# GradNetOT: Learning Optimal Transport Maps with GradNets

arXiv: https://arxiv.org/abs/2507.13191 · HTML: https://mlsanity.com/paper/2507.13191/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 27 references that are in the map.
- First posted on arXiv 17 Jul 2025.

## Builds on (most cited first, 10 of 10)

- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark](https://mlsanity.com/paper/2106.01954/) (2021)
- [Convex Potential Flows: Universal Probability Distributions with Optimal Transport and Convex Optimization](https://mlsanity.com/paper/2012.05942/) (2020)
- [Supervised Training of Conditional Monge Maps](https://mlsanity.com/paper/2206.14262/) (2022)
- [Riemannian Convex Potential Maps](https://mlsanity.com/paper/2106.10272/) (2021)
- [Input Convex Gradient Networks](https://mlsanity.com/paper/2111.12187/) (2021)
- [Learning Gradients of Convex Functions with Monotone Gradient Networks](https://mlsanity.com/paper/2301.10862/) (2023)
- [High-dimensional Optimal Density Control with Wasserstein Metric Matching](https://mlsanity.com/paper/2307.13135/) (2023)

## Abstract (arXiv)

Monotone gradient functions play a central role in solving the Monge formulation of the optimal transport (OT) problem, which arises in modern applications ranging from fluid dynamics to robot swarm control. When the transport cost is the squared Euclidean distance, Brenier's theorem guarantees that the unique optimal transport map satisfies a Monge-Amp`ere equation and is the gradient of a convex function. In [arXiv:2301.10862] [arXiv:2404.07361], we proposed Monotone Gradient Networks (mGradNets), neural networks that directly parameterize the space of monotone gradient maps. In this work, we leverage mGradNets to directly learn the optimal transport mapping by minimizing a training loss function defined using the Monge-Amp`ere equation. We empirically show that the structural bias of mGradNets facilitates the learning of optimal transport maps across both image morphing tasks and high-dimensional OT problems.