# On Variance Reduction in Learning Mean Flows

arXiv: https://arxiv.org/abs/2605.09235 · HTML: https://mlsanity.com/paper/2605.09235/

## Key facts

- Cited by 1 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 28 of its 39 references that are in the map.
- First posted on arXiv 10 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, 28 of 28)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (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)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Bootstrap your own latent: A new approach to self-supervised Learning](https://mlsanity.com/paper/2006.07733/) (2020)
- [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)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [One-step Diffusion with Distribution Matching Distillation](https://mlsanity.com/paper/2311.18828/) (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)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Improved Distribution Matching Distillation for Fast Image Synthesis](https://mlsanity.com/paper/2405.14867/) (2024)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Mean Flows for One-step Generative Modeling](https://mlsanity.com/paper/2505.13447/) (2025)
- [One Step Diffusion via Shortcut Models](https://mlsanity.com/paper/2410.12557/) (2024)
- [Improved Mean Flows: On the Challenges of Fastforward Generative Models](https://mlsanity.com/paper/2512.02012/) (2025)
- [Flow map matching with stochastic interpolants: A mathematical framework for consistency models](https://mlsanity.com/paper/2406.07507/) (2024)
- [Improving the Training of Rectified Flows](https://mlsanity.com/paper/2405.20320/) (2024)
- [On the Closed-Form of Flow Matching: Generalization Does Not Arise from Target Stochasticity](https://mlsanity.com/paper/2506.03719/) (2025)
- [Decoupled MeanFlow: Turning Flow Models into Flow Maps for Accelerated Sampling](https://mlsanity.com/paper/2510.24474/) (2025)
- [Stable Velocity: A Variance Perspective on Flow Matching](https://mlsanity.com/paper/2602.05435/) (2026)
- [Modular MeanFlow: Towards Stable and Scalable One-Step Generative Modeling](https://mlsanity.com/paper/2508.17426/) (2025)
- [Temporal Pair Consistency for Variance-Reduced Flow Matching](https://mlsanity.com/paper/2602.04908/) (2026)

## Built on by (most cited first, 1 of 1)

- [Posterior sampling by source-space MCMC via prior-based few-step transport maps](https://mlsanity.com/paper/2610.01034/) (2026)

## Abstract (arXiv)

One-step generative modeling has emerged as a leading approach for amortizing the inference cost of diffusion and flow-matching models. Among distillation-free methods, MeanFlow training is notoriously unstable, with non-decreasing loss and unbounded gradient variance. In this work, we establish a theory that attributes this pathology to a misuse of the conditional velocity field. We show that the conditional velocity plays two distinct statistical roles in the loss: both as an unbiased regression target and as a Monte Carlo control variate in a Jacobi-vector product, with the original MeanFlow loss assigning the wrong coefficient to the latter. We derive the optimal coefficient in closed form and show that a family of fixes in concurrent works corresponds to different practical realizations of the same optimum. A controlled sweep of this coefficient on two-dimensional benchmarks and on a latent Diffusion Transformer recovers the predicted bias-variance ordering. Our DiT experiment also reveals a quantitative FID-MSE landscape mismatch. Specifically, although the gradient-MSE is minimized at an interior coefficient value near β!=!0.94, the coefficient that minimizes FID prefers to use conditional velocity directly at the unbiased corner. Our analysis therefore explains why MeanFlow is unstable and unifies its concurrent remedies, and shows that the variance-optimal coefficient need not coincide with the quality-optimal one.