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, discontinuities and noise in the source function, we consider nontrivial domains, and we investigate how the method performs in higher dimensions. We investigate the convergence numerically and present error estimates based on a stability result. 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：Monge-Ampère方程是一类完全非线性偏微分方程，在分析学、几何学及应用科学中具有基础性重要意义。本文采用神经网络求解与Monge-Ampère方程相关的Dirichlet问题，并证明利用深度输入凸神经网络的假设能够找到唯一的凸解。在分析过程中，我们研究了源函数中奇异性、间断性及噪声的影响，考虑了非平凡区域，并探究了该方法在高维空间中的表现。我们通过数值方法验证了收敛性，并基于稳定性结果给出了误差估计。此外，我们将该方法与另一种采用标准前馈网络并附加凸性缺失惩罚损失函数的替代方案进行了比较。