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.

翻译：本文旨在分析利用具有无限宽度的两层神经网络求解带有Neumann边界条件的高维Poisson-Neumann偏微分方程（PDEs）的数值格式。通过使用Barron关于解的表示（结合概率测度），利用定义神经网络参数的$2$ Wasserstein空间上的梯度曲线动力学最小化能量。受Bach和Chizat工作的启发，我们证明：若梯度曲线收敛，则所表示的函数即为所考虑的椭圆方程的解。文中给出了数值实验以展示该方法的潜力。