The Monge-Amp\`ere equation is a fully nonlinear partial differential equation (PDE) of fundamental importance in analysis, geometry and in the applied sciences. In this paper we solve the Dirichlet problem associated with the Monge-Amp\`ere equation using neural networks and we show that an ansatz using deep input convex neural networks can be used to find the unique convex solution. As part of our analysis we study the effect of singularities and noise in the source function, we consider nontrivial domains, and we investigate how the method performs in higher dimensions. We also compare this method to an alternative approach in which standard feed-forward networks are used together with a loss function which penalizes lack of convexity.

翻译：蒙古- 安培- 安培- 电子方程式( Monge- Amp- Amp- ere 等式) 是一个在分析、几何和应用科学中具有根本重要性的完全非线性部分方程式( PDE ) 。 在本文中, 我们用神经网络来解决与蒙古- 安培- ere 等式相关的迪里切特问题, 我们显示, 使用深层输入内分泌神经网络的 ansatz 来寻找独特的二次曲线解决方案。 作为我们分析的一部分, 我们研究源函数中奇特和噪音的影响, 我们考虑非三角域, 并调查该方法如何在更高层面发挥作用。 我们还将这种方法与另一种方法相比较, 即标准进料推进网络同时使用一种损失功能, 从而惩罚缺乏二次曲线。