Flow Matching

Learning a velocity field that smoothly transports noise into real data, then generating samples by integrating along it.

Flow matching was proposed by Meta's Lipman and colleagues in 2022 (published at ICLR 2023), for training continuous normalizing flows, a class of generative model that continuously deforms noise into data. The method defines a path between a noise sample and a real sample, most commonly straight-line interpolation, and trains a network to predict the velocity at every point along that path, the velocity field, with a loss that is just the mean squared error between predicted and target velocity; training needs no simulation of the full trajectory. To generate, start from noise and numerically integrate along the network's predicted velocity; for robot actions, a handful to about ten steps is usually enough (π0 uses 10, GR00T N1 uses 4). It belongs to the same family as diffusion models, both turning noise into data step by step, and diffusion can be viewed as one specific path within flow matching; a straight-line path tends to train faster and need fewer sampling steps, in the same spirit as the contemporaneous rectified-flow idea. In robotics, π0 uses it to generate action chunks in its action expert, and GR00T N1 and others use it too.

ExampleAt inference, π0 starts from Gaussian noise and integrates 10 steps, each of size 0.1, to turn noise into a 50-step-long action chunk; during training, the loss is just the squared difference between the network's predicted velocity field and the target velocity.

Also called

FM, Conditional Flow Matching, CFM

Related

Diffusion Model · Velocity Field · Rectified Flow · Flow Matching Loss · π0 · Action Expert

Sources

Flow Matching for Generative Modeling (arXiv 2210.02747)
π0: A Vision-Language-Action Flow Model for General Robot Control (arXiv 2410.24164)
GR00T N1: An Open Foundation Model for Generalist Humanoid Robots (arXiv 2503.14734)