Lagrangian Flow Matching: A Least-Action Framework for Principled Path Design

2026-05-14

AI Summary

This work addresses the limitations of existing flow matching methods, which rely on straight-line trajectories and struggle to capture complex dynamical behaviors. By leveraging the principle of least action, the authors generalize the Lagrangian formalism to construct probability paths and associated velocity fields that satisfy both the continuity equation and endpoint constraints. This approach embeds flow matching within a classical mechanics framework, thereby unifying and extending prior methods such as optimal transport and diffusion-based paths. The resulting static variational objective eliminates the need for trajectory simulation and enables direct optimization. Empirical results demonstrate that the proposed method yields physically meaningful dynamical evolutions and achieves performance on par with state-of-the-art conditional flow matching models.

Abstract

Flow matching trains a neural velocity field by regression against a target velocity associated with a prescribed probability path connecting a simple initial distribution to the data distribution. A central design choice is the path itself. Existing constructions, including rectified and optimal-transport-based paths, transport samples along straight lines between coupled endpoints and thus cover only a narrow class of dynamics. We observe that this corresponds to the simplest case of the least-action principle in classical mechanics, in which the kinetic Lagrangian yields free-particle straight-line trajectories. Building on this observation, we propose Lagrangian flow matching, a physics-based framework in which the probability path and velocity field are determined by minimizing the action of a general Lagrangian subject to the continuity equation and the prescribed endpoints. We show that this dynamic problem admits an equivalent static optimal transport (OT) formulation, yielding a family of simulation-free training objectives that recover OT-based flow matching as the kinetic special case and the trigonometric variance-preserving diffusion path as the harmonic-oscillator case. More general Lagrangians give rise to new probability paths and velocity fields, and numerical experiments show that they induce meaningful changes in the learned dynamics while remaining competitive with existing conditional flow matching models.

Problem

• flow matching
• probability path
• least-action principle
• optimal transport
• Lagrangian dynamics

Innovation

• Lagrangian flow matching
• least-action principle
• optimal transport
• probability path design
• neural velocity field