This paper proposed a hybrid backpropagation neural network combined with an improved particle swarm optimization (PSO) algorithm to solve the Monge–Ampère equation subject to transport boundary conditions, a challenging nonlinear partial differential equation with broad applications in optimal transport and geometric analysis. The proposed method approximates the solution using a neural network, where the loss function incorporates both the residual of the Monge–Ampère equation and the boundary conditions. Traditional gradient-based optimization techniques often struggle with local minima; hence we integrate an improved PSO algorithm to enhance global exploration while retaining the local search capability of backpropagation. The improved PSO introduces additional mechanisms to avoid premature convergence and refine the search process, thereby achieving a more robust optimization framework. Furthermore, the neural network architecture is designed as a fully connected network with multiple hidden layers, employing activation functions tailored to the problem’s nonlinearity. Our approach demonstrates significant potential in addressing the computational difficulties associated with the Monge–Ampère equation, particularly in handling its high nonlinearity and complex boundary conditions. The hybrid methodology not only improves solution accuracy but also offers a generalizable framework for solving similar high-dimensional nonlinear problems. Numerical experiments and theoretical analysis will be presented to validate the effectiveness of the proposed method, highlighting its advantages over conventional techniques in terms of convergence and precision. This work contributes to the growing field of machine learning-assisted numerical methods for partial differential equations, providing a practical and efficient alternative for problems where traditional solvers fall short.