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

Source: [Beckmann Transport Models: From Autonomous Flows to One-Step Maps](<https://www.sota2.com/research/paper/6a717260bc6f0f0705845e4b>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2026-08-15T05:00:01.149Z

Authors: Lee Cheuk-Kit, Florentin Coeurdoux, Yuyuan Chen, Sophia Tang, Peter Potaptchik, Yilun Du, Michael Samuel Albergo, Eric Vanden-Eijnden

Published: 2026-08-03T04:45:52.000Z

## Abstract

We propose an instantiation of flow matching that relies on a time-independent velocity field (an \\emph{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.

## Recorded benchmark results

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2608.01692>)

[PDF](<https://arxiv.org/pdf/2608.01692.pdf>)