Divergence is Uncertainty: A Closed-Form Posterior Covariance for Flow Matching

TL;DR

This paper introduces a closed-form expression for quantifying uncertainty in flow matching generative models, enabling efficient, post-hoc uncertainty estimation without retraining or architectural changes.

Contribution

It extends Tweedie's formula to flow matching, deriving an exact posterior covariance expression computable on pre-trained models, reducing computational costs significantly.

Findings

Uncertainty maps focus on digit boundaries in MNIST.

Scalar uncertainty correlates with prediction error.

Method requires roughly 10,000 times less compute than ensembling.

Abstract

Flow matching has become a leading framework for generative modeling, but quantifying the uncertainty of its samples remains an open problem. Existing approaches retrain the model with auxiliary variance heads, maintain costly ensembles, or propagate approximate covariance through many integration steps, trading off training cost, inference cost, or accuracy. We show that none of these trade-offs is necessary. By extending Tweedie's formula from the denoising setting to the flow matching interpolant, we derive an exact, closed-form expression for the posterior covariance at every point along the generative trajectory. The result depends on a single quantity, namely the divergence of the learned velocity field, which can be computed post-hoc on any pre-trained flow matching model, requiring no retraining and no architectural modification. For one-step generators such as MeanFlow, the same formula yields the end-to-end generation uncertainty in a single forward pass, eliminating the multi-step variance propagation required by all prior methods. Experiments on MNIST confirm that the resulting per-pixel uncertainty maps are semantically meaningful, concentrating on digit boundaries where inter-sample variation is highest, and that the scalar uncertainty score tracks actual prediction error, all at roughly $10^4 \times$ less total compute than ensembling or Monte Carlo dropout.