A Regularity Theory for Static Schr"odinger Equations on ℝd in Spectral Barron Spaces
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Abstract (arXiv)
Spectral Barron spaces have received considerable interest recently as it is the natural function space for approximation theory of two-layer neural networks with a dimension-free convergence rate. In this paper we study the regularity of solutions to the whole-space static Schr"odinger equation in spectral Barron spaces. We prove that if the source of the equation lies in the spectral Barron space Bs(ℝd) and the potential function admitting a non-negative lower bound decomposes as a positive constant plus a function in Bs(ℝd), then the solution lies in the spectral Barron space Bs+2(ℝd).
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References · 11
- 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 583
- 2017Solving high-dimensional partial differential equations using deep learningcited by 556
- 2014Breaking the Curse of Dimensionality with Convex Neural Networkscited by 331
- 2017Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equationscited by 191
- 2021Sharp Bounds on the Approximation Rates, Metric Entropy, and n-widths of Shallow Neural Networkscited by 86
- 2020Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learningcited by 77
- 2021Near-Minimax Optimal Estimation With Shallow ReLU Neural Networkscited by 41
- 2021On the Representation of Solutions to Elliptic PDEs in Barron Spacescited by 25
- 2021Parametric Complexity Bounds for Approximating PDEs with Neural Networkscited by 19
- 2021A Priori Generalization Error Analysis of Two-Layer Neural Networks for Solving High Dimensional Schr"odinger Eigenvalue Problemscited by 19
- 2020Some observations on high-dimensional partial differential equations with Barron datacited by 16
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