# Velocity Scheduled Flow Matching

arXiv: https://arxiv.org/abs/2607.11442 · HTML: https://mlsanity.com/paper/2607.11442/

## 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 26 of its 31 references that are in the map.
- First posted on arXiv 13 Jul 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, 26 of 26)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [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)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [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)
- [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)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Understanding Diffusion Objectives as the ELBO with Simple Data Augmentation](https://mlsanity.com/paper/2303.00848/) (2023)
- [Perception Prioritized Training of Diffusion Models](https://mlsanity.com/paper/2204.00227/) (2022)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [Efficient Diffusion Training via Min-SNR Weighting Strategy](https://mlsanity.com/paper/2303.09556/) (2023)
- [SE(3)-Stochastic Flow Matching for Protein Backbone Generation](https://mlsanity.com/paper/2310.02391/) (2023)
- [Equivariant flow matching](https://mlsanity.com/paper/2306.15030/) (2023)
- [Consistency Flow Matching: Defining Straight Flows with Velocity Consistency](https://mlsanity.com/paper/2407.02398/) (2024)
- [On Kinetic Optimal Probability Paths for Generative Models](https://mlsanity.com/paper/2306.06626/) (2023)
- [SplineFlow: Flow Matching for Dynamical Systems with B-Spline Interpolants](https://mlsanity.com/paper/2601.23072/) (2026)
- [Deep generative models as the probability transformation functions](https://mlsanity.com/paper/2506.17171/) (2025)
- [Dense-Jump Flow Matching with Non-Uniform Time Scheduling for Robotic Policies: Mitigating Multi-Step Inference Degradation](https://mlsanity.com/paper/2509.13574/) (2025)

## Built on by (most cited first, 2 of 2)

- [Model-Aware Schedules Improve Generation via Fiberwise Optimal Transport](https://mlsanity.com/paper/2609.11842/) (2026)
- [What Does a Stream Model Buy You in Flow Matching?](https://mlsanity.com/paper/2609.34123/) (2026)

## Abstract (arXiv)

Flow matching trains a neural network to regress the conditional velocity along a linear interpolant between noise and data, and the number of network evaluations~(NFE) sets the cost of sampling. The straight-line interpolant carries an implicit choice: the sample moves at constant speed throughout the trajectory. We relax this choice and introduce Velocity Scheduled Flow Matching~(VSFM), which replaces the conditional target x1 - x0 with v(t)(x1 - x0) for any nonnegative profile v:[0,1]→ℝ≥ 0 satisfying int01 v,dt = 1. We study six polynomial profiles drawn from motion planning. The first use of VSFM is at inference time: a pretrained linear flow-matching model can be sampled under any admissible profile by integrating its ODE on a non-uniform τ-schedule, with no retraining and no additional computation; on CIFAR-10 this lowers FID by up to 19.8%. Training from scratch under a braking profile gives a further reduction of 17.4% at 4~NFE. Both gains follow from the local truncation error of the Euler integrator on the induced grid.