A Kinetic Energy Perspective of Flow Matching
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arXiv
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| Main Authors: | , , , , , , , , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866910271120867328 |
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| author | Li, Ziyun Hu, Huancheng Lim, Soon Hoe Li, Xuyu Gao, Fei Diao, Enmao Ding, Zezhen Vazirgiannis, Michalis Bostrom, Henrik |
| author_facet | Li, Ziyun Hu, Huancheng Lim, Soon Hoe Li, Xuyu Gao, Fei Diao, Enmao Ding, Zezhen Vazirgiannis, Michalis Bostrom, Henrik |
| contents | Flow-based generative models can be viewed through a physics lens: sampling transports a particle from noise to data by integrating a learned 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 land in sparse representation regions. We further provide theoretical guarantees linking trajectory energy to data sparsity. Paradoxically, this correlation is non-monotonic. At sufficiently high energy, generation can degenerate into memorization. Leveraging the closed-form formula 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_07928 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Kinetic Energy Perspective of Flow Matching Li, Ziyun Hu, Huancheng Lim, Soon Hoe Li, Xuyu Gao, Fei Diao, Enmao Ding, Zezhen Vazirgiannis, Michalis Bostrom, Henrik Machine Learning Artificial Intelligence Flow-based generative models can be viewed through a physics lens: sampling transports a particle from noise to data by integrating a learned 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 land in sparse representation regions. We further provide theoretical guarantees linking trajectory energy to data sparsity. Paradoxically, this correlation is non-monotonic. At sufficiently high energy, generation can degenerate into memorization. Leveraging the closed-form formula 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. |
| title | A Kinetic Energy Perspective of Flow Matching |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2602.07928 |