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arXiv · 2608.01692

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

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

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Lee Cheuk-Kit, Florentin Coeurdoux, Yuyuan Chen, Sophia Tang, Peter Potaptchik, Yilun Du, Michael Samuel Albergo, Eric Vanden-Eijnden. 2026-08-03. Beckmann Transport Models: From Autonomous Flows to One-Step Maps. https://arxiv.org/abs/2608.01692

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