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[2602.19381] Regularity of Second-Order Elliptic PDEs in Spectral Barron Spaces
Summary
This paper establishes a regularity theorem for second-order elliptic PDEs in spectral Barron spaces, demonstrating that solutions can achieve higher regularity under certain conditions.
Why It Matters
Understanding the regularity of solutions to elliptic PDEs is crucial in both mathematics and machine learning, particularly in approximating solutions with neural networks. This research could enhance the application of neural networks in solving complex mathematical problems.
Key Takeaways
- Regularity theorem established for second-order elliptic PDEs.
- Solutions gain two additional orders of Barron regularity under mild conditions.
- Identifies a class of PDEs approximable by two-layer neural networks.
- Neural network width is independent of spatial dimensions.
- Implications for both mathematical analysis and machine learning applications.
Mathematics > Analysis of PDEs arXiv:2602.19381 (math) [Submitted on 22 Feb 2026] Title:Regularity of Second-Order Elliptic PDEs in Spectral Barron Spaces Authors:Ziang Chen, Liqiang Huang, Mengxuan Yang, Shengxuan Zhou View a PDF of the paper titled Regularity of Second-Order Elliptic PDEs in Spectral Barron Spaces, by Ziang Chen and 3 other authors View PDF HTML (experimental) Abstract:We establish a regularity theorem for second-order elliptic PDEs on $\mathbb{R}^{d}$ in spectral Barron spaces. Under mild ellipticity and smallness assumptions, the solution gains two additional orders of Barron regularity. As a corollary, we identify a class of PDEs whose solutions can be approximated by two-layer neural networks with cosine activation functions, where the width of the neural network is independent of the spatial dimension. Subjects: Analysis of PDEs (math.AP); Machine Learning (cs.LG); Numerical Analysis (math.NA) Cite as: arXiv:2602.19381 [math.AP] (or arXiv:2602.19381v1 [math.AP] for this version) https://doi.org/10.48550/arXiv.2602.19381 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Liqiang Huang [view email] [v1] Sun, 22 Feb 2026 23:29:57 UTC (22 KB) Full-text links: Access Paper: View a PDF of the paper titled Regularity of Second-Order Elliptic PDEs in Spectral Barron Spaces, by Ziang Chen and 3 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: math.AP < prev | next >...