Résumé
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 by probability measures over parameter space, the energy minimization problem is reformulated as a constrained optimization problem over measures. Following ideas from Wasserstein gradient flows, we introduce and analyze a constrained gradient curve in the space of probability measures. We prove existence of such curves and show that, whenever the dynamics converges, 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 dimension.
