Notes on Flow Matching

These are informal working notes I wrote while getting comfortable with the flow-matching framework. Nothing here is novel — it is a personal re-derivation of the standard results, kept as a reference.

From diffusion to flows

Diffusion models define a stochastic forward process that gradually destroys data, then learn to reverse it. Flow matching takes a cleaner view: directly regress a vector field v_t(x) whose induced flow transports a simple base distribution (usually a Gaussian) to the data distribution. If p_t is the marginal density along the path, the continuity equation

∂_t p_t + ∇ · (p_t v_t) = 0

tells us exactly which vector fields are consistent with the marginals.

Conditional flow matching

The marginal objective is intractable, but conditioning on a data sample x_1 makes it simple. Pick a conditional path — e.g. the straight line from the optimal-transport family,

x_t = (1 - t) x_0 + t x_1,   u_t(x | x_1) = x_1 - x_0

and train v_θ(t, x_t) to predict the conditional velocity x_1 - x_0. Averaging over (x_0, x_1) pairs recovers the correct marginal field. This is the whole trick: simulation-free training with a regression loss.

What I’m thinking about next

• How much can test-time compute (extra refinement loops over intermediate hidden states) substitute for more sampling steps?
• Where in the DiT backbone does the trajectory become “decided”, and can early tokens be frozen to save compute?
• Connections between rectified-flow reflow and hidden-state refinement.