# Divergence-Suppressing Couplings for Rectified Flow

arXiv: https://arxiv.org/abs/2605.17733 · HTML: https://mlsanity.com/paper/2605.17733/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 20 of its 24 references that are in the map.
- First posted on arXiv 18 May 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, 20 of 20)

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- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [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)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [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)
- [Movie Gen: A Cast of Media Foundation Models](https://mlsanity.com/paper/2410.13720/) (2024)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Fast Sampling of Diffusion Models with Exponential Integrator](https://mlsanity.com/paper/2204.13902/) (2022)
- [InstaFlow: One Step is Enough for High-Quality Diffusion-Based Text-to-Image Generation](https://mlsanity.com/paper/2309.06380/) (2023)
- [Voicebox: Text-Guided Multilingual Universal Speech Generation at Scale](https://mlsanity.com/paper/2306.15687/) (2023)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [SE(3)-Stochastic Flow Matching for Protein Backbone Generation](https://mlsanity.com/paper/2310.02391/) (2023)
- [Improving Flow Matching by Aligning Flow Divergence](https://mlsanity.com/paper/2602.00869/) (2026)
- [Training-Free Refinement of Flow Matching with Divergence-based Sampling](https://mlsanity.com/paper/2604.04646/) (2026)

## Abstract (arXiv)

The promise of Rectified Flow rests on producing self-generated couplings whose trajectories are straight, or nearly so. In practice, trajectories generated by the base flow model can bend and intertwine, and the resulting coupling inherits this distortion. In this paper, we identify that such trajectory entanglement is often associated with regions of nonzero divergence in the learned velocity field, where local expansion or contraction distorts trajectories and steers particles away from their ideal endpoints. We then propose divergence-suppressing couplings for Rectified Flow, an offline correction that attenuate the divergent component of the learned velocity during coupling generation. The correction is paid only once per coupling pair and amortized over training, so deployment runs plain Euler at identical wall-clock cost to standard Rectified Flow. Empirically, this offline modification yields consistent improvements on 2D synthetic benchmarks and on image generation.