
eprint = {2610.21077},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21077},
}
RIS
TY - UNPB
TI - Adaptive Randomized Neural Networks for PDEs: Structure-Aware Feature Redistribution and Residual-Driven Block Enrichment
AU - You Yang
AU - Fei Wang
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21077
UR - https://arxiv.org/abs/2610.21077
AB - 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.
ER -