Convex Physics Informed Neural Networks for the Monge-Ampère Optimal Transport Problem
This addresses logistics optimization for material transport, but it is incremental as it applies an existing neural network method to a specific equation with convexity constraints.
The paper tackles the optimal transport problem in logistics by solving the Monge-Ampère equation using a physics-informed neural network method, with convex neural networks enforcing solution convexity and boundary conditions in the loss function, achieving suitable approximations in numerical experiments.
Optimal transportation of raw material from suppliers to customers is an issue arising in logistics that is addressed here with a continuous model relying on optimal transport theory. A physics informed neuralnetwork method is advocated here for the solution of the corresponding generalized Monge-Amp`ere equation. Convex neural networks are advocated to enforce the convexity of the solution to the Monge-Ampère equation 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. Numerical experiments illustrate the solution to the optimal transport problem in several configurations, and sensitivity analyses are performed.