We propose a numerical method for approximating smooth, convex solutions to the generalized Monge-Ampère equation, a fully nonlinear partial differential equation with significant applications in optimal transport, geometric optics, and differential geometry. This method is built on a least-squares formulation of the partial differential equation, which enables an iterative scheme that decouples the nonlinear and differential components of the problem. The framework introduced in [1] leads to the solution of two minimization subproblems at each iteration: a local, nonlinear problem solved pointwise, and a linear variational problem of biharmonic type. For the latter, we employ the Deep Ritz method [2], which, unlike Physics-Informed Neural Networks, preserves the variational structure of the problem and does not impose additional regularity conditions on the input data. Convexity of the solution is enforced by using input convex neural networks [3]. Furthermore, the Deep Ritz method allows for control over the approximation error in the H² norm. We show the effectiveness of this approach through applications to the classical Dirichlet problem for the Monge-Ampère equation, the mean curvature (Minkowski) problem, and, ultimately, the optimal transport problem. Numerical experiments highlight the method’s capacity to achieve accurate solutions with controlled error in only a few iterations, showing its potential as a hybrid tool that integrates classical numerical analysis with neural network techniques.