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This paper proposes an adaptive variant of randomized neural networks (a fixed random-feature network with a least-squares output layer) for solving PDEs. It redistributes features across regions and adds residual-driven enrichments, with theoretical error bounds for linear problems and empirical tests on localized, multiscale, and nonlinear space-time problems. For ML readers, the main question is whether the adaptivity and error guarantees make randomized networks a practical alternative to gradient-trained PINNs. The guarantees are conditional, so the evidence is still limited to the reported test problems.
Adaptive Randomized Neural Networks for PDEs: Structure-Aware Feature Redistribution and Residual-Driven Block Enrichment
arXiv:2610.21077v1 Announce Type: cross 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.
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.
Numerical Analysis (math.NA); Machine Learning (cs.LG)
arXiv:2610.21077 [math.NA]
(or arXiv:2610.21077v1 [math.NA] for this version)
https://doi.org/10.48550/arXiv.2610.21077 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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