Flow matching models center on designing a probability path from noise to data. Current options are limited: optimal transport paths move samples along straight lines, and rectified flow also follows straight lines. Mathematically, these paths are the same thing—free-particle motion under a minimized kinetic-energy Lagrangian.
Du, Zhang, and Li observed that in classical mechanics this is only a special case—the simplest scenario of the principle of least action: a particle subject to no force moves in a straight line at constant velocity. But real particles are subject to various forces, so trajectories can be curved, oscillatory, or spiral.
Lagrangian Flow Matching generalizes the problem: under the constraints of the continuity equation and endpoint conditions, minimize the action of a general Lagrangian. Optimal transport paths are the special case of a kinetic-energy Lagrangian (free particle motion). Trigonometric variance-preserving diffusion paths are the special case of a harmonic oscillator Lagrangian (particle under a linear restoring force).
The elegance of this framework is that the dynamic problem has an equivalent static optimal transport formulation—meaning the training objective remains simulation-free (no need to simulate the entire path during training). More general Lagrangians yield new probability paths and velocity fields. Numerical experiments show that different Lagrangians do change the learned dynamics and remain competitive in generation quality.
Open Questions
- Do the new paths actually outperform straight-line paths on real high-dimensional generation tasks (e.g., images, protein structures)?
- What are the selection criteria among different Lagrangians—is there a general principle telling you which physical model to use?
- The theoretical framework provides "possibilities" but no guidance on "which one to choose."
References
1. Du, S., Zhang, J., & Li, Y. (2026). *Lagrangian Flow Matching: A Least-Action Framework for Principled Path Design*. arXiv:2605.15419 [cs.LG]. 2. Lipman, Y., et al. (2023). *Flow Matching for Generative Modeling*. ICLR. 3. Liu, X., et al. (2023). *Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow*. ICLR.