Mean Flows for One-step Generative Modeling
Authors: Zhengyang Geng
Mingyang Deng
Xingjian Bai
J Zico Kolter
Kaiming He
Venue: NeurIPS 2025 oral ()
Keywords: Generative Models
Primary area: deep_learning

Abstract: We propose a principled and effective framework for one-step generative modeling. We introduce the notion of average velocity to characterize flow fields, in contrast to instantaneous velocity modeled by Flow Matching methods. A well-defined identity between average and instantaneous velocities is derived and used to guide neural network training. Our method, termed the \textit{MeanFlow} model, is self-contained and requires no pre-training, distillation, or curriculum learning. MeanFlow demonstrates strong empirical performance: it achieves an FID of 3.43 with a single function evaluation (1-NFE) on ImageNet 256$\times$256 trained from scratch, significantly outperforming previous state-of-the-art one-step diffusion/flow models. Our study substantially narrows the gap between one-step diffusion/flow models and their multi-step predecessors, and we hope it will motivate future research to revisit the foundations of these powerful models.

Mean Flows for One-step Generative Modeling
Zhengyang Geng!* Mingyang Deng"
Xingjian Bai
J. Zico Kolter'
Kaiming He
CMU
MIT
Abstract
We propose a principled and effective framework for one-step generative model- ing. We introduce the notion of average velocity characterize flow fields, in contrast to instantaneous velocity modeled by Flow Matching methods. well- defined identity between average and instantaneous velocities is derived and used to guide neural network training. Our method, termed the MeanFlow model, is self-contained and requires no pre-training, distillation, or curriculum learning. McanFllw demonstrates strong empirical performance: it achieves an FID of 3,43 with single function evaluation (I-NFE) ImageNet x26 trained from scratch, significantly outperforming previous state-of-the-an one-step dif- fusion/flow models. Our study substantially narrows the gap between one-step diffusion/flow models and their multi-step predecessors, and we hope it will moti- future research to revisit the foundations of these powerful models. Our code is available at https://gitubb.comggmma /wwna/sssssieeeeaaaooo.
35
B: 131M
1CT-XL
M: 308M
25
L: 45.M
1-step FiD
XL: 675M
Figure 1: One-step generation on ImageNet 256x256 from scratch. Our MeanFlow (MF) model achieves significantly better generation quality than previous state-of-the-art one-step dif- fusion/flow methods. Here, iCT [43], Shortcut [12], and our MF all 1-NFE generation, while IMM's 1-step result [52] involves 2-NFE guid- ance. Detailed numbers in Tab. 2. Images shown are generated by our 1-NFE model.
20
Shortcut XL
10
MF-8
IL
MF -
MF XL
Training Compute (GFLOPs, log-ecale)
1 Introduction
The goal of generative modeling is to transform prior distribution into the data distribution. Flow Matching [28, 1, 30] provides an intuitive and conceptually simple framework for constructing flow paths that transport one distribution to another. Closely related to diffusion models [42, 44, 19], Flow Matching focuses on the velocity fields that guide model training. Since its introduction, Flow Matching has widespread adoption in modern generative modeling [10, 33, 35].
Both Flow Matching and diffusion models perform iterative sampling during generation. Recent re- cewtooowwaddggerrririi
search has paid significant attention to few-step-and and in particular, one-step, models. Pioneering this direction, Consistency Models [46, 43, 14, 31] introduce a consistency con- straint to network outputs for inputs sampled along the same path. Despite encouraging results, the consistency constraint imposed as a property
network's 's behavior, while the properties underlying ground-truth field that should guide learning remain unknown. Consequently, training can be unstable and requires a carefully designed "discretization curriculum" [46, 43, 14] to progressively constrain the time domain.
Work partly done when visiting MIT.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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In this work, we propose a principled and effective framework, termed MeanFlow, for one-step generation. The core idea is introduce a new ground field representing the average velocity. in contrast to the instantaneous velocity typically modeled in Flow Matching. Average velocity is defined as the ratio of displacement to a time interval, with displacement given by the time integral of the instantaneous velocity. Solely from this definition, we derive a well-defined, intrinsic relation between the average and instantaneous velocities, which naturally serves as a principled basis for guiding network training.
Building on this fundamental concept, we train a neural network to directly model the average velocity field. We introduce a loss function that encourages the network to satisfy the intrinsic relation between average and instantaneous velocities. No extra consistency heuristic needed. The existence of the ground-truth target field ensures that the optimal solution is, in principle, independent of the specific network, which in practice can lead to more robust and stable training. We further show that our framework can naturally incorporate classifier-free guidance (CFG) [18] into the target field, incurring no additional cost at sampling time when guidance is used.
Our MeanFlow models demonstrate strong empirical performance in one-step generative modeling. On ImageNet 256x256 [6], our method achieves FID of 3.43 using I-NFE (Number of Function Evaluations) generation. This result significantly outperforms previous state-of-the-art methods in its class by a relative margin of 50% to 70% (Fig. 1). In addition, our method stands as a self- contained generative model: it is trained entirely from scratch, without any pre- pre-training, distillation, or curriculum learning. Our study largely closes the gap between one-step diffusion/flow /flow models and their multi-step predecessors, and we hope it will inspire future work to reconsider the foundations of these powerful models.
2 Related Work
Diffusion and Flow Matching, Over the past decade, diffusion models [42, 44, 19, 45] have been developed into a highly successful framework for generative modeling. These models progressively add noise to clean data and train neural network reverse this process. This procedure involves solving stochastic differential equations (SDE), which can be reformulated as probability flow ordinary differential equations (ODE) [45, 22]. Flow Matching methods [28, 1, 30] extend this framework by modeling the velocity fields that define flow paths between distributions. Flow Matching can also be viewed form of continuous-time Normalizing Flows [36].
Few-step Diffusion/Flow Models. Reducing sampling steps has become an important consideration
from both practical and theoretical perspectives. One approach is to distill pre-trained many-step diffusion model into a few-step model, e.g., [39, 13, 41] or score distillation [32, 50, 53]. Early explorations into training few-step models [46] are built upon the evolution of distillation based methods. Meanwhile, Consistency Models (CM) [46] are developed as a standalone generative model that does not require distillation. These models impose consistency constraints different time steps, encouraging them to produce the same endpoints along the trajectory. Various consistency models and training strategies [46, 43, 14, 31, 49, 16) have been investigated.
network outputs
Several recent works have focused on characterizing diffusion-//low-baaa quantitics with respect to two time-dependent variables. Consistency Trajectory Models [23] extend to two time variables, using a training-time ODE solver for target computation, which computationally expensive. In [2], Flow Map is defined as the flow's integral between two time steps. Compared to the average velocity we are based on, [23, 2] are analogous to displacement. Shortcut Models [12] introduce a self-consistence loss function in addition to Flow Matching, which captures relationships between the flows at different discrete time intervals. Inductive Moment Matching [52] models the self-consistency of stochastic interpolants different time steps.
3 Background: Flow Matching
Flow Matching [28, 30, 1] is family of generative models that learn to match the flows, represented by velocity fields, between two probabilistic distributions. Formally, given data x ~ Puaaa (z) and prior e ~ (e), a flow path can be constructed as 2, = a,æ + bye with time i, where a, and b, are predefined schedules. The velocity is defined as U, = 2 = a/æ: + D.e. where denotes the time derivative. This velocity is referred to as the conditional velocity in [28]. denoted by v = 0,(2 x).
See Fig. 2 left. A commonly used schedule is a, = 1 t and b, which leads to U2

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Figure 2: Velocity fields in Flow Matching [28]. Left: con- ditional flows [28]. given 2; can arise from different (x, c) pairs, resulting in different conditional velocities Ut. Right: marginal flows [28], obtained by marginalizing over all possi- ble conditional velocities. The marginal velocity field serves as the underlying ground truth field for network training. All velocities shown here essentially instantaneous veloe- ities. Illustration follows [11]. (Gray dots: samples from prior; red dots: samples from data.)
0(2t, t)
Because given 2, and its i can arise from different x and c, Flow Matching essentially models the expectation over all possibilities, called the marginal velocity [28] (Fig. 2 right):
v(2,, t) A (p.((a)).))e))
(1)
A neural network 0g parameterized by 0 is learned to fit the marginal velocity field: CFM Ex.p.(s.) [/ (ze; f) o(ze; I)1 Although computing this loss function is infeasible due to the marginalization in Eq. (1), it is proposed to instead evaluate the conditional Flow Matching loss [28]: (0) Ee,,,,e|lu0(
(2 =)1??. where the target U, is the conditional velocity. Minimizing LCFM is equivalent to minimizing LPM [28].
Given marginal velocity field v(ze, l), samples are generated by solving an ODE for 2y:
dt t - ((2,
starting from 21 6 Pprior- The solution can be written as: 2, = [2,,,,, T)dT, where we use to denote another time step. In practice, this integral is approximated numerically over discrete time steps. For example, the Euler method, a first-order ODE solver, computes each step as: 20,11 =t; + (41+1 (.) 4.). Higher-order solvers can also be applied.
is worth noting that even when the conditional flows are designed be straight ("rectified") [28, 30], the marginal velocity field (Eq. (1)) typically induces a curved trajectory. See Fig. 2 for illustration. We also emphasize that this staistraigh is not only a result of neural network approximation, but rather arises from the underlying ground-truth marginal velocity field. When applying coarse discretizations over curved trajectories, numerical ODE solvers lead to inaccurate results.
4
MeanFlow Models
4.1
Mean Flows
The core idea of our approach is to introduce a new field representing average velocity, whereas the velocity modeled in Flow Matching represents the instantaneous velocity.
Average Velocity, We define average velocity as the displacement between two time steps t and (obtained by integration) divided by the time interval. Formally, the average velocity u is:
/ u(2r, T)dr.
u(2,, r.
To emphasize the conceptual difference, throughout this paper, we use the notation u to denote average velocity, and v to denote instantaneous velocity. u(2,, t) is field that is jointly dependent on (r, t). The field of u illustrated in Fig. 3. Note that in general, the average velocity is the result of a functional of the instantaneous velocity o: that is, u = Fly] A - f udr. It a field induced by v. not depending on any neural network. Conceptually, just as the instantaneous velocity v serves as the ground-trule field in Flow Matching, the average velocity in our formulation provides an underlying ground-trum field for learning.
By definition, the field of u satisfies certain boundary conditions and "consistency" constraints (generalizing the terminology of [46]). As r-+l, we have: lim u = v. Moreover, form of "consistency" is naturally satisfied: taking one larger step over [r, 4] is "consistent" with taking two smaller consecutive steps over [r, s] and [8, I). for any intermediate time s. To see this, observe that (t r)u(z,, r, (s- r)u(za, 8) + (t- s)u(ze, s, t), which follows directly from the additivity
3

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((r)a(snd)
((2,r,t)
u(2,)çt)
t= 0.5
t= 0.7
t= 1.0
Figure 3: The field of average velocity u(z, t).. Leftmost: While the instantaneous velocity v determines the tangent direction the path, the average velocity u(2, r, defined in Eq. (3), is generally not aligned with v. The average velocity is aligned with the displacement, which is (L- r), 1). Right three subplots: The field u(z,r, is conditioned on both r and and shown here for t = 0.5. 0.7, and 1.0.
of the integral: J vdr = vdr J vdr. Thus, a network that accurately approximates the true expected to satisfy the consistency relation inherently, without the need for explicit constraints.
The ultimate aim of our MeanFlow model will be to approximate the average velocity using a neural network ug(2,, r, t). This has the notable advantage that, assuming we approximate this quantity accurately. we can approximate the entire flow path using single evaluation of ug(e, 0, 1). In other words, and as will also demonstrate empirically, the approach much more amenable to single or few-step generation, as it does not need to explicitly approximate time integral at inference time, which was required when modeling instantaneous velocity. However, directly using the average velocity defined by Eq. (3) as ground truth for training network is intractable, as it requires evaluating an integral during training. Our key insight is that the definitional equation of average velocity can be manipulated to construct optimization target that is ultimately amenable to training, even when only the instantaneous velocity is accessible.
The MeanFlow Identity. To have a formulation amenable to training, rewrite Eq. (3) as:
(t (t-r)u(z,,,,)) r, o(zr,T)dr. r)dT.
(4)
Now we differentiate both sides with respect to I, treating r as independent of This leads to: r, r,
di (1 - ((4-))u(z,,,1) di f. o(zr:T)dT = r. +) (t- " v(ze,)) (5)
d
di
where the manipulation of the left hand side employs the product rule and the right hand side uses the fundamental theorem of calculus?. Rearranging terms, we obtain the identity:
v( () (t r) d dl u r,
u(ze; I'
(6)
average vel.
instant. vel.
time derivative
We refer this equation as the "MeanFlow Identity", which describes the relation between and u. It is easy to show that Eq. (6) and Eq. (4) are equivalent (see Appendix B.3).
The right hand side of Eq. (6) provides "target" form for u(=,, r, L), which will leverage construct a loss function to train a neural network. To serve as a suitable target, we must also further decompose the time derivative term, which discuss
d Computing Time Derivative. To compute the u term in Eq. (6), note that d denotes total derivative, which can be expanded in terms of partial derivatives:
d
dzg 0.u
dr 0.
dt 0.u.
di u( r,t
di
di
dl
With de = v(2e, (see Eq. (2)), dr = 0, and
1, we have another relation between u and v:
de
d u( r, v(2e, t)0gu + d.u.
(8)
dt
IIf depends on t, the Leibniz rule [26] gives: a J v(3r, r)dr o(ze, 1) v(zr,r)

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This equation shows that the total derivative is given by the Jacobian- product (JVP) between [0_u, d, 0.u] (the Jacobian matrix of the function u) and the tangent vector [e, 0, 1]. In modern libraries, this can be efficiently computed by the jvp interface, such as torch. func. jvp in PyTorch or jax. jvp in JAX, which we discuss later.
Training with Average Velocity. Up to this point, the formulations are independent of any network parameterization. We now introduce a model to learn u. Formally, we parameterize a network ug and encourage it to satisfy the MeanFlow Identity (Eq. (6)). Specifically, we minimize this objective:
( E uo (2, r, sg(Ung))
where
Vegt v(2,, - (1 - (n(2z, 1)a_uo + 0,ua),
(10)
The term Megi serves as the effective regression target, which is driven by Eq. (6). This target uses the instantaneous velocity v as the only ground-truth signal; no integral computation needed. While the target should involve derivatives of (that is, Ou), they are replaced by their parameterized counterparts (that is, Sug). the loss function, a noopggradiett (sg) operation is applied on the target Ugg. following common practice [46, 43, 14, 31, 12]: in our case, it eliminates the need for "double backpropagation" through the Jacobian- vector product, thereby avoiding higher-order optimization. Despite these practices for optimizability, if us were to achieve zero loss, it is easy to show that it would satisfy the McanFlow Identity (Eq. (6)), and thus satisfy the original definition (Eq. (3)).
The velocity v(2e, t) in Eq. (10) is the marginal velocity in Flow Matching [28] (see Fig. 2 right). We follow [28] to replace it with the conditional velocity (Fig. 2 left). With this, the target is:
Uegt U, - (1 (v.00uv + dzua).
(11)
Recall that = afx + b/c is the conditional velocity [28), and by default, c- x.
Pseudocode for minimizing the loss function Eq. (9) is presented in Alg. 1. Overall, our method is conceptually simple: behaves similarly to Flow Matching, with the key difference that the matching target is modified by -(--) (v,0_ua Oeua), arising from our consideration of the average velocity. In particular, note that if we were to restrict to the condition / = r, then the second term vanishes, and the method would exactly match standard Flow Matching.
In Alg. 1, the jvp operation is highly efficient. In essence, computing di a via jvp requires only a single backward pass, similar to standard back- d propagation in neural networks. Because ät is part of the target ligt and thus subject stopgrad (w.r.. 0), the backpropagation for neural network optimization (w.rt. 0) treats de u as a constant, incurring no higher-order gradient computation. Consequently, jvp introduces only single ex- tra backward pass, and its cost is comparable to that of backpropagation. In our JAX implemen- tation of Alg. 1, the overhead is less than 20% of the total training time (see appendix).
Algorithm 1 MeanFlow: Training.
Note: in PoTorch and JAX, jup returns the function output JVP.
fn r, t): function to predict x: training batch
t, r - sample_te (_r randn like (x)
z (1 - t) * *
= e
u, dudt jup(fn, (z, I, t), (v, o, 1))
u_tgt v - r) dudt error u stopgrad
Sampling. Sampling using a MeanFlow model is performed simply by replacing the time inte- gral with the average velocity:
loss = (eroo)
- (t- r),(
(12)
In the case of 1-step sampling, we simply have 20 21 u(=1, 0, 1), where 2 e Pprio( (e). Alg. provides the pseudocode. Although one- step sampling is the main focus on this work, we emphasize that few step sampling is also straightforward given this equation.
Algorithm MeanFlow: 1-step Sampling
randn (x__xxx_e_____ shape)
fn r=0, t-1)
Relation to Prior Work. While related to previous one-step generative models [46, 43, 14, 31, 49, 23, 12, 52], our method provides more principled framework. At the core of our method is the
5

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functional relationship between two underlying fields u and which naturally leads to the MeanFlow Identity that must satisfy (Eq. (6)). This identity does not depend on the introduction of neural networks. In contrast, prior works typically rely on extra consistency constraints, imposed on the behavior of the neural network. Consistency Models [46, 43, 14, 31] are focused on paths anchored at the data side: in our notations, this corresponds to fixing = 0 for any f. As result, Consistency Models are conditioned on a single time variable, unlike ours. On the other hand, the Shortcut [12] and IMM [52] models are conditioned on two time variables: they introduce additional two-time self-consistency constraints. In contrast, our method is solely driven by the definition of average velocity, and the MeanFlow Identity (Eq. (6)) used for training is naturally derived from this definition, with no extra assumption.
4.2 Mean Flows with Guidance
Our method naturally supports classifier-free guidance (CFG) [18]. Rather than naïvely applying CFG at sampling time, which would double NFE, we treat CFG property of the underlying ground-truth fields. This formulation allows us to enjoy the benefits of CFG while maintaining the 1-NFE behavior during sampling.
Ground-truth Fields. We construct a new ground-truth field cel:;
e (2 c) a w c) + (1 (1- w) v(2,,)).
(13)
which is linear combination of class-conditional and a class unconditional field:
t t c) - Ep.(olle and v(2,, t) _ Ee(v(2t. cc],
(14)
where 82 is the conditional velocity [28] (more precisely, sample-conditional velocity this context).
Following the spirit of MeanFlow, we introduce the average velocity oe corresponding to o As per the MeanFlow Identity (Eq. (6)), e6 satisfies:
d de (2,, r, c) (2r,t) - ((t-)) " (
(15)
dt
Again, yefs and u are underlying ground-truth fields that do not depend on neural networks. Here, yd as defined in Eq. (13), can be rewritten as:
(2,, | c) = w v(2r, t | c) + (1 w) () t,
(16)
where we leverage the relation?: v(z,,)) () d 2z, 1), as well as (21, 1) u 1,,,,
Training with Guidance. With Eq. (15) and Eq. (16), we construct a network and its learning target. We directly parameterize u by function ug cfg Based on Eq. (15), obtain the objective:
[())= |8|| de (zt, r, c) sg("ig))) 1|2- where Uiy = 02 - (t r) U10: uo ofe + 0 0.u0
(17)
(18)
This formulation is similar to Eq. (9), with the only difference that it has a modified ū,;
- we, + (1 w) ( t,,,
(19)
which is driven by Eq. (16): the term v( | cc) Eq. (16), which the marginal velocity, is replaced by the cample-)conditionn the no-CFG case in Eq. (9).
velocity ut, following [28]. If = 1, this loss function degenerates to
To expose the network off in Eq. (17) to clasninnconiiiional inputs, we drop the class condition with cf8 10% probability, following [18]. Driven by similar motivation, we can also expose ug (z,, t, t) in Eq. (19) to both class- unaondinconi and class-conditional versions: the details are in Appendix B.I.
Single-NFE Sampling with CFG. In our formulation, ug directly models uf which is the average velocity induced by the CFG velocity gef (Eq. (13)). As a result. no linear combination is required during sampling: we directly use ugg for one-step sampling (see Alg. 2), with only a single NFE, This formulation preserves the desirable single-NFE behavior.
Observv that: u t) & Ee of t c)] Ee((r( c)]+ v( v(ze

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4.3 Design Decisions
Loss Metrics, In Eq. (9), the metric considered is the squared L2 loss. Following [46, 43, 14], we investigate different loss metrics. In general, we consider the loss function in the form of C = = where A denotes the regression error. It can be proven (see [14]) that minimizing || 1 is equivalent to minimizing the squared L.2 loss | | with "adapted loss weights". Details are in the appendix. In practice, set the weight as 1/( 2 + c)² where p
y and 0 (e.g., 10-3). The adaptively weighted loss is =g(w)-C. C, with = I pf = 0.5, this is similar to the Pseudo-Huu Huber loss in [43]. We compare different p values in experiments.
Sampling Time Steps (r, t). We sample the two time steps (r, from a predefined distribution. We investigate two types of distributions: (i) uniform distribution, (4(0, 1), and (ii) logit-normal (lognorm) distribution [10], where a sample is first drawn from a normal distribution N (μ, a) and then mapped to (0, 1) using the logistic function. Given a sampled pair, we assign the larger value to t and the smaller to r. We set a certain portion of random samples with r =
Conditioning on (r,t). We positional embedding [48] to encode the time variables, which then combined and provided as the conditioning of the neural network. We note that although the field is parameterized by uz(2,, r, r), it is not necessary for the network directly condition on (r, t)- For example, we can let the network directly condition on (f, At), with At = t r. In this case, we have ua (+s r, A net(-,t,t r) where net is the network. The JVP computation is always w.E.t the function up r), We compare different forms of conditioning in experiments.
5 Experiments
Experiment Setting, We conduct our major experiments on ImageNet [6] generation at 256 resolution. We evaluate Fréchet Inception Distance (FID) [17] on 50K generated images. We examine the number of function evaluations (NFE) and study I-NFE generation by default. Following [34, 12, 52], we implement our models on the latent space of pre-trained VAE tokenizer [37]. For 256x256 images, the tokenizer produces a latent space of 32 x +2x32x4 x4, which is the input to the model. Our models are all trained from scratch. Implementation details are in Appendix A.
In our ablation study, we use the ViT-B/4 architecture (namely, "Base" size with a patch size of 4) [8] as developed in [34], trained for 80 epochs (400K iterations). As reference, DiT-B/4 in [34] has 68.4 FID, and SiT-B/4 [33] (in our reproduction) has 58.9 FID, both using 250-NFE sampling.
5.1 Ablation Study
We investigate the model properties in Tab. 1, analyzed next:
From Flow Matching to Mean Flows. Our method can be viewed as Flow Matching with a modified target (Alg. 1), and it reduces to standard Flow Matching when r always equals t. Tab. la compares the ratio of sampling r f I. A 0% ratio of r * (reducing to Flow Matching) fails produce reasonable results for I-NFE generation. A non-zero ratio of * t enables McanFlow to take effect, yielding meaningful results under 1-NFE generation. We observe that the model balances learning the instantancous velocity (r = f) ys. propagating into # t via the modified target. Here, the optimal FID achieved at ratio of 25%, and a ratio of 100% also yields a valid result.
JVP Computation. The JVP operation Eq. (8) serves as the core relation that connects all (r, t) coordinates. In Tab. 1b, we conduct destructive comparison which incorrect JVP computation is intentionally performed. It shows that meaningful results are achieved only when the J JVP computation correct. Notably, the JVP tangent along Ò_ u is d-dimensional, where d is the data dimension (here, 32x32x4), and the tangents along 0, u and d.u are dimenssiia.... Nevertheless, these two time variables determine the field u, and their roles are therefore critical even though they are only one-dimensional.
Conditioning on (r, As discussed in Sec. 4.3, we can represent ug(2, r, t) by various forms of explicit positional embedding, e.g., uo(,r,, 1) A net(;, 1, - r). Tab. 1c compares these variants. Tab. 1c shows that all variants of tr,t) embeddings studied yield meaningful 1-NFE results, demon-
7

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% of rifh
FID, I-NFE
jup langent
FID, I-NFE
pos. embed
FID, 1-NFE
0% (= FM)
328.91
(2, 0,
61.06
(1,r)
61.75
25%
61.06
(v, 0,
268.06
(t, tt)
61.06
50%
63.14
(v, 1,
329.22
rt,r, 63.98
100%
67.32
(v, 1,1)
137.96
1-r t-ro
63.13
(a) Ratio of sampling rtt. The 0% en- try reduces to the standard Flow Match- ing baseline.
(b) JVP computation. The correct jvp tangent is (v,0, for Jacobian (0.u, 0,u, deu).
(c) Positional embedding. The network is conditioned the embeddings ap- plied the specified variables.
r sampler
FID, I-NFE
P
FID, I-NFE
FID, 1-NFE
uniform(0, 1)
65.90
0.0
79.75
1.0 (w/o cfg)
61.06
lognorm(-0.2. -002, 1.0)
63.83
0.5
63.98
1.5
33.33
lognorm(- (-02, 1.2)
64.72
1.0
61.06
2.0
20.15
lognorm( -04, 1.0)
61.06
1.5
66.57
3.0
15.53
lognorm(-O.44 1.2)
61.79
69.19
20.75
(d) Time samplers. and r are sampled from the specific sampler.
(e) Loss metries. p-0 is squared L2
(1) CFG scale. Our method supports 1-NFE CFG sampling.
loss. 005. is Pseudo-Huber loss.
Table 1: Ablation study on 1-NFE ImageNett 256 x 256 generation. FID-50K is evaluated. Default configurations are marked in gray B/4 backbone, 80-epoch training from scratch.
- strating the effectiveness of MeanFlow as a framework. Embedding t r), that is, time and interval, achieves the best result, while directly embedding (r, performs almost as well. Notably, even embedding only the interval t - yields reasonable results.
Time Samplers. Prior work [10] has shown that the distribution used sample influences the generation quality. We study the distribution used to sample (r, t) in Tab. 1d. Note that (r, are first sampled independently, followed by petpproccssing step that enforces > by swapping and then caps the proportion of r o to a specified ratio. Tab. 1d reports that a logit-normal sampler performs the best, consistent with observations Flow Matching [10].
Loss Metrics. It has been reported [43] that the choice of loss metrics strongly impacts the perfor- mance of few-/one-step generation. We study this aspect in Tab. le. Our loss metric is implemented via adaptive loss weighting [14] with power (Sec. 4.3). Tab. le shows that p = achieves the best result, whereas p 0.5 (similar to Pseudo-Huber loss [43]) also performs competitively. The standard squared L2 loss (here, p 0) underperforms compared to other settings, but still produces meaningful results, consistent with observations in [43].
Guidance Scale. Tab. 1f reports the results with CFG. Consistent with observations in multi-step generation [34], CFG substantially improves generation quality in our 1-NFE setting too. We emphasize that our CFG formulation (Sec. 4.2) naturally supports I-NFE sampling.
Scalability. Fig. 4 presents the 1-NFE FID results of MeanFlow across larger model sizes and different training durations. Consistent with the behavior of Transformer-based diffusion/floww models (DiT [34] and SiT [33]), MeanFlow models exhibit promising scalability for 1-NFE generation.
5.2 Comparisons with Prior Work
ImageNet 256x256 Comparisons. In Fig. 1 we compare with previous one-step diffusion/flow models, which are also summarized in Tab. (left). Overall, MeanFlow largely outperforms previous methods in its class: it achieves 3.43 FID, which is an over 50% relative improvement IMM's one-step result of 7.77 [52]; we compare only 1-NFE (not just one-step) generation, MeanFlow has nearly 70% relative improvement vs. the previous of-te-of-thearr (10.60, Shortcut [12]). Our method largely closes the gap between one-step and many-step diffusion/flow models.
In 2-NFE generation, our method achieves an FID of 2.20 (Tab. 2, bottom left). This result is on par with the leading baselines many-step diffusion/flow models, namely, DìT [34] (FID 2.27) and SiT [33] (FID 15), both having NFE of 250x2 (Tab. 2, right), under the same XL/2 backbone. Our

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XU2
Figure 4: Scalability of MeanFlow models on ImageNet 2 I-NFE generation FID is reported. All models are trained from scratch. CFG is applied while maintaining the I-NFE sampling behavior. Our method exhibits promis- ing scalability with respect model size.
1 FD
20
220
200
200
240
method
params
NFE
FID
method
paramss
NFE
FID
GANss
I-NFE diffusion/flow from scratch
BigGAN [4]
112M
6.95
iCT-XI-/2 [431*
675M
34.24
GigaGAN (21)
569M
1
3.45
Shortcut-XLa XL/2 [12]
675M
10.60
StyleGAN (40)
166M
2.30
MeanFlow-BA,
131M
1
6.17
autoregressivennsskinng
McanFlow-M/2
308M
5.01
AR w/ VQGAN (9)
227M
1024
26.52
MaskGIT [5]
227M
6.18
MeanFlow-L/2
459M
3.84
VAR-d30 [47)
2B
10x2
1.92
MeanFlow -LL2
676M
3,43
MAR-H (27)
943M
256x2
1.55
2-NFE diffusion/flow from scratch
diffusion/flow
ADM
554M
250x2
10.94
iCT-XL//2 [43]*
675M
2
20.30
LDM-4G [37]
400M
250x2
3.60
IMM-X1/2 [52]
675M
1x2
7.77
SimDiff (20)
2B
512x2
2.77
McanFlow 2o/2//
676M
2
2.93
DIT-XL/2 [34]
675M
250x2
2.27
SiT-XL/2 (33)
675M
250x2
2.06
MeanFlow-XI/24
676M
2
2.20
SiT-XL/2+REPA
675M
250x2
1.42
Table Class-conditional generation ImageNet-256 256. All entries are reported with CFG,
when applicable. Left: 1-NFE and 2-NFE diffusion/flow models trained from scratch. Right: Other families of generative models as reference. In both tables, "x2" indicates that CFG incurs an NFE of 2 per sampling step. Our MeanFlow models are all trained for 240 epochs, except that "MeenFlow-XI..
trained for more epochs and with configurations selected for longer training, specified in appendix. 1: iCT [43] results reported by [52].
results suggest that few few-step diffusion/flow models can rival their many-step predecessors, Orthogonal improvements, such REPA [51], applicable, which left for future work.
Notably, our method is self-contained and trained entirely from scratch. achieves the strong results without using any pre-training, distillation, or the curriculum learning adopted in [43, 14, 31].
CIFAR-10 Comparisons. We report unconditional generation results on CIFAR-10 [25] (32x32) in Tab. 3. FID-50K is reported with 1-NFE sampling. All entries are with the same U-net [38] developed from [44] (~55M), applied directly the pixel space. All other competitors are with the EDM- style pre-conditioner [22], and ours has no preconditioner. Implementation details are in the appendix. On this dataset, our method is competitive with prior approaches.
method
precond
NFE
FID
iCT [43]
EDM
1 2.83
ECT [14]
EDM
I
3.60
sCT [31]
EDM
2.97
IMM [52]
EDM
3.20
MeanFlow
none
1 2.92
Table 3: Unconditional CIFAR-10.
Conclusion
We have presented MeanFlow, principled and effective framework for one-step generation. Broadly speaking, the scenario considered this work is related multi-seal simulation problems in physics that may involve a range of scales, lengths. and resolutions, in space and time. Numerical simulations are inherently limited by the ability computers to resolve the range of scales. Our formulation involves describing the underlying quantity at coarser levels of granularity, common theme that underlies many important applications in physics. We hope that our work will bridge research across generative modeling, simulation, and dynamical systems in related fields.

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Acknowledgments
We greatly thank Google TPU Research Cloud (TRC) for granting us access to TPUs. Zhengyang Geng is partially supported by funding from the Bosch Center for AI. Zico Kolter gratefully acknowl- edges Bosch's funding for the lab. Mingyang Deng and Xingjian Bai are partially supported by the MIT-IBM Watson AI Lab funding award. We thank Runqian Wang, Qiao Sun, Zhicheng Jiang, Hanhong Zhao, Yiyang Lu, and Xianbang Wang for their help with the implementation of JAX and TPU.
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Justification:
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Appendix
A Implementation
Table 4: Configurations on ImageNet 256x256. B/4 is our ablation model.
configs
B/4
B/2
M/2
I.2
XI.22
Xl./+
params (M)
131
131
308
459
676
676
FLOPs (G)
5.6
23.1
80.9
119.0
119.0
depth
12
12
16
24
28
28
hidden dim
768
768
1024
1024
1152
1152
heads
12
12
16
16
16
16
patch size
4x4
2x2
2x2
2x2
2x2
2x2
epochs
80
240
240
240
240
1000
batch size
256
dropout
0.0
optimizer
Adam [24]
Ir schedule
constant
Ir
0.0001
Adam (B1, .2))
(0.9, 0.95)
weight decay ema decay
0.0
0.9999
ratio of myat
Tab. 1a
25%
(r, f) cond
Tab. lc
(f, r) lognorm(-O0 00.4 1.0)
(r, sampler
Tab. 1d
p for adaptive weight
Tab.
1.0
CFG effective seale w/
Tab.
2.0
2.5
2.5
CFG w. Eq. (21)
w
1.0
1.0
0.2
0.2
1.0
CFG Eq. (21)
w/w
CFG cls-cond drop [18]
0.1
[18]
CFG triggered if in:
0.00 1.0)
[0.0,
[0.0, 1.0)
10.0,
[0.0, 0.75)
[0.3, 0.8
ImageNet 256 We use a standard VAE tokenizer to extract the latent representations." The latent size is 32x32x4, which is the input to the model. The backbone architectures follow DiT [34], which are based on VIT [8] with adaLN-Zero [34] for conditioning. To condition on two time variables (e.g,, (), we apply positional embedding on each time variable, followed by 2-layer MIP, and summed. We keep the DiT architecture blocks untouched, while architectural improvements are orthogonal and possible. The configuration specifics are Tab. 4.
We experiment with class-unconditional generation on 11FAR-10. Our implementation follows standard Flow Matching practice [29]. The input to the model is 32x32x3 the pixel space. The network is a U-net [38] developed from [44] (~~55M), which is commonly used by other baselines we compare. We apply positional embedding on the two time variables (here, (t, - and concatenate them for conditioning. We do not use any EDM preconditioner [22].
CIFAR-10.
We use Adam with learning rate 0.0006, batch size 1024, (31, (32) = (0.9, 0.999), dropout 0.2, weight decay 0, and EMA decay of 0.99995. The model is trained for 800K iterations (with 10K warm-up [15]). The (r, t) sampler lognorm(- (200, 2.0). The ratio of sampling for adaptive weighting is 0.75. Our data augmentation setup follows [22], with vertical flipping and rotation disabled.
is 75%. The power p
httt:::/hhugiuggtccc.o//oaau fone-ftmmse flax
21

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Table 5: Improved CFG for MeanFlow. is defined in Eq. (20), whose goal is to enable both caas-conniiiioal u² c) and class-unconditional (.) appear in the target. In this table, we fix the effective guidance scale w', given by w' = w/ k), as 2.0. Accordingly, for different k wi by = (1-x)-w' If 0, it falls back to the CFG case in Eq. (19) (see also Tab. 1f). Similar to standard CFG's practice of randomly dropping class conditions [18], we observe that mixing condiioondii and laaas-unconditioall u in our target improves generation quality.
K
FID, 1-NFE
0.0
20.15
0.5
19.15
0.8
19.10
values,
0.9
18.63
0.95
19.17
B
Additional Technical Details
B.1 Improved CFG for MeanFlow
In Sec. 4.2, have discussed how naturally extend method supporting CFG. The only change needed is to revise the target by Eq. (19). We notice that only the class-unconditional u presented in Eq. (19). the original CFG [18]. it standard practice to mix class-conditional and dlassunncndditionll predictions, approached by random dropping. We observe that a similar idea can be applied to our regression target well.
Formally, we introduce a mixing scale and rewrite Eq. (16) as:
oe 4 cc) w v(2, c) + "
(1-W-R) (f (z,, (z,, 1,1).
(20)
Here, the role of is to mix with u c) the right hand side. We can show that Eq. (20) satisfies the original CFG formulation (Eq. (13)) with the effective guidance scale of w' =
w 1-x* leveraging the
relation (Z,, t) (2e, t) and 2 (2,, t) (2,, t,t) that have used for deriving Eq. (16).
With this, Eq. (19) is rewritten as
ū - w (c-x) K
up de ( Ku (Zz, 4,11, (1-W-x)
(21)
sample Us
els-cond outpur
els-uncond outpur
The loss function the same as defined in Eq. (17).
The influence of introducing is explored in Tab. 5, where we fix the effective guidance scale w' as 2.0 and vary x. It shows that mixing by can further improve generation quality. We note that the ablation in Tab. If in the main paper did not involve this improvement, i.e., was set as O.
B.2 Loss Metrics
The squared L2 loss is given by L || where A uy "ig denotes the regression error. Generally, one can adopt the powered L2 loss L, = | where y a user-specified hyper- parameter. Minimizing this loss is equivalent to minimizing adaptively weighted squared L2 loss
(see [14]): de
This can be viewed as weighting the squared L,2 loss d _ = > (3) d||ا|
de
(| 1 by loss-adaptive weight À x | 2 2(7-1) practice, we follow [14] and weight by:
w 1/(||| + c)),
(22)
where p - and is a small constant to avoid division by zero. p to the Pseudo- Sseudo-Huber loss in [43]. The adaptively weighted loss is eg(w). L, where sg denotes the stop-grae aoo-ggradeent operator.
= 0.5, this similar
B.3 On the Sufficiency of the MeanFlow Identity
In the main paper, starting from the definitional Eq. (4), we have derived the MeanFlow Identity Eq. (6). This indicates that "Eq. (4) = Eq. (6)", that is, Eq. (6) necessary condition for Eq. (4). Next, we show that it is also sufficient condition, that is, "Eq. (6) = Eq. (4)".
In general, equality of derivatives does imply equality of integrals: they may differ by a constant. In our case, we show that the constant is canceled out. Consider a "displacement field" written as:
S(Z, r, (t r) r,t).
(23)
22

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Figure 5: 1-NFE Generation Results. We show curated examples of fass-conditional generation ImageNet 256 using our 1-NFE model (MeanFlow-X1//2, 3.43 FID).
we treat an arbitrary function, then in general, equality of derivatives can only lead to equality of integrals, up to some constants:
d dt S(2,r,t)
t(2,,t) u((,))
S(z,r,t) + o(2r:)) + C2.
(24)
However, the definition of S gives S| = 0, and we also have f udr = 0 when = which gives C1 C2. This indicates "Eq. (6) Eq. (4)".
We note that this sufficiency is consequence of modeling the average velocity u, rather than directly the displacement S. Enforcing the equality of derivatives on the displacement, for example, _ = v, does not automatically yield S J vdr. If we were to parameterize S directly, extra boundary condition I 0 is needed. Our formulation can automatically satisfy this condition.
B.4 Analysis on Jacobian- Vector Product (JVP) Computation
While JVP computation can be seen as a concern some methods, is very lightweight in ours. In our case, as the computed product (between the Jacobian matrix and the tangent vector) is subject stop-gradient (Eq. (10)), it is treated w.r.t. O. Consequently, it does not add any overhead to the o-backpropagation.
constant when conducting SGD using backpropagation
As such, the extra computation incurred by computing this product is the "backward" pass (formally, forward-mode auto-diff) performed by the jvp operation. This "backward" is analogous to the standard backward pass used for optimizing neural networks (w.r.1. 0)-in fact, in some deep learning libraries, such as JAX [3), they use similar interfaces to compute the standard backpropagation (w.rt. 8). In our case, backpropagate to the input variables, not to the parameters O. This overhead is small.
even less costly than a standard backpropagation, as it only needs to
We benchmark the overhead caused by JVP in ablation model, B/4. We implement the model in JAX and benchmark on v4-8 TPUs. We compare MeanFlow with its Flow Matching counterpart, where the only overhead is the backward pass incurred by JVP; notice that the forward pass of JVP the standard forward propagation, which Flow Matching (or any typical objective function) also needs to perform. our benchmarking, the training cost is 0.045 sec/iter using Flow Matching, and 0.052 sec/iter iter using MeanFlow, which is merely 16% wall-clock overhead.
C Qualitative Results
Fig. 5 shows curated generation examples on ImageNet 256x256 using our 1-NFE model.
23


Reviews:
[{"review_id": "FojlNGQ6zt", "reviewer": "Reviewer_AbQD", "summary": "This paper introduces \"MeanFlow,\" a novel framework for one-step generative modeling. \nThe core idea is to move away from modeling the instantaneous velocity, as in Flow Matching, and instead model the average velocity over a time interval. \nThe authors derive a \"MeanFlow Identity\", a differential equation that connects the average velocity to the instantaneous velocity. \nThis identity is then used to construct a training objective that allows a neural network to learn the average velocity field directly, using only the ground-truth instantaneous velocity as a signal. \nThe key benefit is that a single evaluation of the learned average velocity network can approximate the entire flow from noise to data, enabling high-quality 1-NFE (Number of Function Evaluations) generation. \nThe authors demonstrate state-of-the-art results for one-step generation on ImageNet 256x256, significantly outperforming previous methods.", "questions": "N/A", "limitations": "N/A", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 4, "originality": 4, "strengths_and_weaknesses": "[Strength]\n1. The primary strength of this work is its outstanding performance. An FID of 3.43 with a single function evaluation on ImageNet 256x256 is a remarkable achievement among one-step generative models. The 2-NFE result of 2.20 is also highly competitive, rivaling many-step diffusion models. These results alone make the paper a significant contribution.\n2. The idea of modeling average velocity is intuitive and theoretically clean. The derivation of the \"MeanFlow Identity\" (Eq. 6) from the fundamental definition of an integral appears sound and provides a solid theoretical underpinning for the method.\n3. The proposed model is trained from scratch and does not require pre-trained teacher models, distillation, or the complex \"discretization curriculum\" that some consistency models rely on. This simplifies the training pipeline considerably.\n\n[Weakness]\nNo major flaws are presented in this paper. I only have several suggestions here:\n1. Limitaiton section. The paper presents the MeanFlow model as a flawless solution. However, no method is perfect. The authors explicitly answer NO to the question of discussing limitations in their checklist. This is a significant omission. A thorough scientific paper should discuss potential failure modes.\n2. MeanFlow relies on the instantaneous velocity of flow matching, i.e., v_t = \\epsilon - x. However, is it the best choice?", "was_revised": true, "final_justification": "No concerns remained.", "rebuttal": "Dear Reviewer,\n\nWe sincerely thank Reviewer AbQD for the thoughtful and encouraging review and for highlighting the strengths of our contribution in performance, clarity, and simplicity. Below, we address your valuable comments and suggestions regarding limitations and our choice of the velocity term.\n\n**Q1**: The Limitations section is missing.\n\nWe fully agree with this point and will add a Limitations section. Specifically, we plan to acknowledge the following aspects:\n\n1) Performance Gap to Multi-Step Models. Despite achieving state-of-the-art performance among 1-NFE models, there remains a notable quality gap compared to multi-step diffusion/flow models. For example, MeanFlow achieves an FID of 3.43 at 1-NFE, whereas multi-step models such as SiT and MAR-H achieve 2.06 and 1.55, respectively. Bridging this gap remains an important direction for future research.\n2) Scope of experiments. Our current experiments focus on image datasets. Robotics and other modalities present compelling future directions. Especially, motion synthesis for robotics requires real-time and accurate control, where the one-step nature of MeanFlow could prove advantageous.\n3) Dependence on VAE Latent Space. MeanFlow's high-resolution image generation experiments were conducted in the latent space of a Variational Auto-Encoder (VAE). While beneficial for scalability, it inherits limitations from the VAE itself. Investigating ways to extend MeanFlow beyond VAE latent spaces could help unlock further improvements.\n\n\n**Q2**: Is $\\epsilon - x$ the best choice?\n\nThank you for the insightful question! Our core identity (Eq. 6) is indeed agnostic to the specific choice of $v_t$. We adopt the OT-FM formulation primarily due to its practical advantages in stability, scalability, and empirical success. Notably, popular open-weight models such as Stable Diffusion 3 and the recent Flux models have also adopted OT-FM formulations. That said, we agree that exploring alternative velocity choices for potentially faster convergence or improved performance remains an exciting and worthwhile direction for future work.\n\nThank you again for reviewing our work and providing invaluable feedback!\n\nSincerely, \\\nThe Authors"}, {"review_id": "febnYjVIBd", "reviewer": "Reviewer_8EtS", "summary": "This paper proposes MeanFlow, a one-step generative modeling framework built on the concept of an average velocity field in diffusion/flow-based generative models. Instead of modeling the instantaneous velocity as in standard Flow Matching, the authors define an average velocity. To do this, the authors derive a fundamental “MeanFlow Identity” which provides a principled training objective for a neural network to approximate the true average velocity field without needing to explicitly compute any time integrals during training.", "questions": "- Could the authors provide more detailed comparisons regarding the training complexity and computational cost relative to existing one-step baseline methods? While FLOPs and timing details are briefly presented in Table 4 and Appendix B.4, a direct and explicit comparison to baseline methods would further clarify practical feasibility and advantages.\n\n- Could the authors report performance using other generative metrics such as Inception Score, sFID, Precision, and Recall? These metrics would offer deeper insights into aspects like diversity, mode coverage, and overall perceptual quality of generated samples. Additionally, evaluating the method with FID-DINO may provide a more accurate assessment of semantic fidelity and visual realism.\n\n- As mentioned in the weaknesses, could the authors clarify whether substituting Eq. 11 into Eq. 10 maintains theoretical equivalence?", "limitations": "There is no explicit limitation or potential societal impact mentioned in the paper.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\n- Novel Theoretical Contribution: The introduction of average velocity fields and the MeanFlow Identity is a clear conceptual innovation. It provides a new theoretical lens on one-step generation. Importantly, this framework generalizes and unifies prior ideas. For example, it inherently enforces trajectory consistency (one big step vs. multiple small steps) as a property of the ground-truth field, instead of imposing consistency as an external loss.\n\n- MeanFlow does not rely on distilling a pre-trained multi-step model. This makes the approach conceptually cleaner and potentially easier to reproduce. This principled foundation likely contributes to stable and robust training. The authors report that no special tricks beyond the derived loss were required.\n\n- The paper demonstrates impressive results for one-step generative models. These results are obtained from scratch (no pre-trained models) and using a consistent architecture for fair comparison. The experiments cover both high-resolution image generation (ImageNet 256) and low-resolution CIFAR-10, showing that MeanFlow scales across datasets.\n\n- Impact and Relevance: This work is timely and impactful for the generative modeling community. Closing the gap to multi-step diffusion with a one-step model is a longstanding challenge – it has direct implications for faster generative model deployment and could spur new research into flow-based modeling.\n\n**Weaknesses**\n- Theoretical rigor: In standard flow-matching frameworks, the gradients of the flow matching loss and conditional matching loss have been rigorously shown to be equivalent. However, it remains unclear whether such equivalence also holds for the formulations proposed in this paper (Eqs. 9, 10, and 11). Although empirical results indicate strong performance, the theoretical correctness and justification of these equations are not fully established. Clarifying this point would significantly strengthen the theoretical grounding of the method.\n\n- Limited evaluate metric: The performance is currently evaluated exclusively using FID scores, which limits the understanding of the method’s quality along other important dimensions such as sample diversity, mode coverage.", "was_revised": true, "final_justification": "The authors have addressed most of my concerns in the rebuttal, and the paper appears to be highly significant in the field of generative models.\nTherefore, I am maintaining my positive score.", "rebuttal": "Dear Reviewer,\n\nThank you for your valuable review to our work! We are grateful for the constructive and detailed review and affirmation of our contributions, simplicity, and impressive results. Your suggestions on reporting broader metrics, clarifying gradient equivalence, and cost comparisons are very helpful, which we will discuss below.\n\n**Q1**: Training complexity and computational cost relative to one-step baselines.\n\nWe fully agree that clarifying the computational cost is important. We will include further cost analysis and comparison against strong one-step baselines. Specifically, MeanFlow introduces a roughly 20% increase in training time compared to Flow Matching (FM), close to the extra computation of 16% by Shortcut models. Notably, it does not increase peak memory relative to Flow Matching. We will provide detailed comparisons in our revised manuscript. Thank you for your comments!\n\n**Q2**: Could the authors report performance using other generative metrics such as Inception Score, sFID, Precision, and Recall?\n\nThank you for this excellent suggestion. We agree that evaluating additional metrics will provide deeper insight into the performance of our method, particularly in terms of diversity, mode coverage, and perceptual quality. We will incorporate these metrics into the revised paper. Our results for MeanFlow alongside the DiT and SiT baselines are summarized below for ease of comparison.\n\n| Model             | FID$\\downarrow$  | IS$\\uparrow$ | sFID$\\downarrow$ | Precision$\\uparrow$ | Recall$\\uparrow$ |\n|-------------------|------|--------|------|-----------|--------|\n| MeanFlow-XL/2 (NFE=1)   | 3.42 | 247.50 | 6.43 | 0.78      | 0.55   |\n| MeanFlow-XL/2 (NFE=2)   | 2.20 | 269.37 | 4.78 | 0.79      | 0.60   |\n| DiT-XL/2 (NFE=250x2) | 2.27 | 278.24 | 4.60 | 0.83    | 0.57  |\n| SiT-XL/2 (NFE=250x2) | 2.06 | 277.50 | 4.49 | 0.83    | 0.59  |\n\nThese results highlight that MeanFlow achieves competitive performance across reported metrics, approaching multi-step models in quality and diversity. We hope this could serve as a better baseline for future studies.\n\n**Q3**: Could the authors discuss the equivalence of gradients for substitutions in Eqs. (9–11)?\n\nThank you very much for this constructive comment on improving the theoretical rigor of our work. Indeed, this substitution preserves the minimizers of our objective and yields unbiased gradients. The proof shares similar techniques to those in FM [1]. Given that the gradient using the conditional velocity field is unbiased for optimization, it therefore follows the theoretical soundness of Flow Matching. We will provide this justification in the revised version. We appreciate the valuable suggestions for improvement.\n\n[1] Lipman, Yaron, et al. \"Flow matching guide and code.\" arXiv preprint arXiv:2412.06264 (2024).\n\nThank you again for your invaluable comments and feedback for improving our work!\n\nSincerely, \\\nThe Authors"}, {"review_id": "xzPNPBNi5o", "reviewer": "Reviewer_ENCq", "summary": "This paper proposes a novel training objective to improve one-step generation in the flow matching framework. The core idea is to define the average velocity u as the integral of marginal velocity v over a segment (r, t), and to express u in terms of v and du/dt, which leads to a new loss formulation. The method enables one-step generation with a single, unified model trained from scratch—without requiring curriculum learning or distillation. The approach achieves state-of-the-art FID scores on ImageNet 256×256.", "questions": "## Questions\n1. Could the authors elaborate on the training stability of the model and compare it with the consistency models? For example, does it consistently converge under different random initializations? Is it particularly sensitive to certain hyperparameters, such as the batch size?\n2. Given that the proposed training objective involves additional terms, how much slower is the training process compared to standard flow matching when aiming for comparable FID?\n3. In Figure 2 and Equation (6), average velocity is described as the marginal velocity minus its derivative. However, in practice, v is defined as e - x (i.e., conditional flow matching velocity). Could the authors provide more theoretical or intuitive justification for why this formulation remains stable during training?\n4. The use of Jacobian-vector products (JVPs) seems potentially memory-intensive. I wonder whether this could pose a scalability bottleneck, especially as the data or model dimensionality increases. Could the authors discuss how this affects scalability in practice, and whether any memory optimizations or approximations were considered?\n5. In the ablation study, the reported FID scores appear significantly higher than the main results. Why?", "limitations": "The limitations of the work are reflected in the identified weaknesses and the questions posed.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "## Strengths\n1. The proposed method is simple yet effective.\n2. The performance is strong and competitive.\n3. The method is scalable in terms of both model size and image resolution.\n4.  The paper is well-written and easy to follow.\n5.  Compared to distillation-based approaches, the method avoids many heuristic components \n6.  It naturally supports classifier-free guidance (CFG), enhancing its applicability in conditional generation.\n\n## Minor Weaknesses\n- The method cannot be used with classifier-free guidance (CFG) sampling. (If this is incorrect, please clarify.)", "was_revised": true, "final_justification": "Final Justification:\nAfter carefully considering the rebuttal and discussions, I found that my main concerns were fully addressed:\n- The authors provided convincing clarifications regarding training stability and the memory issue, which resolved my earlier doubts.\n- The theoretical explanation for substituting conditional velocity for marginal velocity was clear and persuasive.\n- No significant unresolved issues remain from my original review.\n\nGiven that all major points of concern have been satisfactorily resolved, I have decided to maintain my original score.", "rebuttal": "Dear Reviewer,\n\nThank you very much for your clear and encouraging review, and for highlighting the simplicity and clarity of our work. The questions on training stability, speed, theoretical justification, and memory cost will convincingly help to strengthen our paper. In the following, we respond to each of your comments in detail.\n\n**Q1**: Could the authors elaborate on the training stability of the model and compare it with the consistency models? Does it consistently converge under different random initializations? Is it particularly sensitive to certain hyperparameters, such as the batch size?\n\nWe appreciate your interest in the stability of our training procedure. In our extensive experiments, MeanFlow consistently converged across different random initializations and seeds. Additionally, MeanFlow exhibited low sensitivity to hyperparameters, including batch size. Although our experiments primarily follow the DiT and SiT baselines with a batch size of 256, we observed stable training dynamics with larger batch sizes such as 512 and 1024. This further highlights MeanFlow's stability.\n\n**Q2**: How much slower is the training process compared to standard flow matching?\n\nThank you for your question regarding the computational efficiency. MeanFlow introduces a roughly 20% increase in training time compared to Flow Matching (FM). This additional computational overhead remains comparable with that observed in other state-of-the-art methods; for example, Shortcut Models incur approximately 16% additional computation compared to standard diffusion models. \nWe appreciate your thoughtful feedback to better benchmark our methods.\n\n**Q3**: Could the authors provide more theoretical or intuitive justification for why this formulation using conditional velocity remains stable during training?\n\nWe are grateful for your suggestion to clarify this theoretical aspect. Indeed, substituting conditional velocity for marginal velocity is theoretically sound under the standard assumptions of Flow Matching. Specifically, this substitution preserves the minimizers of our objective and yields unbiased gradients. The proof shares similar techniques to those in FM [1]. Given that the gradient using the conditional velocity field is unbiased for optimization, it therefore follows the theoretical soundness of Flow Matching. We will include a theoretical note in the revised manuscript to clarify and emphasize this justification. We appreciate this valuable recommendation.\n\n[1] Lipman, Yaron, et al. \"Flow matching guide and code.\" arXiv preprint arXiv:2412.06264 (2024).\n\n**Q4**: JVP memory and scalability. The use of Jacobian-vector products (JVPs) seems potentially memory-intensive. Could the authors discuss how this affects scalability in practice and whether any memory optimizations were considered?\n\nThank you for raising the important point of scalability and memory efficiency. MeanFlow does not backpropagate through the JVP path, which means it theoretically avoids additional memory overhead for storing activations.\nPractically, through computational graph compilation techniques in either JAX or PyTorch, we have successfully matched the memory usage of standard FM. \nTo support reproducibility and community engagement, we will release our implementations in both frameworks.\n\n**Q5**: Why are the ablation numbers higher than the main results?\n\nThank you for bringing attention to this clarification. \nOur ablation studies used a smaller and faster backbone (Diffusion Transformer with patch size 4) for development. For comparison, flow-matching and diffusion models of comparable size and compute yield similar FIDs (e.g., DiT-B/4 at 68.4 FID, SiT-B/4 at 58.9 FID). We will explicitly clarify this in the text and cross-reference the baselines of comparable scales to avoid confusion.\n\n**Q6**: Does MeanFlow support CFG sampling?\n\nWe appreciate your insightful question on CFG sampling. Currently, MeanFlow incorporates guidance during training but does not explicitly utilize CFG sampling at test time. Exploring CFG-based inference is an exciting direction for future work, which we will clarify in the updated paper.\n\nThank you once again for your invaluable feedback and suggestions to improve our work!\n\nSincerely, \\\nThe Authors"}, {"review_id": "vF03EJXxti", "reviewer": "Reviewer_LhKn", "summary": "This paper presents MeanFlow, a novel one-step generative modeling framework. It introduces the concept of average velocity to characterize flow fields, contrasting with the instantaneous velocity modeled by traditional Flow Matching methods. By deriving a well-defined relationship between average and instantaneous velocities, the authors establish a theoretical foundation for training neural networks to model the average velocity field. MeanFlow achieves impressive empirical results outperforming previous state-of-the-art one-step diffusion/flow models and narrowing the gap between one-step and multi-step models.", "questions": "1. The paper mentions that MeanFlow incurs additional training time due to the computation of the JVP, even with the use of stop-gradient. Could the authors provide a detailed comparison of the training time between MeanFlow and other methods? \n2. The paper primarily focuses on the quality of generated samples using metrics like FID but does not provide an in-depth analysis of the diversity of the generated samples. Could the authors report additional metrics that assess the diversity of the generated samples, such as the Inception Score (IS)? This would provide a more comprehensive evaluation of MeanFlow's performance in terms of both quality and diversity, giving a clearer picture of its capabilities and limitations.", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### Strengths:\n1. The introduction of the average velocity concept and the derivation of the MeanFlow Identity offer a new perspective and theoretical foundation for one-step generative modeling.\n2. The model can be trained from scratch without relying on pre-training, distillation, or curriculum learning, simplifying the training process.\n3. Experimental results indicate that MeanFlow models scale well with increasing model size and training duration, suggesting potential for further performance improvements.\n4. The paper provides thorough theoretical analysis and derivation, supporting the validity and effectiveness of the proposed method.\n### Weaknesses:\n1. Compared to Flow Matching, MeanFlow incurs additional training time due to the computation of the Jacobian-vector product, which might be a limitation in resource-constrained settings.\n2. The paper focuses on generation quality but lacks in-depth analysis and discussion on the diversity of the generated samples.", "was_revised": true, "final_justification": "The authors have solved my problem, and I recommend that this paper be accepted.", "rebuttal": "Dear Reviewer,\n\nWe are grateful to Reviewer LhKn for the thorough and constructive review and for concrete suggestions on training time and more comprehensive metrics for diversity and quality. We reply to your comments and questions below.\n\n**Q1**: MeanFlow incurs additional training time due to the computation of the JVP Could the authors compare the training time between MeanFlow and other methods?\n\nThank you for sharing your comments on the training cost analysis! We fully agree that clarifying the computational cost is important. Specifically, MeanFlow introduces roughly 20% increase in training time compared to Flow Matching (FM), close to the extra computation of 16% by Shortcut Models. Notably, it does not increase peak memory relative to Flow Matching. We will provide further comparisons in our revised paper and release our code. Thank you for your suggestions!\n\n**Q2**: Could the authors report additional metrics that assess the diversity of the generated samples, such as the Inception Score (IS)?\n\nThank you for this excellent suggestion! We agree that evaluating additional metrics will provide deeper insights into our method’s performance, regarding both quality and diversity. We summarized our evaluation results for MeanFlow alongside the DiT and SiT baselines below.\n\n| Model             | FID$\\downarrow$  | IS$\\uparrow$ | sFID$\\downarrow$ | Precision$\\uparrow$ | Recall$\\uparrow$ |\n|-------------------|------|--------|------|-----------|--------|\n| MeanFlow-XL/2 (NFE=1)   | 3.42 | 247.50 | 6.43 | 0.78      | 0.55   |\n| MeanFlow-XL/2 (NFE=2)   | 2.20 | 269.37 | 4.78 | 0.79      | 0.60   |\n| DiT-XL/2 (NFE=250x2) | 2.27 | 278.24 | 4.60 | 0.83    | 0.57  |\n| SiT-XL/2 (NFE=250x2) | 2.06 | 277.50 | 4.49 | 0.83    | 0.59  |\n\nThese results highlight that MeanFlow achieves competitive performance, approaching multi-step models in both quality and diversity. We hope this could serve as a better baseline for future studies.\n\nThank you once again for your invaluable feedback and suggestions to improve our work!\n\nSincerely, \\\nThe Authors"}, {"review_id": "kBiToDqkkF", "reviewer": "Reviewer_7gPv", "summary": "The paper proposes to directly parameterize the network to output the mean velocity of a displacement map, and uses fundamental theorem of calculus to constrain the derivative of the mean velocity to match that of the flow-matching conditional velocity. The method leverages JVP to compute the time derivative w.r.t. the model, and uses the conditional velocity $v_t$ as supervision signal for JVP calculation.", "questions": "Besides the weaknesses above, the authors need to thoroughly discuss this work's relation to previous works because I feel the paper does not pay proper attention to prior methods and the originality is a bit oversold.", "limitations": "There are no direct negative societal impacts, although such technologies wider ethical implications can be discussed more.", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 2, "strengths_and_weaknesses": "The paper are notable in several aspects ranging from formulation to empirical results, listed below\n1. MeanFlow proposes a simple single loss, which reduces to the vanilla flow matching objective when $t=r$. The conception simplicity is easy to grasp and the L2 loss/stop-gradient makes it stable for optimization.\n2. It achieves strong 1-step FID training from scratch. \n\nHowever, I have major concerns in its exposition in relation to many related works.\n1. To my understanding, this mean field formulation is equivalent to continuous-time consistency models [1] maybe generalizing it to the consistency trajectory case. Specifically, the authors define the flow map as the 2-time indexed Euler sampler $f_\\theta(x_t,t,r) = x_t + (r-t)u_\\theta(x_t,t,r)$ and set it equal to the ODE integral of groundtruth flow $x_t + \\int_t^r u_\\tau d\\tau$. Then, time-derivative is taken on both sides so that $\\frac{d}{dt}f_\\theta(x_t,t,r) = 0$. This is exactly the same as continuous-time consistency model objective where you want to penalize $||\\frac{d}{dt}f_\\theta(x_t,t,r) ||^2$. This formulation is similarly discussed in [2], which explicitly uses a 2-time indexed version. Furthermore, the parameterization used in this paper is Euler sampler, which is a special case of flow maps in general. I feel that such connections with the previous works are inadequately discussed, and by avoiding these discussions the novelty of this work is somewhat intentionally inflated.\n\n2. The time derivative of the flow map, $u_t - u_\\theta(x_t,t,r) + (r-t)[\\frac{\\partial}{\\partial x_t} u_\\theta(x_t,t,r)^T u_t + \\frac{\\partial}{\\partial t} u_\\theta(x_t,t,r) ]$, contains the marginal velocity $u_t$. However, the authors simply substitutes $u_t$ with the conditional velocity $v_t$ without justification. Importantly, this is also equivalent to the proposal of continuous-time consistency models trained from scratch which has already proposed the stop-gradient trick to stablize training. Directly substituting $u_t$ with $v_t$ also breaks the theoretically soundness of the approach, as discussed in [2] in detail, because the two objectives (with $u_t$ and with $v_t$) are NOT equivalent in expectation over the training distribution. The authors are encouraged to justify this choice. I suppose that this discrepancy is also why stop-gradient needs to be employed, because enabling gradient through the JVP path results in minimizing an improper objective.\n\n3. Stop-gradient also makes the objective less theoretically sound because it's difficult to prove its attractivenss towards the groundtruth marginal flow map [2].\n\n4. This paper also seem to intentionally avoid comparing with few-step performance given that it naturally employs a jumping capability from any time t to any time r. Does the performance improve with more steps? If so, why not compare with other types of methods with one- and few-step capabilities such as diffusion pre-training + distillation [3-5] and autoregressive models (VAR [6]). The authors seem to avoid comparing with the best results achieved by other models which can be misleading for the community.\n\n5. Besides stop-gradient, the paper also proposes other tricks such as adaptive weighting, which amounts to minimizing L1 or Pseudo-huber loss. This makes the optimum of the new objective even less clear for analysis. If only L2 is used, what is the performance of the XL model? Is there a noticeable gap? How much does the adaptive weighting affect the best performance (this gap is not clear from the ablation study)\n\n\n\n[1] Lu, Cheng, and Yang Song. \"Simplifying, stabilizing and scaling continuous-time consistency models.\" arXiv preprint arXiv:2410.11081 (2024).\n\n[2] Boffi, Nicholas M., Michael S. Albergo, and Eric Vanden-Eijnden. \"Flow Map Matching.\" arXiv preprint arXiv:2406.07507 (2024).\n\n[3] Yin, Tianwei, et al. \"One-step diffusion with distribution matching distillation.\" Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. 2024.\n\n[4] Zhou, Mingyuan, et al. \"Score identity distillation: Exponentially fast distillation of pretrained diffusion models for one-step generation.\" Forty-first International Conference on Machine Learning. 2024.\n\n[5] Zhou, Mingyuan, et al. \"Adversarial Score identity Distillation: Rapidly Surpassing the Teacher in One Step.\" arXiv preprint arXiv:2410.14919 (2024).\n\n[6] Tian, Keyu, et al. \"Visual autoregressive modeling: Scalable image generation via next-scale prediction.\" Advances in neural information processing systems 37 (2024): 84839-84865.", "was_revised": true, "final_justification": "I will keep my score because the exposition does not sufficiently address many prior works with similar ideas", "rebuttal": "Dear Reviewer,\n\nWe sincerely thank Reviewer 7gPv for your detailed and constructive review and for sharing theoretical considerations. Your comments are invaluable to us in improving the clarity of our work. We would like to address your feedback in detail below.\n\n**Q1**: Relationship to continuous-time consistency models and Flow Map Matching. This mean field formulation is equivalent to continuous-time consistency models.\n\nWe genuinely appreciate your suggestion to discuss these theoretical connections in depth, including suggested works [1-6]. Indeed, continuous-time Consistency Models and Flow Map Matching represent valuable contributions to efficient generative modeling. \n\nContinuous-time consistency Models learn a single temporal line of $r=0$, whereas MeanFlows model the two-dimensional plane $(t, r)$ via the identity. \nAs a 2-time-indexed function, MeanFlow cannot be cast to \"equivalent to\" continuous‑time consistency models. \nUnlike existing consistency models (e.g., sCM [1]), MeanFlow is effectively trained from scratch without pretraining, distillation, or curriculum learning. Our formulation also enables us to incorporate CFG during training. \n\nRegarding flow map matching, it learns a 2-time-indexed function, corresponding to the displacement to the endpoint. \nAs a displacement function, flow map matching requires the derivative matching with the velocity field and backpropagation through the time derivatives of displacement.\n\nIn contrast to both, MeanFlow introduces a simple concept of average velocity. It starts from the definition of average velocity and derives the identity that the ground truth field should follow, based on the Fundamental Theorem of Calculus. \nThus, although MeanFlow shares some conceptual foundations with these models, it remains fundamentally distinct. We are grateful for your comments and hope that all these works are complementary to each other and contribute to the community.\n\n**Q2**: The authors are encouraged to justify the use of the conditional velocity field.\n\nThank you for prompting further clarification here! Using conditional velocity instead of marginal velocity is theoretically valid in Eq. (17) and follows analogous reasoning from Flow Matching (FM) [7]. Under standard FM assumptions, this substitution retains the minimizer and provides unbiased gradient estimates. We fully agree with the importance of explicitly clarifying this point, and we will include the theoretical justification in our revised manuscript.\n\n[7] Lipman, Yaron, et al. \"Flow matching guide and code.\" arXiv preprint arXiv:2412.06264 (2024).\n\n**Q3**: Stop-gradient also makes the objective less theoretically sound because it's difficult to prove its attractiveness.\n\nWe appreciate the question on the `stop-gradient` operator. MeanFlow is optimizing for the identity that the ground truth field should satisfy. As discussed in the paper, despite these optimization practices, it does not alter the minimizer. \nIt is easy to show that it would meet the MeanFlow Identity (Eq. (6)), and thus satisfy the original definition. \nUnlike in representation learning contexts, where the minimizer properties can become ambiguous, MeanFlow’s minimizer remains clearly defined.\nNonetheless, we acknowledge your point on exploring a deeper theoretical understanding of optimization dynamics and look forward to future research into it.\n\n**Q4**: Why not compare with the few-step model by distillation and VAR? Does the performance improve with more steps?\n\nWe acknowledge the suggestion to include distillation-based methods [3,4,5] and will provide relevant discussions comparing these approaches, emphasizing their distinct goals and strengths. These methods largely close the gap between the quality and diversity between 1-NFE/few-NFE models and the pretrained multi-step models, whereas MeanFlow focuses on training 1-NFE models completely from scratch.  \n\nWhile higher-step evaluations are generally outside our primary focus, we do observe modest performance improvements at higher steps; for example, achieving an FID of 1.89 at 4-NFE compared to our 2-NFE results. However, such gains beyond 2-NFE are less relevant to our core research interest. We agree with your comment to discuss distillation-based approaches [3,4,5], highlighting their complementary strengths. VAR [6], as another valuable paradigm, is already included in our comparison Table. 2. Thank you for emphasizing distillation-based methods and the VAR models.\n\n**Q5**: Besides stop-gradient, the paper also proposes other tricks such as adaptive weighting. If only L2 is used, what is the performance on larger models?\n\nWe sincerely appreciate your inquiry regarding adaptive weighting.\nTo clarify, adaptive weighting itself is not proposed in this work, but rather a standard practice widely adopted in baseline methods like iCT, ECT, and sCM.\nWe used it to ensure fair comparisons and comprehensive ablations.\n\nAdditional experiments conducted using only the L2 loss yielded an FID of 7.14 for the B/2 model under the full training budget, close to the FID of 6.17 achieved with adaptive weighting, confirming the robustness of our primary approach. Results on larger-scale experiments align consistently with our ablation studies.\n\nThank you once again for your constructive and invaluable feedback to improve the work together!\n\nSincerely, \\\nThe Authors"}]

Author rebuttal:

Meta-review: This work proposes a principled and effective framework for one-step generative modeling. The key idea is to introduce average velocity to characterize flow fields. The paper demonstrates clear methodology and strong empirical performance, making it a foundational contribution that establishes a strong benchmark for one-step diffusion/flow models and provides inspiration for future research.

During the rebuttal and discussion period, most reviewers were inclined to accept the paper despite Reviewer 7gPv’s concerns. The main issue raised was that the contribution at the idea level appears limited, with the approach resembling a repackaging of continuous-time consistency trajectory models (CTMs). However, the authors provided a strong rebuttal that clarified the novelty and positioning of their work. They are encouraged to further discuss the relation of their method to broader related works, such as CTMs, in the final version. In summary, this work presents a novel approach to one-step generative modeling, offers solid experimental validation, and addresses a problem of clear relevance to the community. I therefore recommend acceptance and strongly support this decision.

Moreover, the combination of a principled theoretical foundation, conceptual clarity, and state-of-the-art empirical results (FID 3.43 with 1-NFE on ImageNet 256×256) makes this work an exceptional contribution of broad interest to the community. Given its potential to inspire new directions in one-step generative modeling and its clear impact, I recommend that it be highlighted as an oral presentation.
Decision: Accept (oral)