High-Order Flow Matching: Unified Framework and Sharp Statistical Rates

Flow matching is an emerging generative modeling framework that learns continuous-time dynamics to map noise into data. To enhance expressiveness and sampling efficiency, recent works have explored incorporating high-order trajectory information. Despite the empirical success, a holistic theoretical foundation is still lacking. We present a unified framework for standard and high-order flow matching that incorporates trajectory derivatives up to an arbitrary order K. Our key innovation is establishing the marginalization technique that converts the intractable K-order loss into a simple conditional regression with exact gradients and identifying the consistency constraint. We establish sharp statistical rates of the K-order flow matching implemented with transformer networks. With nsamples, flow matching estimates nonparametric distributions at a rate eO(n Θ(1/d)), matching minimax lower bounds up to logarithmic factors.