  This paper proposes a novel method, Explicit Flow Matching (ExFM), for
training and analyzing flow-based generative models. ExFM leverages a
theoretically grounded loss function, ExFM loss (a tractable form of Flow
Matching (FM) loss), to demonstrably reduce variance during training, leading
to faster convergence and more stable learning. Based on theoretical analysis
of these formulas, we derived exact expressions for the vector field (and score
in stochastic cases) for model examples (in particular, for separating multiple
exponents), and in some simple cases, exact solutions for trajectories. In
addition, we also investigated simple cases of diffusion generative models by
adding a stochastic term and obtained an explicit form of the expression for
score. While the paper emphasizes the theoretical underpinnings of ExFM, it
also showcases its effectiveness through numerical experiments on various
datasets, including high-dimensional ones. Compared to traditional FM methods,
ExFM achieves superior performance in terms of both learning speed and final
outcomes.
