# Flow Matching: Markov Kernels, Stochastic Processes and Transport Plans

arXiv: https://arxiv.org/abs/2501.16839 · HTML: https://mlsanity.com/paper/2501.16839/

## Key facts

- Cited by 17 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 10 of its 47 references that are in the map.
- First posted on arXiv 28 Jan 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, 10 of 10)

- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Rectified Flow: A Marginal Preserving Approach to Optimal Transport](https://mlsanity.com/paper/2209.14577/) (2022)
- [Flow Priors for Linear Inverse Problems via Iterative Corrupted Trajectory Matching](https://mlsanity.com/paper/2405.18816/) (2024)
- [Consistent Diffusion Meets Tweedie: Training Exact Ambient Diffusion Models with Noisy Data](https://mlsanity.com/paper/2404.10177/) (2024)
- [Conditional Wasserstein Distances with Applications in Bayesian OT Flow Matching](https://mlsanity.com/paper/2403.18705/) (2024)
- [Conditional Optimal Transport on Function Spaces](https://mlsanity.com/paper/2311.05672/) (2023)
- [Dynamic Conditional Optimal Transport through Simulation-Free Flows](https://mlsanity.com/paper/2404.04240/) (2024)
- [Understanding the training of infinitely deep and wide ResNets with Conditional Optimal Transport](https://mlsanity.com/paper/2403.12887/) (2024)

## Built on by (most cited first, 17 of 17)

- [On the Relation between Rectified Flows and Optimal Transport](https://mlsanity.com/paper/2505.19712/) (2025)
- [Telegrapher's Generative Model via Kac Flows](https://mlsanity.com/paper/2506.20641/) (2025)
- [Spherical Flows for Sampling Categorical Data](https://mlsanity.com/paper/2605.05629/) (2026)
- [A Kinetic Energy Perspective of Flow Matching](https://mlsanity.com/paper/2602.07928/) (2026)
- [Generalized Wasserstein Flow Matching: Transport Plans, Everywhere, All at Once](https://mlsanity.com/paper/2605.08424/) (2026)
- [A Jump-Diffusion Framework for Irregular Time Series Generation](https://mlsanity.com/paper/2505.23215/) (2025)
- [Provable Mixed-Noise Learning with Flow-Matching](https://mlsanity.com/paper/2508.18122/) (2025)
- [On The Hidden Biases of Flow Matching Samplers](https://mlsanity.com/paper/2512.16768/) (2025)
- [Position-Blind Ptychography: Viability of image reconstruction via data-driven variational inference](https://mlsanity.com/paper/2509.25269/) (2025)
- [Limit Points of Reflow with Minibatch Optimal Transport](https://mlsanity.com/paper/2608.07042/) (2026)
- [Adapting Noise to Data: Generative Flows from 1D Processes](https://mlsanity.com/paper/2510.12636/) (2025)
- [Estimating Orbital Parameters of Direct Imaging Exoplanet Using Neural Network](https://mlsanity.com/paper/2510.17459/) (2025)
- [Sampling via Stochastic Interpolants by Langevin-based Velocity and Initialization Estimation in Flow ODEs](https://mlsanity.com/paper/2601.08527/) (2026)
- [Regularity of Solutions to Beckmann's Parametric Optimal Transport](https://mlsanity.com/paper/2603.19755/) (2026)
- [A neural operator view on U-Nets for inverse imaging problems](https://mlsanity.com/paper/2608.05839/) (2026)
- [Gromov-Monge Flow Matching for Equivariant Graph Generation](https://mlsanity.com/paper/2608.26961/) (2026)
- [Flow Matching for Fast Posterior Sampling in Bayesian Inverse Problems](https://mlsanity.com/paper/2610.02377/) (2026)

## Abstract (arXiv)

Among generative neural models, flow matching techniques stand out for their simple applicability and good scaling properties. Here, velocity fields of curves connecting a simple latent and a target distribution are learned. Then the corresponding ordinary differential equation can be used to sample from a target distribution, starting in samples from the latent one. This paper reviews from a mathematical point of view different techniques to learn the velocity fields of absolutely continuous curves in the Wasserstein geometry. We show how the velocity fields can be characterized and learned via i) transport plans (couplings) between latent and target distributions, ii) Markov kernels and iii) stochastic processes, where the latter two include the coupling approach, but are in general broader. Besides this main goal, we show how flow matching can be used for solving Bayesian inverse problems, where the definition of conditional Wasserstein distances plays a central role. Finally, we briefly address continuous normalizing flows and score matching techniques, which approach the learning of velocity fields of curves from other directions.