Heat equations in spectral Barron spaces

[2608.26657] Heat equations in spectral Barron spaces Skip to main content Search Submit Donate Log in Mathematics > Analysis of PDEs arXiv:2608.26657 (math) [Submitted on 27 Aug 2026 ( v1 ), last revised 28 Aug 2026 (this version, v2)] Title: Heat equations in spectral Barron spaces Authors: Mourad Choulli , Shuai Lu , Hiroshi Takase View PDF HTML (experimental) Abstract: Spectral Barron spaces, characterized by an \(L^1\)-based Fourier-Lebesgue norm, have earned significant attention in approximation theory due to their remarkable capacity to represent functions via shallow neural networks with controlled complexity. Meanwhile, recent theoretical advances have firmly established an intrinsic and profound connection between these function spaces and the regularity theory of elliptic partial differential equations. Building upon this foundational interplay, the present work undertakes a systematic and comprehensive investigation into the well-posedness of heat equations formulated within the spectral Barron spaces framework. Specifically, we rigorously establish the core aspects of well-posedness, including the existence, uniqueness, and stability of solutions, under suitable assumptions on the source terms and conductivity coefficients. We also investigate a typical parabolic inverse problem, namely the backward heat equation, for which we derive a logarithmic conditional stability estimate. To the best of our knowledge, this constitutes the first stability estimate for inverse problems within the spectral Barron space setting. Moreover, we extend our analytical results to address the more intricate setting of time-fractional heat equations, which govern anomalous diffusion phenomena and introduce nonlocal temporal memory effects. In this extended context, we provide a characterization of the corresponding heat kernels, deriving decay estimates, regularity properties, thereby enriching the theoretical landscape of evolutionary PDEs within the spectral Barron spaces setting. Subjects: Analysis of PDEs (math.AP) Cite as: arXiv:2608.26657 [math.AP] (or arXiv:2608.26657v2 [math.AP] for this version) https://doi.org/10.48550/arXiv.2608.26657 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Hiroshi Takase [ view email ] [v1] Thu, 27 Aug 2026 06:11:54 UTC (24 KB) [v2] Fri, 28 Aug 2026 12:13:11 UTC (24 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.AP < prev | next > new | recent | 2026-08 Change to browse by: math References & Citations SciX 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 (opens in new tab) (opens in new tab) (opens in new tab)

Related documents

• Heat equations in spectral Barron spaces#779382
• Stability of the reconstruction of the heat reflection coefficient in the phonon transport equation#614889
• Stability of the reconstruction of the heat reflection coefficient in the phonon transport equation#613922
• Fast automated adjoints for spectral PDE solvers#120150
• Fast automated adjoints for spectral PDE solvers.#14924