# Improving Flow Matching by Aligning Flow Divergence

arXiv: https://arxiv.org/abs/2602.00869 · HTML: https://mlsanity.com/paper/2602.00869/

## 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 36 of its 65 references that are in the map.
- First posted on arXiv 31 Jan 2026.

## 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 36)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Taming Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2012.09841/) (2020)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Towards Accurate Generative Models of Video: A New Metric & Challenges](https://mlsanity.com/paper/1812.01717/) (2018)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [All are Worth Words: A ViT Backbone for Diffusion Models](https://mlsanity.com/paper/2209.12152/) (2022)
- [Improved Techniques for Training Consistency Models](https://mlsanity.com/paper/2310.14189/) (2023)
- [One Step Diffusion via Shortcut Models](https://mlsanity.com/paper/2410.12557/) (2024)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [Scaling Autoregressive Video Models](https://mlsanity.com/paper/1906.02634/) (2019)
- [SE(3)-Stochastic Flow Matching for Protein Backbone Generation](https://mlsanity.com/paper/2310.02391/) (2023)
- [Self-Supervised Visual Planning with Temporal Skip Connections](https://mlsanity.com/paper/1710.05268/) (2017)
- [Diffusion Models for Video Prediction and Infilling](https://mlsanity.com/paper/2206.07696/) (2022)
- [Dirichlet Flow Matching with Applications to DNA Sequence Design](https://mlsanity.com/paper/2402.05841/) (2024)
- [AlphaFold Meets Flow Matching for Generating Protein Ensembles](https://mlsanity.com/paper/2402.04845/) (2024)
- [Stochastic Latent Residual Video Prediction](https://mlsanity.com/paper/2002.09219/) (2020)
- [Latent Video Transformer](https://mlsanity.com/paper/2006.10704/) (2020)
- [Maximum Likelihood Training for Score-Based Diffusion ODEs by High-Order Denoising Score Matching](https://mlsanity.com/paper/2206.08265/) (2022)
- [Transformation-based Adversarial Video Prediction on Large-Scale Data](https://mlsanity.com/paper/2003.04035/) (2020)
- [Probabilistic Forecasting with Stochastic Interpolants and F"ollmer Processes](https://mlsanity.com/paper/2403.13724/) (2024)
- [Estimating High Order Gradients of the Data Distribution by Denoising](https://mlsanity.com/paper/2111.04726/) (2021)
- [Efficient Video Prediction via Sparsely Conditioned Flow Matching](https://mlsanity.com/paper/2211.14575/) (2022)
- [Flood Prediction Using Machine Learning Models: Literature Review](https://mlsanity.com/paper/1908.02781/) (2019)

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

- [Posterior Augmented Flow Matching](https://mlsanity.com/paper/2605.00825/) (2026)
- [Second Order Drifting Models](https://mlsanity.com/paper/2608.07924/) (2026)
- [When Can Conditional Flow Matching Replace Pointwise Negative Log-Likelihood?](https://mlsanity.com/paper/2608.28010/) (2026)
- [PathGuide: Dynamic Classifier-Free Guidance via On-Policy Transport Alignment](https://mlsanity.com/paper/2608.29107/) (2026)
- [Generative Modeling with Flux Matching](https://mlsanity.com/paper/2605.07319/) (2026)
- [StAD: Stein Amortized Divergence for Fast Likelihoods with Diffusion and Flow](https://mlsanity.com/paper/2605.16486/) (2026)
- [Divergence-Suppressing Couplings for Rectified Flow](https://mlsanity.com/paper/2605.17733/) (2026)
- [Flexible Flows for Biological Sequence Design](https://mlsanity.com/paper/2606.10543/) (2026)
- [VTV-FM: Flow Matching through Variational Terminal-Velocity Closure](https://mlsanity.com/paper/2610.00785/) (2026)

## Abstract (arXiv)

Conditional flow matching (CFM) stands out as an efficient, simulation-free approach for training flow-based generative models, achieving remarkable performance for data generation. However, CFM is insufficient to ensure accuracy in learning probability paths. In this paper, we introduce a new partial differential equation characterization for the error between the learned and exact probability paths, along with its solution. We show that the total variation gap between the two probability paths is bounded above by a combination of the CFM loss and an associated divergence loss. This theoretical insight leads to the design of a new objective function that simultaneously matches the flow and its divergence. Our new approach improves the performance of the flow-based generative model by a noticeable margin without sacrificing generation efficiency. We showcase the advantages of this enhanced training approach over CFM on several important benchmark tasks, including generative modeling for dynamical systems, DNA sequences, and videos. Code is available at hrefhttps://github.com/Utah-Math-Data-Science/FlowDivMatchingUtah-Math-Data-Science.