Stanford CME296 Diffusion & Large Vision Models | Spring 2026 | Lecture 3 - Flow matching

Reversed temporal indexing

Unlike diffusion models where t=0 represents clean data, flow matching conventions define t=0 as the initial Gaussian noise distribution and t=1 as the target data distribution.

Probability transport objective

The core goal is transporting the entire probability density from an initial distribution P0 to a target distribution P1 while maintaining the conservation of mass property where total probability sums to one.

Flow and probability path

The flow s_t(x_0) represents the mapping from initial conditions to positions at time t, while the probability path P_t(x) describes the evolving intermediate distributions between noise and data.

Vector fields as velocity instructions

The vector field u_t(x) specifies both direction and speed for particles at specific spatial locations and times, functioning like dynamic routing instructions for self-driving cars.

Score functions as compasses

While the score acts as a compass pointing toward high-density regions, the vector field provides explicit velocity vectors that guide the deterministic transport of samples through space.

Trajectories over sampling

Flow matching focuses on continuous trajectories that map individual points from noise to data, contrasting with score matching's emphasis on navigating density gradients.

Governing ordinary differential equation

Particle trajectories follow the ODE dx/dt = u_t(x), where infinitesimal position changes are determined by the vector field multiplied by the time differential dt.

Lipschitz continuity requirement

Unique trajectories for given initial conditions are mathematically guaranteed only when the vector field satisfies Lipschitz continuity, preventing scenarios where multiple paths originate from the same point.

Conservation via continuity equation

The framework ensures probability density evolves according to the continuity equation, balancing the rate of density change against the divergence of the vector field to prevent mass loss during transport.