MeanFlow

Flow Matching models the instantaneous velocity $v(z_t, t)$ — the tangent of the path at one instant — and recovers a sample by integrating it. MeanFlow models the average velocity over an interval, defined as that integral divided by its width:

$$u(z_t, r, t) \triangleq \frac{1}{t - r}\int_r^t v(z_\tau, \tau)\,d\tau .$$

The two coincide only in the limit $r \to t$. Away from it the average velocity points along the displacement between the two times rather than along the curve, which is exactly what a one-step sampler needs: with $u$ in hand the whole path is $z_r = z_t - (t - r)\,u(z_t, r, t)$, and generation from noise is a single evaluation at $(r, t) = (0, 1)$.

Training on that definition directly would require the integral. The way through is to multiply it out and differentiate with respect to $t$, holding $r$ fixed, which turns the integral into the MeanFlow Identity:

$$u(z_t, r, t) = v(z_t, t) - (t - r)\,\frac{d}{dt}u(z_t, r, t).$$

Every term on the right is available. The instantaneous velocity is the closed-form conditional velocity Flow Matching already regresses onto, and the total derivative expands by the chain rule into

$$\frac{d}{dt}u(z_t, r, t) = v(z_t, t)\,\partial_z u + \partial_t u,$$

which is one Jacobian-vector product of $u_\theta$ along the tangent $(v,\, 0,\, 1)$ — a single extra forward-mode pass, not a second-order solve. The identity becomes a regression target,

$$u_{\mathrm{tgt}} = v_t - (t - r)\bigl(v_t\,\partial_z u_\theta + \partial_t u_\theta\bigr),$$

and the loss is $\lVert u_\theta - \mathrm{sg}(u_{\mathrm{tgt}}) \rVert^2$. The stop-gradient is what keeps the cost at one extra pass: without it the Jacobian-vector product would itself need to be differentiated.

Setting $r = t$ collapses the second term and recovers Flow Matching exactly, so the method is Flow Matching with a modified target rather than a different framework. The paper draws $r \neq t$ for only a quarter of samples; at 0% it is Flow Matching and one-step generation fails outright (FID 328.91 against 61.06).