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

• URL: http://arxiv.org/abs/2608.01692v2
• Date: Tue, 04 Aug 2026 14:02:59 GMT
• Title: Beckmann Transport Models: From Autonomous Flows to One-Step Maps
• Abstract summary: We propose an instantiation of flow matching that relies on a time-independent velocity field to map between two distributions.<n>We also show that the one-step generative map associated with this flow is the unique solution of a simple conservation equation.
• Score: 47.88178310873025
• License: http://creativecommons.org/licenses/by/4.0/
• 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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