  The aim of this article is to analyze numerical schemes using two-layer
neural networks with infinite width for the resolution of the high-dimensional
Poisson-Neumann partial differential equations (PDEs) with Neumann boundary
conditions. Using Barron's representation of the solution with a measure of
probability, the energy is minimized thanks to a gradient curve dynamic on the
$2$ Wasserstein space of parameters defining the neural network. Inspired by
the work from Bach and Chizat, we prove that if the gradient curve converges,
then the represented function is the solution of the elliptic equation
considered. Numerical experiments are given to show the potential of the
method.
