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Particle Dynamics of Flow Matching and Classifier-Free Guidance from a Stagewise Geometry Perspective

arXiv · AI, language, vision and robotics · article · Sep 7, 2026 · UTC

Flow matching, together with classifier-free guidance (CFG), is widely used in generative modeling, yet much of the theoretical understanding remains distribution-wise. Since practical sampling follows individual trajectories, distribution-level guarantees alone do not fully capture how trajectories interact with the data geometry or how guidance reshapes it. To overcome this limitation, we establish a unified stagewise geometric theory of attraction and absorption for both continuous dynamics and explicit Euler discretization. Specifically, with $t\in[0,1]$ running from noise to data, we show

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