A Kinetic-Energy Perspective of Flow Matching

• URL: http://arxiv.org/abs/2602.07928v1
• Date: Sun, 08 Feb 2026 11:51:50 GMT
• Title: A Kinetic-Energy Perspective of Flow Matching
• Authors: Ziyun Li, Huancheng Hu, Soon Hoe Lim, Xuyu Li, Fei Gao, Enmao Diao, Zezhen Ding, Michalis Vazirgiannis, Henrik Bostrom,
• Abstract summary: Flow-based generative models can be viewed through a physics lens.<n>Motivated by classical mechanics, we introduce Kinetic Path Energy (KPE)<n>We show that extreme energies drive trajectories toward near-copies of training examples.
• Score: 23.42786172624299
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Flow-based generative models can be viewed through a physics lens: sampling transports a particle from noise to data by integrating a time-varying velocity field, and each sample corresponds to a trajectory with its own dynamical effort. Motivated by classical mechanics, we introduce Kinetic Path Energy (KPE), an action-like, per-sample diagnostic that measures the accumulated kinetic effort along an Ordinary Differential Equation (ODE) trajectory. Empirically, KPE exhibits two robust correspondences: (i) higher KPE predicts stronger semantic fidelity; (ii) high-KPE trajectories terminate on low-density manifold frontiers. We further provide theoretical guarantees linking trajectory energy to data density. Paradoxically, this correlation is non-monotonic. At sufficiently high energy, generation can degenerate into memorization. Leveraging the closed-form of empirical flow matching, we show that extreme energies drive trajectories toward near-copies of training examples. This yields a Goldilocks principle and motivates Kinetic Trajectory Shaping (KTS), a training-free two-phase inference strategy that boosts early motion and enforces a late-time soft landing, reducing memorization and improving generation quality across benchmark tasks.

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