Flow Matching

Diffusion learns to undo noise step by step. Flow matching asks a simpler question: what velocity field carries the simple distribution to the data one, in a straight line if possible? Train a network to predict that velocity, then integrate it. Straighter paths mean fewer integration steps, which is most of why newer image and video models sample so much faster.

The objective is a plain regression onto a target velocity, with no noise schedule to tune and no variational bound to derive — which is why it has largely displaced the diffusion formulation in new work even though the two are closely related. Path straightness is the design variable: an optimal-transport-shaped path needs far fewer function evaluations at sampling time than a curved one.

Regress v sub _θ (x sub _t , t) onto the conditional velocity of a chosen probability path. With the linear path x sub _t  = (1−t)x sub ₀  + t·x sub ₁  the target is simply x sub ₁  − x sub ₀ , constant along the path — so sampling is integrating an ODE rather than simulating an SDE, and a straight path is exactly integrable in one step.