rectified flow

Curved trajectories are expensive to integrate, because following a bending path accurately requires many small steps. Rectified flow asks for the opposite: transport paths that are as straight as possible, since perfectly straight paths can be traversed in a single Euler step. It trains a velocity field to follow the straight line connecting a noise sample to a data sample, then optionally iterates a procedure that straightens those paths further.

The training target is the constant velocity of the linear interpolation between a noise point and a data point, which is just their difference; regressing the network onto it is a special case of conditional flow matching with the straight-line probability path. A single round already gives reasonably straight marginal trajectories. The reflow step then resamples noise-data pairs by coupling them through the current model and retrains, which provably reduces the transport cost and crowds the trajectories toward straight lines, after which one or two steps suffice to sample, and distillation can push to a single step.

Rectified flow underlies several state-of-the-art large image and video generators trained with a flow-matching objective, where straightness is the lever that turns a many-step diffusion-like model into a few-step one without sacrificing much quality.