# Learning normalizing flows from Entropy-Kantorovich potentials

arXiv: https://arxiv.org/abs/2006.06033 · HTML: https://mlsanity.com/paper/2006.06033/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jun 2020.

## Built on by (most cited first, 18 of 18)

- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021)
- [Plugin Estimation of Smooth Optimal Transport Maps](https://mlsanity.com/paper/2107.12364/) (2021)
- [Energy-guided Entropic Neural Optimal Transport](https://mlsanity.com/paper/2304.06094/) (2023)
- [Optimal transport map estimation in general function spaces](https://mlsanity.com/paper/2212.03722/) (2022)
- [Minimax estimation of discontinuous optimal transport maps: The semi-discrete case](https://mlsanity.com/paper/2301.11302/) (2023)
- [Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein space](https://mlsanity.com/paper/2312.02849/) (2023)
- [Comprehensive Review of Neural Differential Equations for Time Series Analysis](https://mlsanity.com/paper/2502.09885/) (2025)
- [Neural Optimal Transport with Lagrangian Costs](https://mlsanity.com/paper/2406.00288/) (2024)
- [Learning High Dimensional Wasserstein Geodesics](https://mlsanity.com/paper/2102.02992/) (2021)
- [Plug-in estimation of Schr"odinger bridges](https://mlsanity.com/paper/2408.11686/) (2024)
- [Turning Normalizing Flows into Monge Maps with Geodesic Gaussian Preserving Flows](https://mlsanity.com/paper/2209.10873/) (2022)
- [Scalable Simulation-free Entropic Unbalanced Optimal Transport](https://mlsanity.com/paper/2410.02656/) (2024)
- [Devil is in the Details: Density Guidance for Detail-Aware Generation with Flow Models](https://mlsanity.com/paper/2502.05807/) (2025)
- [Simplified Diffusion Schr"odinger Bridge](https://mlsanity.com/paper/2403.14623/) (2024)
- [Normalizing flows as approximations of optimal transport maps via linear-control neural ODEs](https://mlsanity.com/paper/2311.01404/) (2023)

## Abstract (arXiv)

We approach the problem of learning continuous normalizing flows from a dual perspective motivated by entropy-regularized optimal transport, in which continuous normalizing flows are cast as gradients of scalar potential functions. This formulation allows us to train a dual objective comprised only of the scalar potential functions, and removes the burden of explicitly computing normalizing flows during training. After training, the normalizing flow is easily recovered from the potential functions.