A Priori Generalization Error Analysis of Two-Layer Neural Networks for Solving High Dimensional Schr\"odinger Eigenvalue Problems
Numerical Analysis
2021-05-05 v1 Numerical Analysis
Mathematical Physics
Analysis of PDEs
math.MP
Probability
Machine Learning
Abstract
This paper analyzes the generalization error of two-layer neural networks for computing the ground state of the Schr\"odinger operator on a -dimensional hypercube. We prove that the convergence rate of the generalization error is independent of the dimension , under the a priori assumption that the ground state lies in a spectral Barron space. We verify such assumption by proving a new regularity estimate for the ground state in the spectral Barron space. The later is achieved by a fixed point argument based on the Krein-Rutman theorem.
Cite
@article{arxiv.2105.01228,
title = {A Priori Generalization Error Analysis of Two-Layer Neural Networks for Solving High Dimensional Schr\"odinger Eigenvalue Problems},
author = {Jianfeng Lu and Yulong Lu},
journal= {arXiv preprint arXiv:2105.01228},
year = {2021}
}