A PRIORI GENERALIZATION ERROR ANALYSIS OF TWO-LAYER NEURAL NETWORKS FOR SOLVING HIGH DIMENSIONAL SCHRÖDINGER EIGENVALUE PROBLEMS

Abstract

This paper analyzes the generalization error of two-layer neural networks for computing the ground state of the Schrödinger operator on a d-dimensional hypercube with Neumann boundary condition. We prove that the convergence rate of the generalization error is independent of 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 latter is achieved by a fixed point argument based on the Krein-Rutman theorem.