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.

翻译：流匹配（FM）作为一种免模拟生成模型已获得广泛关注。与基于随机微分方程的扩散模型不同，FM采用更简洁的方法，通过求解以正态分布为初始条件的常微分方程来简化样本生成过程。本文讨论了FM在$p$-Wasserstein距离（一种分布差异度量）意义上的收敛性质。我们证明FM能够实现$1 \leq p \leq 2$范围内的极小极大最优收敛速率，这首次为FM能达到与扩散模型相当的收敛速率提供了理论依据。我们的分析通过考察向量场更广泛的均值与方差函数类别，扩展了现有理论框架，并明确了达到这些最优速率所需的具体条件。