Beckmann Transport Models

Beckmann Transport Models (BTM) are a generative-modeling framework introduced by Lee Cheuk-Kit and coauthors in "Beckmann Transport Models: From Autonomous Flows to One-Step Maps" (arXiv:2608.01692, August 2026). BTM instantiates flow matching, the technique of learning a velocity field that transports a simple source distribution to a data distribution, using a time-independent ("autonomous") flow rather than the usual time-dependent one. When the target distribution is singular, concentrated on a lower-dimensional manifold as image and text data typically are, this autonomous flow maps exactly between the two distributions.

The paper's central result is that the one-step generative map associated with the autonomous flow is the unique solution of a simple conservation equation. This equation can be fit directly from data samples, without numerically integrating a learned velocity field or distilling an already-trained diffusion model. The flow and its one-step map therefore have a physical interpretation tied to the flux constraint of Beckmann's classical transportation problem. The authors show the framework recovers, as special cases, the closed-form Poisson-flow generative model and equilibrium matching. They demonstrate one-step generation on ImageNet 256x256.

The same month, Sophia Tang, Shiyi Wang, and coauthors extended the framework to discrete data with Discrete Beckmann Transport Models (arXiv:2609.15903). This version carries any point in a discrete latent space to a fixed vertex of the probability simplex in one step, without a teacher flow or time conditioning. On language modeling and reasoning benchmarks, it reports state-of-the-art few-step performance, including 84.6% accuracy in four function evaluations on Sudoku-Hard.