Beckmann Transport Models: From Autonomous Flows to One-Step Maps
2026-08-03 • Machine Learning
Machine Learning
AI summaryⓘ
The authors introduce a new method using a time-independent flow to transform one data distribution exactly into another when the target data lies on a lower-dimensional surface. They prove that the one-step transformation linked to this flow solves a simple equation, allowing it to be learned directly from data samples. Their approach connects ideas from optimal transport theory and fixes problems found in earlier methods. They test their method on ImageNet images and show it works well.
flow matchingautonomous flowsingular distributiondata manifoldconservation equationoptimal transportBeckmann's transportation problemPoisson-flow generative modelquadratic flow-matching lossImageNet
Authors
Lee Cheuk-Kit, Florentin Coeurdoux, Peter Potaptchik, Yilun Du, Michael Samuel Albergo, Eric Vanden-Eijnden
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.