Flow matching has emerged as a powerful framework for generative modeling, offering computational advantages over diffusion models by leveraging deterministic Ordinary Differential Equations (ODEs) instead of stochastic dynamics. While prior work established the worst case optimality of standard flow matching under Wasserstein distances, the theoretical guarantees for higher-order flow matching - which incorporates acceleration terms to refine sample trajectories - remain unexplored. In this paper, we bridge this gap by proving that higher-order flow matching preserves worst case optimality as a distribution estimator. We derive upper bounds on the estimation error for second-order flow matching, demonstrating that the convergence rates depend polynomially on the smoothness of the target distribution (quantified via Besov spaces) and key parameters of the ODE dynamics. Our analysis employs neural network approximations with carefully controlled depth, width, and sparsity to bound acceleration errors across both small and large time intervals, ultimately unifying these results into a general worst case optimal bound for all time steps.

本研究针对现有生成模型中的流匹配方法，探讨了高阶流匹配在样本轨迹精炼中的理论保障。通过证明高阶流匹配在分布估计中的最坏情况最优性，我们衍生出针对二阶流匹配的估计误差上界，发现收敛率与目标分布的光滑度及ODE动态的关键参数成多项式关系。该分析利用神经网络近似方法，为不同时间区间的加速误差提供了约束，从而统一了结果，形成了针对所有时间步骤的一般最坏情况最优边界。

生成流中高阶轨迹精炼的理论保障