Trajectories, vector fields and probability paths: the intuition behind flow matching
ariG23498 · x · 2026-09-08
A clean conceptual explainer of the three ideas behind flow matching and diffusion models:
- Trajectory: a function of time mapping a moment to a point in space.
- Vector field: a function of time and location giving the velocity at a point.
- Probability path: at each time, a probability density you can sample from.
The interesting part is how they connect: start from a point sampled from a Gaussian and move along the vector field until t=1, where the point should look like it came from the data distribution. Moving on the field alone gives an ODE; adding noise gives an SDE. The author argues much of flow and diffusion modeling boils down to understanding this transportation.
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