  Over the several recent years, there has been a boom in development of Flow
Matching (FM) methods for generative modeling. One intriguing property pursued
by the community is the ability to learn flows with straight trajectories which
realize the Optimal Transport (OT) displacements. Straightness is crucial for
the fast integration (inference) of the learned flow's paths. Unfortunately,
most existing flow straightening methods are based on non-trivial iterative FM
procedures which accumulate the error during training or exploit heuristics
based on minibatch OT. To address these issues, we develop and theoretically
justify the novel Optimal Flow Matching approach which allows recovering the
straight OT displacement for the quadratic transport in just one FM step. The
main idea of our approach is the employment of vector field for FM which are
parameterized by convex functions.
