Adaptive Randomized Neural Networks with Locally Activation Function: Theory and Algorithm for Solving PDEs

[2604.08869] Adaptive Randomized Neural Networks with Locally Activation Function: Theory and Algorithm for Solving PDEs Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Numerical Analysis arXiv:2604.08869 (math) [Submitted on 10 Apr 2026 ( v1 ), last revised 15 Aug 2026 (this version, v2)] Title: Adaptive Randomized Neural Networks with Locally Activation Function: Theory and Algorithm for Solving PDEs Authors: Ran Bi , Weibing Deng View a PDF of the paper titled Adaptive Randomized Neural Networks with Locally Activation Function: Theory and Algorithm for Solving PDEs, by Ran Bi and Weibing Deng View PDF HTML (experimental) Abstract: This paper establishes an approximation theory and develop an adaptive computational framework for randomized neural networks (RaNNs). For RaNNs of the form $\sum_{i=1}^{N} W_i \sigma(A_i\cdot x+B_i)$ with hidden parameters uniformly sampled from a bounded set of scale $M$, we show that the choice $M \asymp N^{p/[2((p-1)(d+1)+p\eta)]}$ yields an expected $W^{k,p}$-approximation error of order $\mathcal{O}\left(N^{-p\eta/[2((p-1)(d+1)+p\eta)]}\right)$, where $\eta$ is associated with the regularity of the target function in a generalized Barron spectral space. This theoretical result demonstrates that lower regularity requires a larger sampling range for the hidden parameters. Motivated by the relationship between $M$ and $\eta$, we introduce an adaptive physics-informed RaNN (PIRaNN) method that which couples parameter sampling with an adaptive partition of unity via local affine scaling. Residual-based a posteriori error indicators and Dörfler marking are used to drive local refinement. Numerical experiments, including the 1D viscous Burgers' equation and 2D/3D problems with localized peaks and L-shaped corner singularities, demonstrate that our method preferentially refines regions exhibiting large gradients and singularities. Compared to standard non-adaptive RaNNs, the adaptive PIRaNN effectively captures complex local features with significantly enhanced accuracy and efficiency. Subjects: Numerical Analysis (math.NA) MSC classes: 68T07, 65N50, 41A25 Cite as: arXiv:2604.08869 [math.NA] (or arXiv:2604.08869v2 [math.NA] for this version) https://doi.org/10.48550/arXiv.2604.08869 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Ran Bi [ view email ] [v1] Fri, 10 Apr 2026 02:07:07 UTC (621 KB) [v2] Sat, 15 Aug 2026 05:34:57 UTC (1,019 KB) Full-text links: Access Paper: View a PDF of the paper titled Adaptive Randomized Neural Networks with Locally Activation Function: Theory and Algorithm for Solving PDEs, by Ran Bi and Weibing Deng View PDF HTML (experimental) TeX Source view license Current browse context: math.NA < prev | next > new | recent | 2026-04 Change to browse by: cs cs.NA math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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