A Lagrangian View of Flow Matching
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.
Related
- A neural characteristic mapping method: Lagrangian PINNs based on flow maps for transport-dominated problems
- RainFlow: Flow Matching for Infrared-Only Quantitative Precipitation Estimation
- Convergence of Roberts flow dynamos in Eulerian and Lagrangian codes
- Probabilistic Simulation of Gas-Liquid Interfaces with Coupled Flow Matching on Spherical Harmonics
- SymADiT: Symmetry-Aware Crystal Generation with Latent Flow Matching
BibTeX
@misc{milanfar2026lagrangian,
title = {{A Lagrangian View of Flow Matching}},
author = {Peyman Milanfar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00198},
eprint = {2609.00198},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00198},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Lagrangian View of Flow Matching
AU - Peyman Milanfar
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.00198
UR - https://arxiv.org/abs/2609.00198
AB - 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.
ER -