A Canonical Optimal Control Theory for Continuous-Time Generative Models

We formulate continuous-time generative modeling as a deterministic optimal control problem posed on the quadratic Wasserstein space of probability measures with finite second moment, and prove that, unlike classical optimal transport or stochastic control formulations, it admits a unique population-level Pontryagin minimizer that induces a canonical velocity field intrinsic to this space. We show that flow matching, rectified flow, and diffusion models in the vanishing-noise limit are provably equivalent population-level relaxations of this same control problem, establishing a new non-identifiability theorem: once the hypothesis class and time discretization are fixed, no deterministic continuous-time objective can yield a strictly better transport solution. We further prove that this equivalence is sharp by identifying a geometric phase transition: when the data distribution does not belong to the space of probability measures with finite second moment, quadratic transport loses coercivity, deterministic objectives become ill-posed, and only entropically regularized diffusion formulations remain well-posed. Finally, we establish stability and discretization guarantees for practical training and sampling pipelines and empirically validate both the equivalence in the transport regime and its breakdown beyond it.