Velocity Field

The function a flow-matching model learns: it tells each sample which direction to move, and how fast, at every timestep.

The velocity field is what flow matching and rectified flow — a family of generative models introduced in 2022 by Lipman et al. and, separately, Liu et al. — actually learn. It's a function v(x, t): given a sample x at time t (0 means pure noise, 1 means real data), it outputs the direction and speed x should move right now. During training, the network regresses the velocity along a predetermined “noise-to-data” path, usually a straight line, so the target velocity is simply data minus noise — there's no need to simulate the full generation process step by step. At inference time, the model starts from random noise and integrates along the velocity field with an ODE solver (such as the Euler method) for a handful of steps to produce a sample. Straighter paths need fewer steps, which is why this approach usually samples faster than traditional diffusion models. In robotics, the action expert in π0 predicts the velocity field for a chunk of future actions.

ExampleAt inference, π0 first samples a chunk of Gaussian noise to serve as the “action chunk.” The action expert predicts a velocity field conditioned on the current image, instruction, and robot state, then integrates it for 10 steps with the forward Euler method, turning the noise into a continuous 50-step action sequence.

Also called

Vector Field, Velocity Vector Field

Related

Flow Matching · Rectified Flow · Flow Matching Loss · Diffusion / Flow Samplers · Action Expert · π0

Sources

Flow Matching for Generative Modeling (arXiv:2210.02747)
Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow (arXiv:2209.03003)
π0: A Vision-Language-Action Flow Model for General Robot Control (arXiv:2410.24164)