A new arXiv paper explores the mathematical properties of functions within Barron spaces, which are tailored for wide ReLU networks with a single hidden layer. The research demonstrates that harmonic functions with Dirichlet boundary data in Barron spaces are not generally Lipschitz continuous or in the Sobolev class $H^2$. However, these functions can be approximated with high accuracy by Barron functions of low norm, particularly in Lebesgue and Sobolev norms with at most two derivatives. This regularity theory has implications for deriving error estimates for Deep Ritz neural PDE solvers. AI
IMPACT Provides theoretical underpinnings for understanding the capabilities and limitations of wide ReLU networks in PDE solving.
RANK_REASON The cluster contains an academic paper detailing mathematical theory and its application to neural networks. [lever_c_demoted from research: ic=1 ai=1.0]
- arXiv
- Barron space
- Deep Ritz Method for Elliptical Multiple Eigenvalue Problems
- Lipschitz condition
- Relu Networks
- Sobolev class $H^2$
- Stephan Wojtowytsch
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