analiza aiqbench
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
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.
Text integral sub licența deschisă a publicației · 164 cuvinte
Subiecte conexe
Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations
arXiv · 30 sept. 2026
Neural Network Approximation of Solutions to Fractional Parabolic Partial Differential Equations
arXiv · 31 iul. 2026
Barron Space Representations for Elliptic PDEs with Homogeneous Boundary Conditions
arXiv · 4 aug. 2026
Spectral Convergence of Random Feature Method in Multiple Dimensions
arXiv · 5 sept. 2026