# Entropy-Controlled Flow Matching

arXiv: https://arxiv.org/abs/2602.22265 · HTML: https://mlsanity.com/paper/2602.22265/

## 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 1 of its 46 references that are in the map.
- First posted on arXiv 25 Feb 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, 1 of 1)

- [Learning High Dimensional Wasserstein Geodesics](https://mlsanity.com/paper/2102.02992/) (2021)

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

- [When Policy Entropy Constraint Fails: Preserving Diversity in Flow-based RLHF via Perceptual Entropy](https://mlsanity.com/paper/2605.12112/) (2026)

## Abstract (arXiv)

Modern vision generators transport a base distribution to data through time-indexed measures, implemented as deterministic flows (ODEs) or stochastic diffusions (SDEs). Despite strong empirical performance, standard flow-matching objectives do not directly control the information geometry of the trajectory, allowing low-entropy bottlenecks that can transiently deplete semantic modes. We propose Entropy-Controlled Flow Matching (ECFM): a constrained variational principle over continuity-equation paths enforcing a global entropy-rate budget d/dt H(mu_t) >= -lambda. ECFM is a convex optimization in Wasserstein space with a KKT/Pontryagin system, and admits a stochastic-control representation equivalent to a Schrodinger bridge with an explicit entropy multiplier. In the pure transport regime, ECFM recovers entropic OT geodesics and Gamma-converges to classical OT as lambda -> 0. We further obtain certificate-style mode-coverage and density-floor guarantees with Lipschitz stability, and construct near-optimal collapse counterexamples for unconstrained flow matching.