A Lagrangian View of Flow Matching

Lipman, Liu

generative modelsoptimal transportpartial differential equations

2609.00198

Builder Relevance

70%

1h ago

Abstract

This paper presents a bottom-up mechanical derivation of Flow Matching grounded in a Lagrangian perspective, emphasizing the conservation of target identity.

Reality Card

Core Claim

The paper establishes a strict invariance condition for optimal single-step generation, leading to a governing quasi-linear advection PDE that explains the straight-line trajectories of Flow Matching.

Method / Result

Solving the governing PDE via the Method of Characteristics yields straight-line trajectories, enabling massive step sizes in generative models.

Limitations

The requirement for distillation to flatten intersecting characteristics may limit the reproducibility of empirical models.

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