Solving the Dirichlet problem for the Monge–Ampère equation using neural networks

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