flow matching

Flow matching is a way to train a continuous generative flow by simple regression, no simulation and no likelihood needed during training. You first pick a probability path, a prescribed sequence of distributions that morphs noise into data over time, and you know the velocity field that would push samples along that path. Then you simply teach a network to output that velocity at every point and time, by minimizing a mean-squared error, exactly as one fits any regression.

The obstacle is that the marginal velocity that transports the whole mixture is itself an intractable expectation. The conditional flow matching insight is that regressing onto a per-sample conditional velocity, the one transporting noise to a single fixed datum, yields the same gradient as regressing onto the true marginal field, so the loss becomes a tractable expectation over pairs of noise and data. Choosing a Gaussian probability path recovers diffusion as a special case, while choosing the straight-line optimal-transport path gives near-linear trajectories that sample in very few steps.

Flow matching has become the training method of choice for large continuous generators because it is stable, simulation-free, and agnostic to the path geometry, letting designers trade likelihood quality against sampling speed by their choice of path.