Theoretical Guarantees for High Order Trajectory Refinement in Generative Flows

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.