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：本文旨在分析采用无限宽度双层神经网络求解具有Neumann边界条件的高维Poisson-Neumann偏微分方程（PDEs）的数值格式。利用Barron表示法以概率测度形式表达解，通过定义神经网络的参数在$2$Wasserstein空间上的梯度曲线动力学实现能量最小化。受Bach与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椭圆方程的解。文中给出数值实验以展示该方法的潜力。