# Flow Diverse and Efficient: Learning Momentum Flow Matching via Stochastic Velocity Field Sampling arXiv: https://arxiv.org/abs/2506.08796 · HTML: https://mlsanity.com/paper/2506.08796/ ## Key facts - Cited by 2 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 37 of its 45 references that are in the map. - First posted on arXiv 10 Jun 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, 30 of 37) - [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020) - [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020) - [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) - [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024) - [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019) - [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021) - [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022) - [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017) - [Decoupled Weight Decay Regularization](https://mlsanity.com/paper/1711.05101/) (2017) - [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022) - [Adversarial Diffusion Distillation](https://mlsanity.com/paper/2311.17042/) (2023) - [On Distillation of Guided Diffusion Models](https://mlsanity.com/paper/2210.03142/) (2022) - [Assessing Generative Models via Precision and Recall](https://mlsanity.com/paper/1806.00035/) (2018) - [InstaFlow: One Step is Enough for High-Quality Diffusion-Based Text-to-Image Generation](https://mlsanity.com/paper/2309.06380/) (2023) - [DeepCache: Accelerating Diffusion Models for Free](https://mlsanity.com/paper/2312.00858/) (2023) - [Discrete Flow Matching](https://mlsanity.com/paper/2407.15595/) (2024) - [Rectified Flow: A Marginal Preserving Approach to Optimal Transport](https://mlsanity.com/paper/2209.14577/) (2022) - [SnapFusion: Text-to-Image Diffusion Model on Mobile Devices within Two Seconds](https://mlsanity.com/paper/2306.00980/) (2023) - [UFOGen: You Forward Once Large Scale Text-to-Image Generation via Diffusion GANs](https://mlsanity.com/paper/2311.09257/) (2023) - [Improving the Training of Rectified Flows](https://mlsanity.com/paper/2405.20320/) (2024) - [DistriFusion: Distributed Parallel Inference for High-Resolution Diffusion Models](https://mlsanity.com/paper/2402.19481/) (2024) - [PeRFlow: Piecewise Rectified Flow as Universal Plug-and-Play Accelerator](https://mlsanity.com/paper/2405.07510/) (2024) - [Rectified Diffusion: Straightness Is Not Your Need in Rectified Flow](https://mlsanity.com/paper/2410.07303/) (2024) - [OmniFlow: Any-to-Any Generation with Multi-Modal Rectified Flows](https://mlsanity.com/paper/2412.01169/) (2024) - [Neural Flow Diffusion Models: Learnable Forward Process for Improved Diffusion Modelling](https://mlsanity.com/paper/2404.12940/) (2024) - [MobileDiffusion: Instant Text-to-Image Generation on Mobile Devices](https://mlsanity.com/paper/2311.16567/) (2023) - [Efficient Diffusion Models: A Comprehensive Survey from Principles to Practices](https://mlsanity.com/paper/2410.11795/) (2024) - [FlowIE: Efficient Image Enhancement via Rectified Flow](https://mlsanity.com/paper/2406.00508/) (2024) - [SEEDS: Exponential SDE Solvers for Fast High-Quality Sampling from Diffusion Models](https://mlsanity.com/paper/2305.14267/) (2023) ## Built on by (most cited first, 2 of 2) - [MAISI-v2: Accelerated 3D High-Resolution Medical Image Synthesis with Rectified Flow and Region-specific Contrastive Loss](https://mlsanity.com/paper/2508.05772/) (2025) - [Deep Neural Networks Inspired by Differential Equations](https://mlsanity.com/paper/2510.09685/) (2025) ## Abstract (arXiv) Recently, the rectified flow (RF) has emerged as the new state-of-the-art among flow-based diffusion models due to its high efficiency advantage in straight path sampling, especially with the amazing images generated by a series of RF models such as Flux 1.0 and SD 3.0. Although a straight-line connection between the noisy and natural data distributions is intuitive, fast, and easy to optimize, it still inevitably leads to: 1) Diversity concerns, which arise since straight-line paths only cover a fairly restricted sampling space. 2) Multi-scale noise modeling concerns, since the straight line flow only needs to optimize the constant velocity field bm v between the two distributions bmπ0 and bmπ1. In this work, we present Discretized-RF, a new family of rectified flow (also called momentum flow models since they refer to the previous velocity component and the random velocity component in each diffusion step), which discretizes the straight path into a series of variable velocity field sub-paths (namely ``momentum fields'') to expand the search space, especially when close to the distribution pnoise. Different from the previous case where noise is directly superimposed on bm x, we introduce noise on the velocity bm v of the sub-path to change its direction in order to improve the diversity and multi-scale noise modeling abilities. Experimental results on several representative datasets demonstrate that learning momentum flow matching by sampling random velocity fields will produce trajectories that are both diverse and efficient, and can consistently generate high-quality and diverse results. Code is available at https://github.com/liuruixun/momentum-fm.