A Lagrangian View of Flow Matching

A new arXiv paper (arXiv:2609.00198v1) presents a Lagrangian, particle-centric derivation of Flow Matching and Rectified Flow, showing that enforcing a strict invariance condition—conservation of target identity—yields a quasi-linear advection PDE whose solution via the Method of Characteristics analytically reproduces the straight-line trajectories of Flow Matching. The authors identify the Jacobian of the denoiser as the primary source of trajectory curvature, explaining why straight-line flows allow large step sizes and why empirical models require distillation to flatten intersecting characteristics.

arXiv:2609.00198v1 Announce Type: new 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.