Two-layers neural networks for Schr"odinger eigenvalue problems

The aim of this article is to analyze numerical schemes using two-layer neural networks with infinite width for the resolution of high-dimensional Schr"odinger eigenvalue problems with smooth interaction potentials and Neumann boundary condition on the unit cube in any dimension. More precisely, any eigenfunction associated to the lowest eigenvalue of the Schr"odinger operator is a unit L 2 norm minimizer of the associated energy. Using Barron's representation of the solution with a probability measure defined on the set of parameter values and following the approach initially suggested by Bach and Chizat [1], the energy is minimized thanks to a constrained gradient curve dynamic on the 2-Wasserstein space of the set of parameter values defining the neural network. We prove 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"odinger operator. At least up to our knowledge, this is the first work where this type of analysis is carried out to deal with the minimization of non-convex functionals.