Source
Boffi, Albergo and Vanden-Eijnden (2024), LIT-793, §3.2–3.7 and App. C. Geng, Deng, Bai, Kolter and He (2025), LIT-784, §4.1, Eqs. 3–6. Frans, Hafner, Levine and Abbeel (2024), LIT-787, §3, Eqs. 3–4. Song, Dhariwal, Chen and Sutskever (2023), LIT-093. Zheng et al. (2025), LIT-783, App. F.1.
The account
One object. Write X_{s,t}(x) for the point the probability-flow ODE reaches at time t from x at time s. Flow map matching (LIT-793) proves it is the unique solution of two equations (Props. 3.5–3.6). In the end time it is Lagrangian: ∂t X{s,t}(x) = b_t(X_{s,t}(x)), with b the velocity. In the start time it is Eulerian: ∂s X{s,t} + b_s·∇X_{s,t} = 0, so the output does not change as the start point slides along its own trajectory. And it composes: X_{t,τ}∘X_{s,t} = X_{s,τ}, “the consistency property, here stated over two times”. A sample in n steps is n applications of the map over a partition of the interval. At the true map every partition gives the same endpoint.
Each named method trains one of these identities.
| Method | What the network outputs | Identity trained on |
|---|---|---|
| Progressive distillation | one step of the new sampler | composition, against two steps of the previous one (LIT-793, App. C.4, Eq. C.23) |
| Shortcut models | s(x, t, d), with x + d·s = X_{t,t+d}(x) | composition at binary steps, s(x, t, 2d) = ½s(x, t, d) + ½s(x′, t+d, d) (LIT-787, Eqs. 3–4), with the velocity as the d = 0 base case |
| Consistency models | the one-time map X_{t,0} | Eulerian; consistency distillation is its discretization, and its continuous-time limit is the Eulerian loss (LIT-793, App. C.2, Eqs. C.12, C.18) |
| MeanFlow | the average velocity u(z, r, t) | Eulerian, written for the displacement |
| Flow map matching, Lagrangian | the two-time map | Lagrangian, against the teacher’s velocity (§3.3) |
MeanFlow in these terms. MeanFlow (LIT-784) defines u(z_t, r, t) as the displacement from t to r divided by t − r (Eq. 3). Its own text says that, in flow map matching’s language, “the Flow Map corresponds to displacement”. So X_{t,r}(z_t) = z_t − (t − r)·u(z_t, r, t). Differentiate that along the trajectory in t, with r held fixed. It vanishes exactly when u = v − (t − r)·du/dt, which is the MeanFlow identity. The identity is the Eulerian equation for the map from t to r. This is the record’s algebra on the paper’s definitions, and rCM (LIT-783, App. F.1) reaches the same place from the other side: adding a second time to continuous-time consistency gives a continuous-time consistency trajectory model, “which MeanFlow is under the rectified-flow schedule”. MeanFlow’s additivity remark, that one step over [r, t] equals two over [r, s] and [s, t], is composition, holding automatically for the true field.
Consistency models are the one-time case. A consistency model (LIT-093) maps every point of a trajectory to its data end, X_{t,0} only. Flow map matching places it as that special case (App. C.2). That is why its multistep sampling has to re-noise and denoise again rather than stop at an intermediate time. LIT-793’s §2 overstates this as consistency models “do not benefit from multistep sampling”. LIT-093 and its successors report that they do, by re-noising.
What it explains
SOTA-206’s dial. The practice says to keep a multi-step option in a few-step model, and its evidence is one set of weights sampled at several step counts. On this account the dial is not a feature added to some models. It is what a two-time map is. Composition makes every partition of [0, 1] reach the same endpoint at the optimum, so the step count is chosen at sampling time, and a learned map’s error is what makes more steps help. Shortcut models (LIT-787) show it most plainly: the step size is an input, the network is trained so that one step of 2d equals two of d, and one set of weights scores 6.9, 13.8 and 20.5 FID at 128, 4 and 1 steps on CelebA-HQ (Table 1). Inductive Moment Matching and MeanFlow also condition on two times and expose the dial directly. Consistency models, with one time, keep it only through re-noising. Progressive distillation trains the map at a single step size per stage and keeps no dial, which is the counter-case in SOTA-206’s Conditions.
What was measured
- The identities are not interchangeable as losses. With a 5.53-FID teacher on CIFAR-10, the Lagrangian loss distils to 7.13 FID at two steps and 6.04 at four. The Eulerian loss reaches 48.32 and 44.35 (LIT-793, Table 1). The authors attribute the gap to the Eulerian loss’s spatial Jacobian of a nearly singular map (§3.4). The record’s view: small models, one dataset, one run each.
- The second time is what does the work in MeanFlow. With r = t on every sample the objective is flow matching, and one-step FID is 328.91. With r ≠ t on 25% of samples it is 61.06 (LIT-784, Table 1a).
What it does not say
- Not which estimator wins. SOTA-448 ranks MeanFlow above consistency training and shortcut models from scratch, on rows copied between papers at different budgets. This account says the three aim at the same object, so their differences are differences of estimator: identity, discretization, weighting and budget. It does not say which, and it does not explain that ranking. rCM’s one figure has the MeanFlow form distilling a large model worse than plain continuous-time consistency (LIT-783, App. F.1).
- Not that the training dynamics agree. The Eulerian form is trained with a stop-gradient on the target. Flow map matching shows the flow map is then a critical point of the update, not the minimizer of a loss (Prop. 3.12). THEORY-124 is about what follows from that.
- Not Inductive Moment Matching. IMM (LIT-775) also learns a two-time sampler from scratch, but matches the distributions reached from t and from a nearby r. Its own §3.1 notes that the minimizer “is not unique and, under mild assumptions, a deterministic minimizer exists”. So it learns a marginal-preserving map, which need not be the ODE’s flow map. Consistency models are its one-particle case (Lemma 1), and there the two accounts meet.
- Not distribution-matching distillers. DMD and its successors train a one-step generator on a divergence to the teacher’s marginals, with no identity of the ODE’s map in the loss. Their one-step generator need not be X_{1,0}.