Purpose Optimal transportation of raw material from suppliers to customers is an issue that arises frequently in logistics. A numerical framework relying on optimal transport theory allows for accurate modeling and efficient performance simulations of optimal transport problems. Design/methodology/approach A physics-informed neural network (PINN) method is advocated for the solution of the corresponding generalized Monge–Ampère equation. Convex neural networks are advocated to enforce the convexity of the solution and obtain a suitable approximation of the optimal transport map. A particular focus is set on the enforcement of transport boundary conditions in the loss function. Findings Numerical experiments illustrate the solution to the optimal transport problem in several configurations and show the accuracy and efficiency of the PINN approach. Sensitivity analyses are performed with respect to the numerical parameters. Originality/value This work presents a numerical framework to accurately model and efficiently perform simulations of optimal transport problems. A physics-informed neural network is applied in a novel way to solve the generalized Monge–Ampère equation together with the so-called transport boundary conditions.