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This research applies physics-informed neural networks with convex constraints to solve the generalized Monge-Ampère equation for optimal transport problems. By enforcing boundary conditions directly in the loss function, the method provides a robust approximation of the optimal transport map without relying on traditional numerical solvers. The work advances machine learning applications in continuous models where physical laws and geometric constraints are central to the solution.
Convex Physics Informed Neural Networks for the Monge-Amp\`ere Optimal Transport Problem
arXiv:2501.10162v3 Announce Type: replace-cross Abstract: 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 neural network 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-Amp\`ere 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.
Abstract: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 neural network method is advocated here for the solution of the corresponding generalized Monge-Ampère 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.
Numerical Analysis (math.NA); Machine Learning (cs.LG)
arXiv:2501.10162 [math.NA]
(or arXiv:2501.10162v3 [math.NA] for this version)
Caboussat A, Peruso A (2026), "Convex physics-informed neural networks for the Monge-Ampère optimal transport problem". Engineering Computations, Vol. 43 No. 8 pp. 3151-3170
https://doi.org/10.1108/EC-09-2024-0883 DOI(s) linking to related resources
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