# Convex Physics Informed Neural Networks for the Monge-Amp`ere Optimal Transport Problem

arXiv: https://arxiv.org/abs/2501.10162 · HTML: https://mlsanity.com/paper/2501.10162/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 33 references that are in the map.
- First posted on arXiv 17 Jan 2025.

## Builds on (most cited first, 11 of 11)

- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Challenges in Training PINNs: A Loss Landscape Perspective](https://mlsanity.com/paper/2402.01868/) (2024)
- [Optimal Control Via Neural Networks: A Convex Approach](https://mlsanity.com/paper/1805.11835/) (2018)
- [Supervised Training of Conditional Monge Maps](https://mlsanity.com/paper/2206.14262/) (2022)
- [VarNet: Variational Neural Networks for the Solution of Partial Differential Equations](https://mlsanity.com/paper/1912.07443/) (2019)
- [Input Convex Gradient Networks](https://mlsanity.com/paper/2111.12187/) (2021)
- [Solving the Dirichlet problem for the Monge-Amp`ere equation using neural networks](https://mlsanity.com/paper/2110.03310/) (2021)
- [Physics Informed Convex Artificial Neural Networks (PICANNs) for Optimal Transport based Density Estimation](https://mlsanity.com/paper/2104.01194/) (2021)

## 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.