A priori generalization error analysis of two-layer neural networks for solving high dimensional Schrödinger eigenvalue problems
Jianfeng Lu, Yulong Lu
Abstract
This paper analyzes the generalization error of two-layer neural networks for computing the ground state of the Schrödinger operator on a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -dimensional hypercube with Neumann boundary condition. We prove that the convergence rate of the generalization error is independent of dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , 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.