# Computing high-dimensional optimal transport by flow neural networks

arXiv: https://arxiv.org/abs/2305.11857 · HTML: https://mlsanity.com/paper/2305.11857/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 51 of its 83 references that are in the map.
- First posted on arXiv 19 May 2023.

## Builds on (most cited first, 30 of 51)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Image-to-Image Translation with Conditional Adversarial Networks](https://mlsanity.com/paper/1611.07004/) (2016)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Taming Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2012.09841/) (2020)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [MINE: Mutual Information Neural Estimation](https://mlsanity.com/paper/1801.04062/) (2018)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [Generative Visual Manipulation on the Natural Image Manifold](https://mlsanity.com/paper/1609.03552/) (2016)
- [A note on the evaluation of generative models](https://mlsanity.com/paper/1511.01844/) (2015)
- [Rectified Flow: A Marginal Preserving Approach to Optimal Transport](https://mlsanity.com/paper/2209.14577/) (2022)
- [Joint Distribution Optimal Transportation for Domain Adaptation](https://mlsanity.com/paper/1705.08848/) (2017)
- [Learning to Discover Cross-Domain Relations with Generative Adversarial Networks](https://mlsanity.com/paper/1703.05192/) (2017)
- [How to train your neural ODE: the world of Jacobian and kinetic regularization](https://mlsanity.com/paper/2002.02798/) (2020)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics](https://mlsanity.com/paper/2002.04461/) (2020)
- [OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport](https://mlsanity.com/paper/2006.00104/) (2020)
- [Wasserstein Fair Classification](https://mlsanity.com/paper/1907.12059/) (2019)
- [Neural Optimal Transport](https://mlsanity.com/paper/2201.12220/) (2022)
- [Telescoping Density-Ratio Estimation](https://mlsanity.com/paper/2006.12204/) (2020)
- [Action Matching: Learning Stochastic Dynamics from Samples](https://mlsanity.com/paper/2210.06662/) (2022)

## Built on by (most cited first, 8 of 8)

- [Convergence of flow-based generative models via proximal gradient descent in Wasserstein space](https://mlsanity.com/paper/2310.17582/) (2023)
- [Flow-based Distributionally Robust Optimization](https://mlsanity.com/paper/2310.19253/) (2023)
- [Density Ratio Estimation with Conditional Probability Paths](https://mlsanity.com/paper/2502.02300/) (2025)
- [Annealing Flow Generative Models Towards Sampling High-Dimensional and Multi-Modal Distributions](https://mlsanity.com/paper/2409.20547/) (2024)
- [Local Flow Matching Generative Models](https://mlsanity.com/paper/2410.02548/) (2024)
- [Generative models for decision-making under distributional shift](https://mlsanity.com/paper/2604.04342/) (2026)
- [High-dimensional Mean-Field Games by Particle-based Flow Matching](https://mlsanity.com/paper/2512.01172/) (2025)
- [Flow-based generative models as iterative algorithms in probability space](https://mlsanity.com/paper/2502.13394/) (2025)

## Abstract (arXiv)

Computing optimal transport (OT) for general high-dimensional data has been a long-standing challenge. Despite much progress, most of the efforts including neural network methods have been focused on the static formulation of the OT problem. The current work proposes to compute the dynamic OT between two arbitrary distributions P and Q by optimizing a flow model, where both distributions are only accessible via finite samples. Our method learns the dynamic OT by finding an invertible flow that minimizes the transport cost. The trained optimal transport flow subsequently allows for performing many downstream tasks, including infinitesimal density ratio estimation (DRE) and domain adaptation by interpolating distributions in the latent space. The effectiveness of the proposed model on high-dimensional data is demonstrated by strong empirical performance on OT baselines, image-to-image translation, and high-dimensional DRE.