Convex Physics Informed Neural Networks for the Monge-Amp`ere Optimal Transport Problem
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Abstract (arXiv)
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.
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References · 11
- 2018Computational Optimal Transportcited by 780
- 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 583
- 2016Input Convex Neural Networkscited by 414
- 2019Variational Physics-Informed Neural Networks For Solving Partial Differential Equationscited by 148
- 2024Challenges in Training PINNs: A Loss Landscape Perspectivecited by 113
- 2018Optimal Control Via Neural Networks: A Convex Approachcited by 85
- 2022Supervised Training of Conditional Monge Mapscited by 61
- 2019VarNet: Variational Neural Networks for the Solution of Partial Differential Equationscited by 35
- 2021Input Convex Gradient Networkscited by 17
- 2021Solving the Dirichlet problem for the Monge-Amp`ere equation using neural networkscited by 3
- 2021Physics Informed Convex Artificial Neural Networks (PICANNs) for Optimal Transport based Density Estimationcited by 2
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