  We study the geometry of conditional optimal transport (COT) and prove a
dynamical formulation which generalizes the Benamou-Brenier Theorem. Equipped
with these tools, we propose a simulation-free flow-based method for
conditional generative modeling. Our method couples an arbitrary source
distribution to a specified target distribution through a triangular COT plan,
and a conditional generative model is obtained by approximating the geodesic
path of measures induced by this COT plan. Our theory and methods are
applicable in infinite-dimensional settings, making them well suited for a wide
class of Bayesian inverse problems. Empirically, we demonstrate that our method
is competitive on several challenging conditional generation tasks, including
an infinite-dimensional inverse problem.
