Flow Matching and Rectified Flow

Training a continuous normalising flow by maximum likelihood requires integrating the ODE inside the training loop, which is why CNFs did not scale. Flow matching instead regresses the velocity field onto the known closed-form velocity of a conditional probability path, so each training step needs one noise sample, one data sample and a scalar t. No ODE is solved during training.

With x_t = (1-t) x_0 + t x_1, the path velocity is constant: dx_t/dt = x_1 - x_0. So the loss is E ||v_theta(x_t, t) - (x_1 - x_0)||^2. The regression label is just the difference of the two endpoint samples, and this per-sample objective has the same gradient as the intractable marginal one.

Sampling integrates an ODE. If the velocity along a trajectory were constant, a single Euler step of size 1 would be exact; the closer to constant it is, the fewer function evaluations a solver needs for a target error. The saving is in solver steps, not in per-step quality, which is a distinction worth keeping straight when reading efficiency claims.

The network at (x_t, t) cannot know which pair generated that point, so it learns the average velocity over all consistent pairs, and an average of straight segments with different endpoints is curved. Reflow exists precisely for this: it re-couples noise and data pairs using the current model and retrains, costing a full generation pass plus training per iteration.

No. Gaussian diffusion is recoverable as one particular choice of conditional probability path inside the flow-matching framework, so the real comparison is between path schedules and parameterisations. Reported wins come from choosing straighter paths plus training details such as SD3's middle-weighted timestep sampling, and matched-compute comparisons remain scarce.

Same problem shape. A dexterous policy must emit a chunk of continuous, multimodal actions at high frequency, which categorical cross-entropy over binned actions handles badly. Regressing a velocity field over a continuous action chunk conditioned on observations is exactly the image-generation setup with a different output space, which is the design in the pi-0 VLA (Black et al., 2024).

Training a continuous normalising flow by maximum likelihood requires integrating the ODE inside the training loop, which is why CNFs did not scale. Flow matching instead regresses the velocity field onto the known closed-form velocity of a conditional probability path, so each training step needs one noise sample, one data sample and a scalar t. No ODE is solved during training.

With x_t = (1-t) x_0 + t x_1, the path velocity is constant: dx_t/dt = x_1 - x_0. So the loss is E ||v_theta(x_t, t) - (x_1 - x_0)||^2. The regression label is just the difference of the two endpoint samples, and this per-sample objective has the same gradient as the intractable marginal one.

Sampling integrates an ODE. If the velocity along a trajectory were constant, a single Euler step of size 1 would be exact; the closer to constant it is, the fewer function evaluations a solver needs for a target error. The saving is in solver steps, not in per-step quality, which is a distinction worth keeping straight when reading efficiency claims.

The network at (x_t, t) cannot know which pair generated that point, so it learns the average velocity over all consistent pairs, and an average of straight segments with different endpoints is curved. Reflow exists precisely for this: it re-couples noise and data pairs using the current model and retrains, costing a full generation pass plus training per iteration.

No. Gaussian diffusion is recoverable as one particular choice of conditional probability path inside the flow-matching framework, so the real comparison is between path schedules and parameterisations. Reported wins come from choosing straighter paths plus training details such as SD3's middle-weighted timestep sampling, and matched-compute comparisons remain scarce.

Same problem shape. A dexterous policy must emit a chunk of continuous, multimodal actions at high frequency, which categorical cross-entropy over binned actions handles badly. Regressing a velocity field over a continuous action chunk conditioned on observations is exactly the image-generation setup with a different output space, which is the design in the pi-0 VLA (Black et al., 2024).