Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers

Zhengchao Wan, Qingsong Wang, Gal Mishne, Yusu Wang

TL;DR: We analyze flow-matching ODE trajectories, proving convergence and quantitatively identifying three stages of evolution. Our theory extends to low-dimensional manifolds—where prior work fails—and provides insights into memorization.

Abstract: Flow matching (FM) models extend ODE sampler based diffusion models into a general framework, significantly reducing sampling steps through learned vector fields. However, the theoretical understanding of FM models, particularly how their sample trajectories interact with underlying data geometry, remains underexplored. A rigorous theoretical analysis of FM ODE is essential for sample quality, stability, and broader applicability. In this paper, we advance the theory of FM models through a comprehensive analysis of sample trajectories. Central to our theory is the discovery that the denoiser, a key component of FM models, guides ODE dynamics through attracting and absorbing behaviors that adapt to the data geometry. We identify and analyze the three stages of ODE evolution: in the initial and intermediate stages, trajectories move toward the mean and local clusters of the data. At the terminal stage, we rigorously establish the convergence of FM ODE under weak assumptions, addressing scenarios where the data lie on a low-dimensional submanifold---cases that previous results could not handle. Our terminal stage analysis offers insights into the memorization phenomenon and establishes equivariance properties of FM ODEs. These findings bridge critical gaps in understanding flow matching models, with practical implications for optimizing sampling strategies and architectures guided by the intrinsic geometry of data.

Lay Summary:

Modern image-generating AI models often use a process called the diffusion model, which gradually transforms random noise into realistic images. While effective, this process can be slow, requiring hundreds of steps to produce high-quality results. Flow Matching builds on this same core idea of noise-to-image transformation but provides a more direct generation path from noise to real data. However, the theoretical foundations of Flow Matching remain incomplete, and we lack rigorous guarantees that the generation path will reliably converge to realistic data.

We analyze the generation path in flow matching and trace its evolution patterns. Our analysis shows that the path first moves toward the center of the data distribution, then shifts toward local clusters, and finally lands on a realistic sample. We also prove that, under broad conditions, the path reliably ends in the right place—even when the data lies on a complex, curved surface in a high-dimensional space.

This work fills a key theoretical gap in the foundations of flow matching and suggests practical implications for improving sampling strategies in generative AI.

Primary Area: Deep Learning->Generative Models and Autoencoders

Keywords: Deep learning theory, Diffusion model, Flow matching, ODE dynamics, Generative model

Submission Number: 7924