# Markovian Flow Matching: Accelerating MCMC with Continuous Normalizing Flows

arXiv: https://arxiv.org/abs/2405.14392 · HTML: https://mlsanity.com/paper/2405.14392/

## Key facts

- Cited by 9 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 43 of its 87 references that are in the map.
- First posted on arXiv 23 May 2024.

## 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, 30 of 43)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Black Box Variational Inference](https://mlsanity.com/paper/1401.0118/) (2013)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluation](https://mlsanity.com/paper/1602.03253/) (2016)
- [Path Integral Sampler: a stochastic control approach for sampling](https://mlsanity.com/paper/2111.15141/) (2021)
- [Stochastic Normalizing Flows](https://mlsanity.com/paper/2002.06707/) (2020)
- [Denoising Diffusion Samplers](https://mlsanity.com/paper/2302.13834/) (2023)
- [Flow-based generative models for Markov chain Monte Carlo in lattice field theory](https://mlsanity.com/paper/1904.12072/) (2019)
- [An optimal control perspective on diffusion-based generative modeling](https://mlsanity.com/paper/2211.01364/) (2022)
- [Flow Annealed Importance Sampling Bootstrap](https://mlsanity.com/paper/2208.01893/) (2022)
- [Measuring Sample Quality with Kernels](https://mlsanity.com/paper/1703.01717/) (2017)
- [Iterated Denoising Energy Matching for Sampling from Boltzmann Densities](https://mlsanity.com/paper/2402.06121/) (2024)
- [Improved sampling via learned diffusions](https://mlsanity.com/paper/2307.01198/) (2023)
- [Theoretical guarantees for sampling and inference in generative models with latent diffusions](https://mlsanity.com/paper/1903.01608/) (2019)
- [Annealed Flow Transport Monte Carlo](https://mlsanity.com/paper/2102.07501/) (2021)
- [NeuTra-lizing Bad Geometry in Hamiltonian Monte Carlo Using Neural Transport](https://mlsanity.com/paper/1903.03704/) (2019)
- [Continual Repeated Annealed Flow Transport Monte Carlo](https://mlsanity.com/paper/2201.13117/) (2022)
- [Particle Denoising Diffusion Sampler](https://mlsanity.com/paper/2402.06320/) (2024)
- [Diffusion Generative Flow Samplers: Improving learning signals through partial trajectory optimization](https://mlsanity.com/paper/2310.02679/) (2023)
- [Bayesian Learning via Neural Schr"odinger-F"ollmer Flows](https://mlsanity.com/paper/2111.10510/) (2021)
- [Score-Based Diffusion meets Annealed Importance Sampling](https://mlsanity.com/paper/2208.07698/) (2022)
- [Target Score Matching](https://mlsanity.com/paper/2402.08667/) (2024)

## Built on by (most cited first, 9 of 9)

- [Outsourced diffusion sampling: Efficient posterior inference in latent spaces of generative models](https://mlsanity.com/paper/2502.06999/) (2025)
- [Annealing Flow Generative Models Towards Sampling High-Dimensional and Multi-Modal Distributions](https://mlsanity.com/paper/2409.20547/) (2024)
- [Amortized Bayesian Workflow](https://mlsanity.com/paper/2409.04332/) (2024)
- [Geodesic Slice Sampler for Multimodal Distributions with Strong Curvature](https://mlsanity.com/paper/2502.21190/) (2025)
- [LatentFlow: A General Framework for Conditioning Stochastic Processes](https://mlsanity.com/paper/2607.12922/) (2026)
- [Empirical evaluation of normalizing flows in Markov Chain Monte Carlo](https://mlsanity.com/paper/2412.17136/) (2024)
- [Tempered Guided Diffusion](https://mlsanity.com/paper/2605.03712/) (2026)
- [Entropic Mirror Monte Carlo](https://mlsanity.com/paper/2602.03165/) (2026)
- [GRPO-QPS: Target-Preserving Reinforcement Learning for Quantum Posterior Sampling](https://mlsanity.com/paper/2609.14711/) (2026)

## Abstract (arXiv)

Continuous normalizing flows (CNFs) learn the probability path between a reference distribution and a target distribution by modeling the vector field generating said path using neural networks. Recently, Lipman et al. (2022) introduced a simple and inexpensive method for training CNFs in generative modeling, termed flow matching (FM). In this paper, we repurpose this method for probabilistic inference by incorporating Markovian sampling methods in evaluating the FM objective, and using the learned CNF to improve Monte Carlo sampling. Specifically, we propose an adaptive Markov chain Monte Carlo (MCMC) algorithm, which combines a local Markov transition kernel with a non-local, flow-informed transition kernel, defined using a CNF. This CNF is adapted on-the-fly using samples from the Markov chain, which are used to specify the probability path for the FM objective. Our method also includes an adaptive tempering mechanism that allows the discovery of multiple modes in the target distribution. Under mild assumptions, we establish convergence of our method to a local optimum of the FM objective. We then benchmark our approach on several synthetic and real-world examples, achieving similar performance to other state-of-the-art methods, but often at a significantly lower computational cost.