A Physics-Informed Neural Network Approach for Solving the Monge-Ampère Partial Differential Equation

Abstract

This paper presents a comparative study of classical numerical and machine learning techniques for solving the nonlinear Monge–Ampère Equation (MAE). A finite difference method (FDM) implemented in MATLAB is first validated using a quadratic exact solution on a square domain. Two neural network approaches are then investigated: a supervised neural network trained on analytical solution data, and a Physics-Informed Neural Network (PINN) implemented in Python using a Streamlit interface. The PINN solves the MAE through an unsupervised learning strategy based on minimizing PDE residuals and boundary losses. The effects of tanh and sigmoid activation functions on convergence and accuracy are also examined. Numerical results show that while FDM provides a reliable benchmark, PINNs offer an effective mesh-free alternative. Compared with classical methods, PINNs trade algorithmic rigor for adaptability and dimensional scalability, offering a modern, mesh-free perspective on PDE approximation. The ReLU/Tanh networks employed here is capable of representing both convex and concave solutions, however the gradient descent procedure can get trapped in the concave branch with the hyperbolic tangent (tanh) activation function, as we discovered in our case. Overall, our study demonstrates the potential of PINNs for solving fully nonlinear partial differential equations.