A Lagrangian View of Flow Matching

• URL: http://arxiv.org/abs/2609.00198v2
• Date: Sat, 05 Sep 2026 02:17:57 GMT
• Title: A Lagrangian View of Flow Matching
• Abstract summary: We present an alternative, bottom-up mechanical derivation grounded in a Lagrangian (particle-centric) perspective.<n>We demonstrate that solving this Partial Differential Equation analytically yields the straight-line trajectories of Flow Matching.
• Score: 17.972940829317455
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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