A Lagrangian View of Flow Matching

2026-08-31

AI Summary

本文从拉格朗日视角出发，通过分析连续去噪器的局部泰勒展开，推导出一种新的流匹配方法，并利用特征线法解决了轨迹曲率问题。

Abstract

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.

Problem

• Lagrangian
• Flow Matching
• Optimal Transport
• continuity equation
• denoiser

Innovation

• Lagrangian Perspective
• Taylor Expansion
• Conservation of Target Identity
• Quasi-linear Advection PDE
• Method of Characteristics