Rectified Flow

A generative model that moves noise toward data along a straight line — the straighter the path, the fewer sampling steps needed.

Rectified flow was proposed by Xingchao Liu, Chengyue Gong, and Qiang Liu in their 2022 paper 'Flow Straight and Fast.' It connects a noise sample and a data sample with a straight line and trains a network to predict the velocity along that line; generation starts from noise and solves an ordinary differential equation (ODE) that follows the predicted velocity to arrive at data. The training objective is a simple least-squares regression, and it's essentially equivalent to the flow matching Lipman and colleagues proposed around the same time when both use straight-line paths. Its other contribution is 'reflow': using a trained model to generate noise-data pairs, then retraining on those pairs, which makes the paths progressively straighter — straight enough that even a single Euler step can produce a decent result, which suits pairing with distillation for few-step or one-step generation. Stability AI's Stable Diffusion 3 trains with rectified flow; in robotics, π0's flow matching uses the same straight-line interpolation path to generate actions.

Exampleπ0 trains by linearly mixing a real action chunk with Gaussian noise according to a time τ, having the action expert predict the velocity between the two; at deployment, it starts from pure noise and integrates 10 Euler steps to get an executable action chunk.

Also called

RF

Related

Flow Matching · Velocity Field · Diffusion Model · One-step Generation · Consistency Model · π0

Sources

Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow (arXiv 2209.03003)
Scaling Rectified Flow Transformers for High-Resolution Image Synthesis (SD3, arXiv 2403.03206)
π0: A Vision-Language-Action Flow Model for General Robot Control (arXiv 2410.24164)