# On the Equivalence of Optimal Transport Problem and Action Matching with Optimal Vector Fields

arXiv: https://arxiv.org/abs/2510.27385 · HTML: https://mlsanity.com/paper/2510.27385/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 3 of its 5 references that are in the map.
- First posted on arXiv 31 Oct 2025.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 3 of 3)

- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Action Matching: Learning Stochastic Dynamics from Samples](https://mlsanity.com/paper/2210.06662/) (2022)
- [Optimal Flow Matching: Learning Straight Trajectories in Just One Step](https://mlsanity.com/paper/2403.13117/) (2024)

## Built on by (most cited first, 1 of 1)

- [Discrete Action Matching: Learning Stochastic Dynamics from Samples via State Graphs](https://mlsanity.com/paper/2610.05071/) (2026)

## Abstract (arXiv)

Flow Matching (FM) method in generative modeling maps arbitrary probability distributions by constructing an interpolation between them and then learning the vector field that defines ODE for this interpolation. Recently, it was shown that FM can be modified to map distributions optimally in terms of the quadratic cost function for any initial interpolation. To achieve this, only specific optimal vector fields, which are typical for solutions of Optimal Transport (OT) problems, need to be considered during FM loss minimization. In this note, we show that considering only optimal vector fields can lead to OT in another approach: Action Matching (AM). Unlike FM, which learns a vector field for a manually chosen interpolation between given distributions, AM learns the vector field that defines ODE for an entire given sequence of distributions.