发表机构
Google（谷歌公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文从拉格朗日视角推导Flow Matching的控制方程，揭示其直线轨迹的数学本质，解释了直线流支持大步长及模型需蒸馏的原因。
AI 中文摘要
现代显式时间生成模型，如Flow Matching（Lipman等人，2023）和Rectified Flow（Liu等人，2023），通常通过最优传输和连续性方程自上而下推导。这种标准欧拉方法聚焦于概率质量的宏观传输。本文提出一种基于拉格朗日（以粒子为中心）视角的自下而上机械推导方法。通过分析连续去噪器的局部泰勒展开，我们推导了最优单步生成所需的严格不变性条件：目标身份的守恒。强制该条件会得到一个控制拟线性平流偏微分方程（PDE）。我们证明，通过特征线法解析求解该PDE可得到Flow Matching的直线轨迹。这种几何视角将去噪器的雅可比分隔为轨迹曲率的主要来源，为直线流为何能支持大步长、以及经验模型为何需要蒸馏以扁平化相交特征线提供了直接数学解释。
英文摘要
Modern explicit-time generative models, such as Flow Matching [Lipman et al., 2023] and Rectified Flow [Liu et al., 2023], are typically derived top-down via Optimal Transport and the continuity equation. This standard Eulerian approach focuses on the macroscopic transport of probability mass. In this paper, we present an alternative, bottom-up mechanical derivation grounded in a Lagrangian (particle-centric) perspective. By analyzing the local Taylor expansion of a continuous denoiser, we motivate a strict invariance condition required for optimal, singlestep generation: the conservation of target identity. Enforcing this condition yields a governing quasi-linear advection Partial Differential Equation (PDE). We demonstrate that solving this PDE via the Method of Characteristics analytically yields the straight-line trajectories of Flow Matching. This geometric perspective isolates the Jacobian of the denoiser as the primary source of trajectory curvature, providing a direct mathematical explanation for why straight-line flows enable massive step sizes, and why empirical models require distillation to flatten intersecting characteristics.
