Two-layers neural networks for Schr{\"o}dinger eigenvalue problems

TL;DR

This paper analyzes the use of two-layer neural networks with infinite width for solving high-dimensional Schrödinger eigenvalue problems, proposing a Wasserstein gradient flow approach and proving convergence to eigenfunctions.

Contribution

It introduces a novel Wasserstein gradient flow framework for neural network-based Schrödinger eigenvalue problem solutions and proves existence and convergence of solutions.

Findings

Proves existence of solutions to the constrained gradient flow.

Shows that convergence yields an eigenfunction of the Schrödinger operator.

First analysis of non-convex functional minimization in this context.

Abstract

The aim of this article is to analyze numerical schemes using two-layer neural networks withinfinite width for the resolution of high-dimensional Schr{\"o}dinger eigenvalue problems with smoothinteraction potentials and Neumann boundary condition on the unit cube in any dimension. Moreprecisely, any eigenfunction associated to the lowest eigenvalue of the Schr{\"o}dinger operator is a unitL 2 norm minimizer of the associated energy. Using Barron's representation of the solution witha probability measure defined on the set of parameter values and following the approach initiallysuggested by Bach and Chizat [1], the energy is minimized thanks to a constrained gradient curvedynamic on the 2-Wasserstein space of the set of parameter values defining the neural network. Weprove the existence of solutions to this constrained gradient curve. Furthermore, we prove that,if it converges, the represented function is then an eigenfunction of the considered Schr{\"o}dingeroperator. At least up to our knowledge, this is the first work where this type of analysis is carriedout to deal with the minimization of non-convex functionals.