# Numerical solution of Poisson partial differential equations in high dimension using two-layer neural networks

arXiv: https://arxiv.org/abs/2305.09408 · HTML: https://mlsanity.com/paper/2305.09408/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 May 2023.

## Built on by (most cited first, 1 of 1)

- [Feature Learning for the High Dimensional Stationary Sch"odinger Equation with Deep Ritz Method](https://mlsanity.com/paper/2607.06369/) (2026)

## Abstract (arXiv)

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.