Lagrangian flow matching reformulates flow matching paths via the least-action principle, recovering optimal-transport and trigonometric diffusion paths as special cases of kinetic and harmonic Lagrangians while enabling new paths.
´ωsinpωtqx 0 `ωcospωtqB. Per-pair kinetic energy.The kinetic energy along ϕ‹ is kωpx0, x1q “ ş1 0 1 2 } 9ϕ‹ t }2 dt. Expanding the square gives } 9ϕ‹ t }2 “ω 2
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Lagrangian Flow Matching: A Least-Action Framework for Principled Path Design
Lagrangian flow matching reformulates flow matching paths via the least-action principle, recovering optimal-transport and trigonometric diffusion paths as special cases of kinetic and harmonic Lagrangians while enabling new paths.