A Priori Generalization Error Analysis of Two-Layer Neural Networks for Solving High Dimensional Schr\"odinger Eigenvalue Problems

TL;DR

This paper provides a dimension-independent analysis of the generalization error for two-layer neural networks applied to high-dimensional Schrödinger eigenvalue problems, under spectral Barron space assumptions.

Contribution

It establishes a priori generalization error bounds that are independent of dimension and verifies the spectral Barron space assumption for the ground state.

Findings

Error convergence rate is dimension-independent

Ground state lies in spectral Barron space under certain conditions

New regularity estimate for the ground state

Abstract

This paper analyzes the generalization error of two-layer neural networks for computing the ground state of the Schr\"odinger operator on a $d$-dimensional hypercube. We prove that the convergence rate of the generalization error is independent of the dimension $d$, 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.