从动能视角分析生成模型轨迹，发现高能量更保真但易记忆，提出新推理策略提升质量。
A Kinetic Energy Perspective of Flow Matching
- 用动能路径能量衡量生成轨迹的物理努力，量化每条样本的动态代价。
- 高动能对应更强语义保真度和稀疏区域分布，但过高的能量导致生成退化为记忆。
- 提出无需训练的分阶段推理方法，平衡早期加速与晚期软着陆，避免记忆偏差。
基于流的生成模型可从物理角度理解：采样过程通过积分学习到的速度场将粒子从噪声推进至数据，每个样本对应一条具有自身动力学代价的轨迹。受经典力学启发，我们引入动能路径能量（KPE），一种类似作用量的、针对每条样本的诊断指标，用于衡量沿常微分方程（ODE）轨迹积累的动能努力。实证表明，KPE具有两个稳健对应关系：(i) 更高的KPE预测更强的语义保真度；(ii) 高KPE轨迹倾向于落在稀疏表示区域。我们进一步提供了理论保证，将轨迹能量与数据稀疏性关联。值得注意的是，这种相关性是非单调的：在足够高的能量下，生成会退化为记忆。利用经验流匹配的闭式公式，我们证明极端能量会驱动轨迹趋向训练样本的近似复制品。这引出了‘恰到好处’原则，并启发了无需训练的两阶段推理策略——动能轨迹塑形（KTS），通过增强早期运动并强制晚期软着陆，减少记忆现象，在多个基准任务上提升生成质量。
原文摘要 · Abstract (English)
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.
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