# Uncertainty in a Single Pass: A Closed-Form Identity for One-Step Flow Matching

arXiv: https://arxiv.org/abs/2605.00941 · HTML: https://mlsanity.com/paper/2605.00941/

## Key facts

- Cited by 3 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 19 of its 28 references that are in the map.
- First posted on arXiv 1 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, 19 of 19)

- [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)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (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)
- [Mean Flows for One-step Generative Modeling](https://mlsanity.com/paper/2505.13447/) (2025)
- [Common Diffusion Noise Schedules and Sample Steps are Flawed](https://mlsanity.com/paper/2305.08891/) (2023)
- [A Survey on Uncertainty Quantification Methods for Deep Learning](https://mlsanity.com/paper/2302.13425/) (2023)
- [A Review of Uncertainty Estimation and its Application in Medical Imaging](https://mlsanity.com/paper/2302.08119/) (2023)
- [BayesDiff: Estimating Pixel-wise Uncertainty in Diffusion via Bayesian Inference](https://mlsanity.com/paper/2310.11142/) (2023)
- [On the Posterior Distribution in Denoising: Application to Uncertainty Quantification](https://mlsanity.com/paper/2309.13598/) (2023)
- [Quantifying and Leveraging Predictive Uncertainty for Medical Image Assessment](https://mlsanity.com/paper/2007.04258/) (2020)
- [ImageFlowNet: Forecasting Multiscale Image-Level Trajectories of Disease Progression with Irregularly-Sampled Longitudinal Medical Images](https://mlsanity.com/paper/2406.14794/) (2024)
- [Learning Patient-Specific Disease Dynamics with Latent Flow Matching for Longitudinal Imaging Generation](https://mlsanity.com/paper/2512.09185/) (2025)
- [Flow Matching with Uncertainty Quantification and Guidance](https://mlsanity.com/paper/2602.10326/) (2026)
- [Training-Free Refinement of Flow Matching with Divergence-based Sampling](https://mlsanity.com/paper/2604.04646/) (2026)

## Built on by (most cited first, 3 of 3)

- [The Geometry of Flow-Matching Uncertainty: A Cost-free Uncertainty Proxy and Its Application in Flow-based VLA Failure Detection](https://mlsanity.com/paper/2607.27933/) (2026)
- [When does training on downscaled images yield the same gradients?](https://mlsanity.com/paper/2608.04448/) (2026)
- [Posterior Flow Matching](https://mlsanity.com/paper/2610.13994/) (2026)

## Abstract (arXiv)

Flow matching provides a highly effective framework for generative modeling, yet estimating the uncertainty of its generated samples remains a fundamental challenge. Existing methods rely on auxiliary variance heads, model ensembles, or iterative covariance propagation, which invariably introduce accumulated errors and structural interference along the noise-to-image trajectory. We resolve these limitations by adapting Tweedie's formula to the linear flow matching interpolant, yielding an exact, closed-form identity for the posterior covariance. Because this identity relies exclusively on the divergence of the learned velocity field, it can be evaluated post hoc on pretrained models without any architectural modifications. This analytical formulation proves exceptionally powerful for single-step generative frameworks. By computing the end-to-end posterior covariance in one forward pass, our approach completely bypasses the compounding errors inherent to sequential integration. This mechanism fundamentally enhances robustness, providing highly accurate and stable uncertainty estimates while preventing the numerical degradation typical of multi-step approximations. We validate our framework across benchmark datasets, including CIFAR-10 and real brain MRI scans. The resulting divergence-based uncertainty maps are highly interpretable and tightly correlated with empirical reconstruction errors. Crucially, in medical imaging, this precise localization of structural ambiguity provides essential decision support for high-stakes clinical tasks such as tumor boundary delineation and neurosurgical planning.