Adaptive Weighting Strategies and Loss Balancing For Physics Informed Neural Network Solution of the Monge Ampere Partial Differential Equation

Abstract

In this research we present a comprehensive computational framework for solving the nonlinear Monge-Ampère equation using an adaptive Physics-Informed Neural Network (PINN) approach. The Monge-Ampère equation is a fully nonlinear second-order partial differential equation which plays a fundamental role in optimal transport theory, differential geometry, image processing, and fluid mechanics. Despite its importance, numerical solutions remain challenging due to the equation's inherent nonlinearity, the requirement of convexity, and the existence of multiple solution branches. Traditional numerical methods, including finite differences and finite elements, often struggle with complex geometries, high-dimensional domains, and the enforcement of convexity constraints. We propose an adaptive PINN framework that addresses these limitations through a novel gradient magnitude balancing strategy. The neural network architecture employs a multilayer perceptron with tanh activation functions to approximate the solution on a square domain. Unlike supervised approaches that require labeled data, our unsupervised method incorporates the governing physics directly into the loss function through automatic differentiation, enabling mesh-free computation of the Hessian components. The total loss function comprises three components: a PDE residual loss enforcing the Monge-Ampère equation, a boundary loss enforcing Dirichlet conditions on the boundary, and a convexity loss penalizing negative eigenvalues of the Hessian matrix to ensure positive definiteness. The tanh activation produced an inverted (concave) solution because the convexity penalty wasn't strong enough to guide the optimizer to the correct branch.