Build a flow-matching & DiT image synthesizer

Pick a starting noise value and take Euler steps. None of these four starting points were in the training set — the field being applied was fit once, on four completely different pairs, and now generates from anywhere.

Fitting a constant velocity field $v_\theta \equiv \bar v$ by least squares against training pairs' velocities has a closed form — the mean:

• $\bar v$ — the single constant velocity value being fit, applied identically at every point and every time.
• $N$ — the number of training pairs.
• $x_0^{(i)}$ — the noise sample from the $i$-th training pair.
• $x_1^{(i)}$ — the data sample from the $i$-th training pair.

1. The smallest possible flow-matching model

   This has no dependence on $x$ or $t$ at all — just one number.

2. Real networks condition on x and t, same objective

   Real flow-matching networks condition on both, so different starting points and different times along the path get genuinely different corrections — but the training objective being minimized is the exact same squared error between a predicted velocity and $x_1-x_0$, just with a far more expressive model fitting it.

The trained field doesn't equal any single training pair's velocity — it's their average. Every pair pulled the fit a little differently, and the least-squares solution is the one number that minimizes total squared disagreement across all four, not the best match to any one of them.

Four training pairs: $({-}3,6)$, $({-}1,4)$, $({-}4,8)$, $({-}2,5)$:

1. Each pair's velocity

   Computed exactly as in Chapter 3, one subtraction each ($x_1-x_0$):

   • $(-3,6)$: $6-(-3)=9$
   • $(-1,4)$: $4-(-1)=5$
   • $(-4,8)$: $8-(-4)=12$
   • $(-2,5)$: $5-(-2)=7$
2. Train: average them

   $\bar v = (9+5+12+7)/4 = 8.25$. This is the entire training procedure for this model.

3. Generate from noise the model never trained on

   $x_0 = -2$ (not one of the four training points): $x_1 = -2 + 8.25(1) = 6.25$ — one Euler step, exact, since the trained field is constant.