direction and magnitude over time. This is a time-variant velocity field. Now, imagine that we drop an entire cloud of particles onto this field. In fact, infinitely many. That's our blue shape over there. The continuity equation tells us that the evolution of its density in time is determined by how individual particles are transported by the velocity field according to this ODE. It implies a deep connection between global evolution and local movement. So, if we train a model to predict this velocity field, then at inference time, we could push particles through it. Start from a point X0 sampled from P0, follow the field over time, and eventually reach a destination that is likely to be a well-formed image. Learning a velocity field is the idea behind flow-based methods. Flow matching is the poster child today, but its lineage goes back more than a decade.