A posteriori certification of PDE approximations with particular application to neural networks

[2502.20336] A posteriori certification of PDE approximations with particular application to neural networks Skip to main content Search Submit Donate Log in Mathematics > Numerical Analysis arXiv:2502.20336 (math) [Submitted on 27 Feb 2025 ( v1 ), last revised 28 Aug 2026 (this version, v6)] Title: A posteriori certification of PDE approximations with particular application to neural networks Authors: Lewin Ernst , Nikolaos Rekatsinas , Karsten Urban View PDF HTML (experimental) Abstract: We propose rigorous and efficiently computable lower and upper a posteriori error bounds for given approximations to PDEs on a given domain, which might be geometrically complex. This is done by embedding or enveloping the original domain towards geometrically simpler domains, enabling the use of fast numerical solvers. To this end, we extend and restrict the residual and provide efficient methods to compute those Hahn-Banach extensions. Then, we efficiently compute their Riesz representations on the geometrically simpler domains and obtain the desired a posteriori bounds for which we prove that they are sharp. The resulting bounds control the error in the natural norm induced by a well-posed variational formulation, require only minimal regularity assumptions, and thus remain applicable on complex geometries. The framework is detailed for elliptic as well as parabolic problems. Numerical experiments demonstrate the good quantitative behavior of the derived upper and lower error bounds. A central motivation for this paper comes from physics-informed and related neural-network approximations of PDEs, which are naturally mesh-free and can be evaluated pointwise on complex or parameter-dependent geometries. Nevertheless, the framework applies to any approximation for which the variational residual can be evaluated. Subjects: Numerical Analysis (math.NA) MSC classes: 35J20, 65M15, 68T07 Cite as: arXiv:2502.20336 [math.NA] (or arXiv:2502.20336v6 [math.NA] for this version) https://doi.org/10.48550/arXiv.2502.20336 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Nikolaos Rekatsinas [ view email ] [v1] Thu, 27 Feb 2025 18:05:23 UTC (917 KB) [v2] Wed, 27 Aug 2025 17:41:12 UTC (920 KB) [v3] Wed, 3 Sep 2025 09:20:45 UTC (920 KB) [v4] Tue, 14 Apr 2026 12:17:34 UTC (71 KB) [v5] Fri, 10 Jul 2026 09:38:59 UTC (489 KB) [v6] Fri, 28 Aug 2026 10:36:19 UTC (578 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.NA < prev | next > new | recent | 2025-02 Change to browse by: cs cs.NA math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation 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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