Entropy-Controlled Flow Matching

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 continuityequation paths enforcing a global entropy-rate budget d H(µ ) ≥ −λ. dt t ECFM is a convex optimization in Wasserstein space with a KKT/Pontryagin system, and admits a stochastic-control representation equivalent to a Schrödinger bridge with an explicit entropy multiplier. In the pure transport regime, ECFM recovers entropic OT geodesics and Γ-converges to classical OT as λ → 0. We further obtain certificate-style mode-coverage and density-floor guarantees with Lipschitz stability, and construct nearoptimal collapse counterexamples for unconstrained flow matching.