Smart Flow Matching: On The Theory of Flow Matching Algorithms with
  Applications
Gleb Ryzhakov Svetlana Pavlova Egor Sevriugov Ivan Oseledets
The paper presents the exact formula for the vector field that minimizes the
loss for the standard flow. This formula depends analytically on a given
distribution \rho_0 and an unknown one \rho_1. Based on the presented formula,
a new loss and algorithm for training a vector field model in the style of
Conditional Flow Matching are provided. Our loss, in comparison to the standard
Conditional Flow Matching approach, exhibits smaller variance when evaluated
through Monte Carlo sampling methods. Numerical experiments on synthetic models
and models on tabular data of large dimensions demonstrate better learning
results with the use of the presented algorithm.