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

2026-08-03

AI Summary

This work addresses the problem of constructing an exact one-step generative mapping between a source distribution and a singular target distribution supported on a low-dimensional manifold. To this end, it proposes a flow matching approach based on time-independent (autonomous) velocity fields, which directly learns the desired mapping through a conservation equation and provides a dynamical interpretation of the flux constraint arising in the Beckmann formulation of optimal transport. Theoretically, the paper establishes the equivalence between autonomous flows and one-step mappings, unifying Poisson flow generative models with equilibrium matching under quadratic regression loss, thereby resolving inconsistencies in existing methods. Empirically, the proposed approach demonstrates its effectiveness through high-quality image generation on ImageNet at 256×256 resolution.

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.

Problem

• autonomous flow
• singular distribution
• Beckmann transportation
• one-step generative map
• flow matching

Innovation

• autonomous flow
• flow matching
• Beckmann transport
• one-step map
• singular distributions