Neural Network Approximations of Solutions to Schrödinger Equations

Abstract

Solving eigenvalue problems for high-dimensional Schrödinger operators is a central challenge in numerical analysis, with applications ranging from quantum mechanics to the approximation of complex interacting systems. In this talk, we present a variational approach based on infinite-width two-layer neural networks for the computation of the lowest eigenvalue and associated eigenfunctions of Schrödinger operators with smooth interaction potentials and Neumann boundary conditions on the unit cube.

Using a Barron-type representation of neural network functions as probability measures over parameter space, the energy minimization problem is reformulated as a constrained optimization problem over measures. Drawing on ideas from Wasserstein gradient flows, we introduce and analyze a constrained gradient curve in the space of probability measures. We prove the existence of such curves and show that, whenever the dynamics converge, the limiting represented function is an eigenfunction of the Schrödinger operator. This provides a theoretical foundation for neural-network-based variational methods for non-convex eigenvalue problems in high dimensions.

Virginie Galland Ehrlacher

Virginie Galland Ehrlacher is a mathematician and professor at CERMICS, the applied mathematics laboratory at the École nationale des Ponts et Chaussées. Her work focuses on the development and mathematical analysis of efficient numerical methods for solving large-scale problems, particularly those arising from applications in materials science. She is the principal investigator of the ERC Starting Grant project HighLEAP and was the 2023 recipient of the Irène Joliot-Curie Prize from the Academy of Sciences and Technologies in the “Young Female Scientist” category.