  Flow matching (FM) has gained significant attention as a simulation-free
generative model. Unlike diffusion models, which are based on stochastic
differential equations, FM employs a simpler approach by solving an ordinary
differential equation with an initial condition from a normal distribution,
thus streamlining the sample generation process. This paper discusses the
convergence properties of FM in terms of the $p$-Wasserstein distance, a
measure of distributional discrepancy. We establish that FM can achieve the
minmax optimal convergence rate for $1 \leq p \leq 2$, presenting the first
theoretical evidence that FM can reach convergence rates comparable to those of
diffusion models. Our analysis extends existing frameworks by examining a
broader class of mean and variance functions for the vector fields and
identifies specific conditions necessary to attain these optimal rates.
