# Beckmann Transport Models: From Autonomous Flows to One-Step Maps

arXiv: https://arxiv.org/abs/2608.01692 · HTML: https://mlsanity.com/paper/2608.01692/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 22 of its 28 references that are in the map.
- First posted on arXiv 3 Aug 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 22 of 22)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [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)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers](https://mlsanity.com/paper/2401.08740/) (2024)
- [Mean Flows for One-step Generative Modeling](https://mlsanity.com/paper/2505.13447/) (2025)
- [Improved Techniques for Training Consistency Models](https://mlsanity.com/paper/2310.14189/) (2023)
- [One Step Diffusion via Shortcut Models](https://mlsanity.com/paper/2410.12557/) (2024)
- [Generative Modeling via Drifting](https://mlsanity.com/paper/2602.04770/) (2026)
- [Flow map matching with stochastic interpolants: A mathematical framework for consistency models](https://mlsanity.com/paper/2406.07507/) (2024)
- [How to build a consistency model: Learning flow maps via self-distillation](https://mlsanity.com/paper/2505.18825/) (2025)
- [Poisson Flow Generative Models](https://mlsanity.com/paper/2209.11178/) (2022)
- [PFGM++: Unlocking the Potential of Physics-Inspired Generative Models](https://mlsanity.com/paper/2302.04265/) (2023)
- [Is Noise Conditioning Necessary for Denoising Generative Models?](https://mlsanity.com/paper/2502.13129/) (2025)
- [Equilibrium Matching: Generative Modeling with Implicit Energy-Based Models](https://mlsanity.com/paper/2510.02300/) (2025)
- [Blind denoising diffusion models and the blessings of dimensionality](https://mlsanity.com/paper/2602.09639/) (2026)
- [The Geometry of Noise: Why Diffusion Models Don't Need Noise Conditioning](https://mlsanity.com/paper/2602.18428/) (2026)
- [Learning Optimal Flows for Non-Equilibrium Importance Sampling](https://mlsanity.com/paper/2206.09908/) (2022)
- [Beyond Fixed Horizons: A Theoretical Framework for Adaptive Denoising Diffusions](https://mlsanity.com/paper/2501.19373/) (2025)

## Built on by (most cited first, 2 of 2)

- [Discrete Beckmann Transport Models for One-Step Language Modeling and Reasoning](https://mlsanity.com/paper/2609.15903/) (2026)
- [A statistical theory for blind denoising: minimax estimation of the noise level from a single sample](https://mlsanity.com/paper/2609.33008/) (2026)

## Abstract (arXiv)

We propose an instantiation of flow matching that relies on a time-independent velocity field (an autonomous flow) to exactly map between two distributions, so long as the target is singular, i.e. supported on a lower-dimensional data manifold. We also show that the one-step generative map associated with this flow is the unique solution of a simple conservation equation, which can be used to learn the map directly from samples. These autonomous flows and maps give a dynamical meaning to the flux constraint of Beckmann's transportation problem. Their construction provides a unifying framework that recovers, for instance, the closed-form Poisson-flow generative model and equilibrium matching with a quadratic flow-matching regression loss. We illustrate how this theory corrects inconsistencies in existing methods and demonstrate the effectiveness of the autonomous flow and the one-step map on ImageNet 256x256.