aiqbench analysis
This research establishes a regularity theorem for second-order elliptic PDEs in spectral Barron spaces, demonstrating that solutions gain two additional orders of regularity under mild assumptions. As a direct corollary, the study proves that specific classes of these PDE solutions can be efficiently approximated by two-layer neural networks with cosine activations, where network width remains independent of spatial dimension. This work bridges theoretical analysis of partial differential equations with the practical capabilities of deep learning architectures for high-dimensional problems.
Regularity of Second-Order Elliptic PDEs in Spectral Barron Spaces
arXiv:2602.19381v2 Announce Type: replace-cross 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.
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.
Analysis of PDEs (math.AP); Machine Learning (cs.LG); Numerical Analysis (math.NA)
arXiv:2602.19381 [math.AP]
(or arXiv:2602.19381v2 [math.AP] for this version)
https://doi.org/10.48550/arXiv.2602.19381 Focus to learn more arXiv-issued DOI via DataCite
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