Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers

Zhengchao Wan, Qingsong Wang, Gal Mishne, Yusu Wang
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:62020-62083, 2025.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-wan25e, title = {Elucidating Flow Matching {ODE} Dynamics via Data Geometry and Denoisers}, author = {Wan, Zhengchao and Wang, Qingsong and Mishne, Gal and Wang, Yusu}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {62020--62083}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/wan25e/wan25e.pdf}, url = {https://proceedings.mlr.press/v267/wan25e.html}, 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.} }
Endnote
%0 Conference Paper %T Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers %A Zhengchao Wan %A Qingsong Wang %A Gal Mishne %A Yusu Wang %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-wan25e %I PMLR %P 62020--62083 %U https://proceedings.mlr.press/v267/wan25e.html %V 267 %X 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.
APA
Wan, Z., Wang, Q., Mishne, G. & Wang, Y.. (2025). Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:62020-62083 Available from https://proceedings.mlr.press/v267/wan25e.html.

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