AlphaFlow: Understanding and Improving MeanFlow Models
Authors: Huijie Zhang
Aliaksandr Siarohin
Willi Menapace
Michael Vasilkovsky
Sergey Tulyakov
Qing Qu
Ivan Skorokhodov
Venue: ICLR 2026 Poster (2026)
Keywords: diffusion models
mean flows
mean flow models
few-step diffusion
one-step diffusion
generative models
imagenet
Primary area: generative models
TL;DR: We analyze the recently proposed MeanFlow framework and generalize it AlphaFlow, which obtains new SotA for one/two-step generation on ImageNet 256x256.

Abstract: MeanFlow has recently emerged as a powerful framework for few-step generative modeling trained from scratch, but its success is not yet fully understood. In this work, we show that the MeanFlow objective naturally decomposes into two parts: trajectory flow matching and trajectory consistency. Through gradient analysis, we find that these terms are strongly negatively correlated, causing optimization conflict and slow convergence. Motivated by these insights, we introduce $\alpha$-Flow, a broad family of objectives that unifies trajectory flow matching, Shortcut Model, and MeanFlow under one formulation. By adopting a curriculum strategy that smoothly anneals from trajectory flow matching to MeanFlow, $\alpha$-Flow disentangles the conflicting objectives, and achieves better convergence. When trained from scratch on class-conditional ImageNet-1K 256×256 with vanilla DiT backbones, $\alpha$-Flow consistently outperforms MeanFlow across scales and settings. Our largest $\alpha$-Flow-XL/2+ model achieves new state-of-the-art results using vanilla DiT backbones, with FID scores of 2.58 (1-NFE) and 2.15 (2-NFE). The source code and pre-trained checkpoints are available on \url{https://github.com/snap-research/alphaflow}.

ALPHAFLOW:
UNDERSTANDING AND IMPROVING
MEANFLOW MODELS
Huijie Zhang!
Aliaksandr Siarohin'
Willi Menapace'
Michael Vasilkovsky!
Sergey Tulyakov!
Qing Qu
Tvan Skorokhodoy
SSna Inc. Department of EECS, University of Michigan
Menn (FID: 3.47)
NFB
o- o-Flow
2.95)
MonnFlow
(FID: 2.46)
NFT
o-Flow (FI:: 2.34)
Figure Uncurated samples (seeds 1-8) from the DiT-XL/2 model for MeanFlow Geng et al. 2025a) and a-Flow (our proposed method) produced with (upper) and 2 (lower) sampling steps for ImageNet-IK 256
ABSTRACT
MeanFlow has recently emerged powerful framework for few-step genera- tive modeling trained from scratch, but its success not yet fully understood. In this work, show that the McanFlow objective naturally decomposes into two parts: trajectory flow matching and trajectory consistency. Through gradi- ent analysis, find that these terms strongly negatively correlated, caus- ing optimization conflict and slow convergence. Motivated by these insights, we introduce a-Flow, a broad family of objectives that unifies trajectory flow matching, Shortcut Model, and MeanFFow under one formulation. By adopt- ing a curriculum strategy that smoothly anneals from trajectory flow matching to MeanFlow, a-Flow disentangles the conflicting objectives, and achieves bet- ter convergence. When trained from scratch on class-conditional ImageNet- 256x256 with vanilla DiT backhones, a-Flow consistently outperforms Mean- Flow across scales and settings. Our largest e-Fllw-XL/22 model achieves new state-of-the-art results using vanilla DiT backbones, with FID scores of 2.58 (1- NFE) and 2.15 (2-NFE). The source code and pre-trained checkpoints are avail- able on
thub. com httppp..ooo/oppprrrrearraaaaaaaa
INTRODUCTION
Diffusion models (Sohl-Dickstein et al. (2015) have emerged as the leading paradigm for generative
modeling of visual data (Dhariwal & Nichol 2021 Rombach et al. 2022: Brooks al.) 2024). However, their widespread use is limited by slow inference, as generating high-fidelity samples
Work done during internship Snap Inc.

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LTFM
LMF
LTC
La=1 (CTFM) Co La-s0( (b) a-Flow
(a) McanFlow
Diagrams of training dynamics between MeanFlow and a-Flow. (a) MeanFlow training exhibits a gradient conflict. (b) a-Flow resolves the conflict by first minimizing a trajectory flow matching loss then following annealing trajectory to approach the MeanFlow optimal solution.
Figure 2:
typically requires a large number of denoising steps. This computational bottleneck has spurred extensive research into designing efficient diffusion-based generators that are able to operate in very
few steps while preserving high generation quality (Salimans Ho 2022
Sauer et al.
2024
Song
et al.
2023 Song & Dhariwal 2024 Lu & Song 2025 Geng et al. 2025b Frans et al.
2025 Geng
et al.
2025a).
Early attempts reduce the inference time of diffusion models through distilling a pre-trained multi-
step model into few-step one (Salimans & Ho 2022; Sauer al. (024)). The subsequent devel-
opment of consistency models (Song et al. [023] Song & Dhariwal [2024] Lu & Song (202) en- abled training from scratch for few-step generative models. However, a significant performance gap still remains between existing few-step and multi-step diffusion models. The recently introduced MeanFlow framework (Geng et al. 2025a) enables more stable training and better classifier free guidance (Ho & Salimans| 2022) integration, significantly bridging the gap between few-step and multi-step from-scratch trained diffusion models. Despite its practical success, there still lacks a clear understanding of why MeanFlow performs better, which hinders further improvements and the design of stronger few-step models.
In this work, we provide deeper understanding of why MeanFlow works, revealing that its train- ing objective can be decomposed into two components: trajectory flow matching and trajectory consistency. Our gradient analysis shows that these components strongly negatively corre- lated during training, leading to instability and slow convergence in joint optimization. We further demonstrate that the previous heuristic adoption of border-case flow matching supervision is crucial: it actually acts as surrogate loss for trajectory flow matching and mitigates gradient conflict. How- ever, over 75% MeanFlow's 's computation is spent on this border-case supervision, which is not design more efficient techniques optimize
its primary focus. This raises an open question: MeanFlow objective, without such computational overhead?
Motivated by these observations, we introduce a-Flow, new broad family of objectives for few- step flow models. This framework unifies trajectory flow matching. Shortcut Models Frans et al. (2025), and MeanFlow under single unified formulation. As visualized in Figure 2 by employing a curriculum learning strategy that smoothly transitions from trajectory flow matching to MeanFlow, a-Flow better disentangles the optimization of trajectory flow matching and trajectory consistency, reduces reliance on border-case flow matching supervision, and achieves better convergence.
By training vanilla DiT- (Per-eee & Xie 2023) models from scratch with a-Flow ImageNet-1K 256 we obtain consistently stronger performance across both small- and large-scale settings compared with MeanFlow, for both one-step and few-step generation. Our largest DłT- XL/2+ model establishes new state-of-the-art results among all from-scratch trained models with the vanilla DiT backbone and training pipeline, achieving FID scores of 2.58 (I-NFB) and 2.15 (2-NFE).
conssioondii
2 PRELIMINARIES
Diffusion models and flow matching. Diffusion model (Ho et al. 2020: Song & Ermon 2019 Rombach et al. 2022) define forward process that progressively adds noise to data sample
2

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( over continuous timestep t [0. 1]. Specifically, given training data, the forward process perturbs z into a noisy version 2, = _ + o,E where e ()) 3g and o, are pre-defined Flow matching (Liu et al.
scheduler parameters that depend on t, such that 2 and 21 2023 Lipman et al. 2023) is a deterministic alternative that defines the forward process as a straight- line path between the noise distribution and the data distribution, setting Br 1 and or t. A neural network Ug( 2;, t) is trained to model the ground-truth vector field dz, /t along this trajectory 2e by minimizing the objective:
Lax (0) a,, (0 1x vx
where v, A v(2,, M/x) = dz//dl_ 6 x. To generate a new sample, the probability flow ODE (PF-ODE) dz/dt- = vz (,,,t) t) is solved from & = 1 to t = 0, starting with an initial value 21 (0, I).
One primary challenge of diffusion models is the slow sampling speed. To address this, several methods have been proposed enable high-quality generation with significantly fewer steps.
Consistency model (CM). (Song et al. 2023 enables one-step generation by training neural network fe(z,, t) to directly map the noisy input 2₁ to clean samples æ. The core idea to enforce consistency property at any two nearby timesteps and s, by minimizing the difference between the model's output. Depending on the Al == 8, the training objective can be categorized into:
Discrete-tine Consistency Training (CT) (Geng et al. 2025b Song et al. 2023 Song Dhariwal 2024 minimizes the following discrete time CT loss Ler.:
LCr (0)= Er,,,* [] t) fo- (z., 8)11
where 0 S s s<t 1, 2, 2, At v and fe- == stopgrad (fe). While smaller values of AL reduce the discretization error and improve performance, they might also lead to training instability (Song et al.) 2023) Geng et al. (2025)). This necessitates carefully designed scheduler for At to ensure good performance and stability during training.
Continuous-time CT (Lu & Song 2025 Song et al. 2023) eliminates the discretization error by
the continuous time CT loss Lor.:
Ler, (0) = 2 Søl (zt, 1) dfo- (zr,
dt
Song et al. 20023) theoretically show that VoLeT, (0) = limac-o (0) However, es-
timating afa (2..
df relies on the Jacobian vector product (JVP) operation, which causes potential
issues of scalability and efficiency in modern deep learning frameworks (Wang et al. 2025b Peng
et al. 2025).
Consistency trajectory model (CTM). (Kim et al. 2024 Zhou et al. 2025 Frans et al. 2025
Geng al.) 2025a) generalize Consistency Models (CMs) by training neural network ug 2, r, to enforce consistency across a trajectory from to r with S r S t S 1. This allows jumping from any /. (0, 1] to any < during inference, enabling multi-step generation. To train CTM from scratch:
Shortcut model (Frans et al., 2025) enforces consistency by ensuring that a single "shortcut" step from é to r is consistent with two consecutive shortcut steps of half the size. The training objective is:
E || us 3,t,,, us (Zo,T,3) Csc(O)
(4)
1,
where 2, = 2; (t s) us- (2,, 8, t) and r)/2
MeanFlow (Geng et al.
2025a) trains the model uo (z r,t) to estimate the mean velocity E-r Sr¹ u(zr, d)r,, with training objective given by:
1
3

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conVVrm Plrc) Mare!
Cow App ratio: 75%
Ape ruioo O%- Eppe
Cow
(a) Gradient similarity
(b) Lrf under different Low ratios. (c) Lr under different Law ratios.
Figure 3: MeanFlow training analysis. (a) Shows the cosine similarity between the gradients of two loss pairs (V Lro vs. TTy. and Lre vs. VLew) throughout training. (b) Evaluated Lrew when MeanFlow trained with 0% and 75% of Lav. (c) Evaluated Lre when MeanFlow trained with 0% and 75% of Lyv.
r, + (t dug (Ze, 1) 1 dt
CxF (
E t,
(5)
In practice, McanFlow significantly outperforms other one/few-step diffusion and flow models. Yet, there has been little analysis explaining why it works so effectively. To shed light on this, we analyze MeanFlow training in the next section.
3 ANALYZING MEANFLOW TRAINING
An intriguing aspect of McanFlow is the noise distribution used during training: Geng et al. (0225a) empirically found that the best results are achieved when setting r for 75% of the samples. This might look counter-intuitive, since we are interested in learning the average velocity on [r. inter- val to perform large trajectory leaps during inference, so why spending the majority the training computation on fitting this border case that corresponds to vanilla flow matching supervision? In this section, we show that the MeanFlow loss on its own can interpreted as velocity consistency training with extra flow matching supervision, and analyze the interaction of these two objectives.
3.1 UNDERSTANDING THE OBJECTIVE
Through algebraic manipulations, the original MeanFlow loss Lxe in Eq. 5 be rewritten into the following equivalent form (see Appendix D.I.:
2 (1-r)) up (21, duo (Zi, r,,)
LuF (0)
E t.r.3e
10 r,
E
+C,
dt
(6)
Trajectory flow matching Cere
Trajectory consistency Eu
where C' is constant independent of O. In this decomposition, the first term Lren corresponds to a flow matching loss but with an additional modeling input parameter v, so we refer to as trajectory flow matching. The second term Lrc, denoted trajectory consistency loss, acts as a (t rerweeigtteed continuous consistency loss ! but also without boundary condition (Song al. 2023). This decomposition highlights that the MeanFlow objective be interpreted consistency (trajectory) model with extra flow matching supervision.
An interesting property of this decomposition is that Lrc does not have any boundary condition. In comparison, Sonparison,SSongee et al. (223) enforces such a condition for vanilla consistency models using zp- prediction parameterization: without it, the model would quickly converge a constant output). In the McanFlow case, this collapse does not occur, which suggests that LrFE implicitly provides the boundary condition for Lrc. We hypothesize that the absence of an explicit
trivial solution (e.g..
'Smilarlyy the proof in Remark 10 of Song et al. 2023), minimizing the difference between n0 r,t) and thg (2:-Ac, r, At) At 0.
show that this term is equivalent to

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boundary condition makes Lre easier to optimize and enables significantly larger solution space. Further analysis of the solution space is provided in Appendix E.1
Another important observation here is that trajectory flow matching involves random & which differs from the = case used during training by Geng et al. (2025a). To clarify this distinction, directly compare trajectory flow matching (Crem) with vanilla flow matching, which we denote as Lew when using the u-prediction parameterization:
LIPM
[1 Zt, t,
E
LeW
E
u0 2x
Here, LIFN arises from the decomposition of the MeanFlow loss, while Law corresponds to the objective used in Geng et al. (2025a) for joint training. From this formulation, several observations follow. First, Lew is a "part" of LTFM. active only on the p(t,r t) slice of the joint distribution p(t, Second, the network is independent of r, then marginalizing out yields Lrex Lew, reducing the objective to vanilla flow matching.
3.2 EMPIRICAL ANALYSIS
With the decomposition in Equation (6). how does LFw interact with the two decomposed terms? In this section, analyze the gradients of these losses and examine how extra Lex minimization affects LTH and Lic individually. We conduct detailed experiments by training MeanFlow with the DiT-B/2 (Peebles Xie 2023) architecture on ImageNet- 256² (Deng al., 2009 for 400K steps. Additional experiment settings are in Appendix
We first analyze the training dynamics by measuring the cosine similarity between the gradients V LTEN and V Lre during training. As shown in Figure 3al these two gradients are strongly negatively correlated, with a similarity typically below 00.4. This reveals that optimizing LTFM and Lro jointly inherently difficult. We hypothesize this stems from the fact that Lrc, without any boundary condition, has a very large optimal solution manifold, compared to LITM whose manifold very narrow. Thus the optimization process is getting pulled towards the Lro manifold, distracting from reaching a narrow intersection.
Given this gradient conflict, the question arises: why does joint training with Lew help? We iden- tify two key reasons: First, as a subset of Lres, Law directly reduces LrEx- This empirically confirmed in Figure 3bb where allocating 75% of the training budget to Law significantly lowers the overall LTex compared to pure MeanFlow training. Second, Lew applies only at r = t, where Lrc 0. Consequently, the gradient V interferes less with V V.r than the V VTFF gradient. This is demonstrated in Figure Bal which shows that cos( V Lyx, VLre) is consistently higher than cos( Vrrn, LTc), that strongly negative for more than 95% of the training. Surprisingly, Lre component doesn't 't seem to be affected and can even be lower when allocating 75% of the training budget Lew, as shown Figure Be] Which again hints the fact that Lre is relatively easy to optimize, even near the Lrex optimum.
In conclusion, our analysis reveals three important observations:
Lue can be decomposed into trajectory flow matching LTFK and trajectory consistency LTc objectives, whose gradients are strongly negatively correlated during training.
Lre does not have necessary boundary condition on its own, implying that Lrex serves implicit boundary condition for it.
Lew acts as surrogate loss for Lyvn, but with significantly less gradient conftict with the Trajectory consistency loss Lrc-
a-FLOW MODELS
As we showed in the previous section, the LTEH loss is difficult to optimize jointly with the Lrc- While the introduction of the L loss serves effective surrogate for optimizing LrvE. this approach dedicates a significant portion training to an objective that is not of our primary interest. This raises key question: Can we more efficiently optimize Lren when optimizing Lue without this
5

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1/2
s
fo
0
fe
fe
0
11
(a) Discrete CT
(b) Continuous CT
(c) Shortcut Model
(d) McanFlow
(c) a-Flow
Figure 4: Comparison of training trajectories for various few-step diffusion and flow-based models.
computational overhead? To answer this, we introduce our a-Flow loss, new family of training
objectives for flow-based models.
4.1 a-FLOW: UNIFYING ONE, FRW, AND MANY-STEP FLOW-BASED MODELS
Definition 1. The a-Flow loss La is defined as:
(0)
-
11u0 (a Us.t + (1 a1) ug (2.,r,)) s))11
(8)
2,
where t, E [0, 1] the start and end timestep, is the intermediate timestep: s = a (1 a) t, a E (0, 1] is the consistency step ratio, and z, = + (t- ()-- s), is the trajectory value at this timestep s. Here, Ù, is the "shift velocity" used estimate the intermediate variable z, from 2t.
The a-Flow loss is visualized Figure 4e
Intuitively, enforces trajectory consistency between t and r by introducing an additional which is an interpolation between t, with ratio a. More importantly, this definition generalizes previously introduced training objectives such as trajectory flow matching, Shortcut Model training, and MeanFlow training:
Theorem 1. The o-Fllow loss unifies flow matching, Shortcut Models, and MeanFlow:
(T) = (0) with DzZt = Vt:.
(oc( 1/2 af=112(0) with Ds,t = ug- (21, 8, t,
(O) VoLa-+o(0) with Uz. Us.
Proof sketch: The connection between La and LIFN and Lisc are straight forward. For the non- trivial relationship showing the convergence from (0) to (0), we leverage a first-order Taylor expansion on the term ug (, around = t. This yields:
uo (Z,, ug (z,, r,t) dug (z,,r, (t- 00
dl
Substituting this expansion into the Equation and taking the limit as a - 0 causes the higher- order terms O (a?) to vanish and recover 0 (0). The detailed proof is deferred to Ap- pendix [..2 Furthermore, under Assumption we prove the upper bound of the non-asymptotic distance between VeL. (0) and VoL (0) in Appendix E.3 This bound is linearly related to and converges to 0 as + 0, which aligns with the result of Theorem
Moreover, one considers a--aaaamtrrzation uz 0,t) (z, - So(z,, 1)) /t 20. Ca incor- porates discrete and continuous consistency training as well. With ū..t Vn and = 0:
LeT, (0) (0) for t0,
VoLer. (0) voLa-n(
This theorem reveals that the ratio a is the key hyperparameter that unifies seemingly different methods, which controls the relative position of the intermediate timestep within the (r, t) interval. By annealing a from 1 to O, we obtain a family of models in the interpolation between trajectory flow matching and MeanFlow. Notably, discrete CT is special case of a-Flow with r = 0. Unlike discrete CT, a-Flow requires no complex timestep partitioning: once t and r are sampled, s is immediately determined with fixed a.

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4.2
Q- R--LOW MODELS
The a-Flow loss enables a curriculum learning strategy that progressively transitions from the tra- jectory flow matching MeanFlow objective. As visualize in Figure 2 this approach better dis- entangles the optimization of the trajectory flow matching and consistency losses, could potentially reduce reliance on the flow matching objective, and leads better convergence. The detailed cur- riculum learning can be summarized into three phases:
Trajectory flow matching pretraining (a = 1). speed-up convergence toward narrow Lrex manifold, we prioritize optimizing trajectory flow matching in the early training phase. Addition- ally, as a low-variance objective, trajectory flow matching quickly establishes a reliable noise-to- data mapping, providing a good initialization for subsequent few-step refinement. Notably, this pretraining strategy is aligned with previous diffusion model pretraining strategy applied on con- sisteney model (Geng et al., 20256), while we start from different motivations and generalize it into the a-Flow framework.
a-Flow transition (a E (0, 1)). After initial training establishes a solid foundation with LrEM, we transition to Lup via curriculum learning: progressively decreasing from I to O. This gradual shift, inspired by discrete CT methods (Song et al., 2023), serves two critical purposes: (a) theo- retically, the optimal solution of a- a-Flow smoothly transitions from that of CIFN to that of Lyr decreases (Appendix F.2); (b) curriculum learning steers the training objective from a "high-bias, low-variance" state to the necessary "low-bias, high-variance" state, with theoretical proof show- ing gradient variance increases as a decreases (Appendix (E4). This strategy yields significantly better convergence than directly optimizing the high-variance MeanFlow objective.
MeanFlow fine-tuning (a 0). In the final stage, we focus entirely on the McanFlow training objective. Unlike the original paper, our improved early-stage optimization of trajectory flow matching significantly reduces the need for the flow matching loss (as shown in Table 2 (b)) and achieves significantly better few-step generation quality.
The overall training code of a-Flow is shown in Algorithm [ where we first sample t, r and obtain the from the schedule. Based on whether a = 0 or not, a-Flow will use either Lue or La to train the model. a-Flow applics the same training details as MeanFlow when training Eys (except a lower ratio of flow matching). Below, only show the difference: the schedule of a well the design space of La when a > O.
Algorithm 1 a-Flow: Training.
fa Ey El:: function predict
x: training batch, kt training iterations
t, r eample.t.r
alpha sample.alphe (k)
alpha A x + (1 alphal t randn.like
(1 - 1x
is alpha u, dudt (fn, (zt, I, t), (v, 0, 11) dust
u_tgt
else
(zt,, F, t) = 2t (t - 5) + u_tgt alpha v alpha) fn (ns, r, 5)
Schedule. To schedule the train- ing, we use sigmoid function, = Sigmoid,, =k, (k), which depends on the training iteration The function is defined by its start- ing and ending iterations, kg, kes a temperature parameter y (set to be 25) and clamping value 77. The specific implementation can be found in Algorithm|] Figure/6lpro- vides a visualization of this sched- uler, while Section [5.2 conducts an ablation study over its parameters.
arvor DD- oporad (1_tgt) loss metele (erzor)
Algorithm 2 a-Flow: Curriculum Schedule
k_a, k_ei atart/end schedule fterationa, gamma: Lemperalure parameler k: current iteration, eta: clamping value
scele - 11 (Ke k_)) offaet a aphh = algmot ( a k.k + offset) gamma)
ik_s k_e) / 2 (k_e k_s)
alpha ir alpha > (1- eta) else (0 if alpha eta else alpha!
Clamping value. Geng et al. 2005b) show that when At = t s approaches 0, the performance of few-step CT model will first increase and then decrease. For a-Flow, we observe similar phe-
7

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NFE
NFE
Method
Source
Params
Epochs
FID
FDD
FID
FDD FID
Shortcut-XI.
Frans al.
2025
675M
160
10.60
IMM-XL/2
Zhou el al
2025
676M
3840
8.05
3.88
MeanFlow-XLL
Geng el al.
2025a
676M
240
3.43
2.93
MeanFlow-XLL X
Geng et al Peng
2025a
676M
1000
2.20
FACM-XL/2
2025
675M
S00
250 x
2.07
FACM-XL/2
675M
120 2
9.54
410.4
7.31
362.0
FACM-XL/2
675M
240 2
6.59
327.7
4.73 278.6
Our reproduction
CT-XL/2
676M
240
7.44
324.9
6.22 271.9
MeanFlow-BB
131M
240
6.04
312.3
5.17
232.1
MeanFlow-XXe FLL2
676M
240
3,47
185.8
2.46
108.7
2.26
a-FlowwBB
131M
240
5,40
287.1
5.01 231.8
a-Flow-XIL XL/2
Our methods
676M
240
2.95
164.6
2.34 105.7
2.16
1--Flo-XLL/2
676M
240+60
2.58
148,4 2.15 96.8
1.95
Table I: Class-conditional generation on ImageNet-256x2556 The table reports the results for few-step diffusion/flow matching-based methods trained from scratch. "x2" indicates that FACM requires roughly twice the computation per epoch compared to other methods. For a direct "epoch- to-epoch comparison," a-Flow-XL.2 MeanFlow -L/2 and FACM-XL/2 epochs. a-Flow-XL/2+ is a fine-tuned version of -L/2, trained for extra 60 epochs with a batch size of 1024. FID scores are evaluated with the balanced class sampling (see Appendix J
each trained for 240
nomenon: by training a-Flow with a fixed a, as a approaches 0, the 1-step generation performance will first increase then decrease. Detailed experiments
shown in Table (c). From the exper- iment, the optimal performance is achieved when a = 5 x 10 10-3 Thus, we set a clamping value n 5 10-3 for the schedule. a will be set to 0 when a 77- We also use the same clamping value to set a to 1 when a > 7), as when a is close to 1, LrTM is similar to La but more efficient.
Training objective. In the unifying space of a-Flow loss, all other few-step models set Ü,,: = vt except the shortcut model which uses U..t ug (Zt, 8, f). Additionally, we are interested in seeing whether we need exponential moving average (EMA) for 0 With ablation study in Table 5 (a), we set U, and do not EMA for 0
Adaptive loss weight. MeanFlow (Geng al. 2025al demonstrates the effectiveness of adaptive loss. Basically, let || denote the squared L2 loss. The adaptive loss weight w 1/(||a|| + c) where c 10-3 And the adaptively weighted loss is sg(w)] ||3 Theoretically, we derived an equivalent adaptive loss weight = az [1] +c) for La- We defer the derivation in Appendix H.2 With ablation study in Table (b). we demonstrate the derived adaptive loss weight better than other loss weights.
Classifier-free guidance (CFG). We apply similar CFG training strategy MeanFlow, by setting G. in Equation S with 0s,t = w v(z,, [[x) + k Up (Zt, ([c) + (1 - w - k)
ug- (z,, f|t)), where w, are the guidance scale, up- (.c), ug- (-1) denotes the class-condition
(with class e) and class-unconditional prediction. Detailed settings are deferred to Appendix G
Sampling. We employ both consistency sampling (Song al. 2023) and ODE sampling for two-
step generation. Implementation details are provided in Algorithm 3 Empirically, we observe that consistency sampling outperforms ODE sampling for larger models with better convergence. Consequently, we adopt ODE sampling for all DíT-B/2 architectures and consistency sampling for all DİT-XL/2 architectures, with additional ablation studies on DiT-XL/2 presented in Figure
EXPERIMENTS
In this section, employ a-Flow real image dataset ImageNet- 256 Deng et al. (2009) (Sec- tion 5.1 Section 5.2) and video dataset Kinetics- 700 17 x 256 Carreira et al. (2019) (Section 5.3
We use exactly the same DiT Peebles & Xie (2023) architecture MeanFlow Geng al. (2025a).
For evaluation, we use Fréchet Inception Distance (FID) Heusel et al. (2017). Fréchet DINOv2 Dis- tance (FDD) Oquab et al. (2023). We also use Fréchet Video Distance (FVD) Unterthiner et al.
8

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Schodule
NFE 1
NFE 2
Model
NFE 1
NFE
FID FDD FID FDD
% t Schedule
FID FDD
FID FDD
Constanto.0
44.4 844.1 42.1 836.3
Constanto.0
46.0 879.6 44.3 867.7
0%
Trajectory flow matching iterations
Sigmoid
40.4 822.5 38,9 811.8
Sigmoidok-
44.3 860.3 40.8 826.9
Sigmoid40K-11dd
44.1 846.8 39.9 811.6
25%
Constanto.o
44.4 844.1 42.1 836.3
Sigmoid 100K
42.4 828.0 38.3 795.4
Sigmoidok
40.0 785.4 37.1 782.9
41.3 818.8
38.1 793.1
Constanto.0
43.9 844.1 42.1 836.3
50%
Transition iterations
Sigmoid
40.2 781.0
37.1 775.0
si Sigmoid200k-
41.4 794.4
38.8 796.7
Sigmoid1sok-zaa
150kid110--22200
41.3 818.8
38.1 793.1
75%
Constanto.0
43.1 819.2 38.5 787.6
Sigmold oK-000
40.0 785.4 37.1 782.9
Sigmoid
42.2 810.5 36.2 754.7
(a) Consistency step ratio schedule.
(b) Flow matching ratio.
Table 2: Ablation study on ImageNet- IK 256 for a-Flow-B/2.
2019) for video generation evalution. We evaluate model performance for both 1 and 2 Number of Function Evaluations (NFE=1, NFE=2). We implement our models the latent space of the Stable Diffusion Variational Autoencoder (SD-VAE) More details on the experiments settings are
in Appendix G
COMPARISON WITH BASELINE
In T 1. compare alFlow with previous few-step Diffusion and Flow models, demonstrating
its superior performance for 1-NFE and 2-NFE generation. Across models trained for 240 epochs, a-Flow-XL/2 achieves 2.95 FID (164.6 FDD), representing relative improvement of 15% (12%) over McanFlow-XL/3 and 70% (60%) over FACM-XL/2. Our best model, a- a-Flow-XL/2+, sets new state-of-the-arr 1-NFE generation with an impressive FID of 2.58 (148.4 FDD), compared with all the other few-step Diffusion and Flow models trained over the SD-VA VAE. Furthermore, for 2-NFE generation, a-Flow-XL XL-2++ achieves 2.15 FID (96.8 FDD), outperforms all these baseline methods. It's particularly notable that it surpasses FACM-XL/2's 2.07 FID (achieved with class-balanced sampling) by reaching 1.95 FID with only 23% the training epochs. Uncurated samples, shown in Figure and Appendix [. visually confirm these results. Specifically in Figure c-Flow-XL/2 generates more images with better quality, as highlighted in green.
5.2 ABLATION STUDY
Consistency step ratio schedule. In Table (a), we evaluate our a-Flow framework trained with various sigmoid schedules, as visualized in Figure 6 For these experiments, the flow matching ratio fixed at 25%. We first analyze the impact the trajectory flow matching pretraining du- ration. By fixing h. k, to 100K iterations, we progressively increase k, from OK to 150K. As the pretraining duration increases, a-Flow's performance consistently improves across all metrics. The best-performing schedule, Sigmoid - significantly outperforms the baseline Mean- Flow (Constanto.u). This suggests that optimizing trajectory flow matching is more crucial than optimizing MeanFlow in the early training stages for achieving superior few-step flow modeling. This finding aligns with our empirical analysis, which shows that because the gradients of the tra- jectory flow matching and consistency losses conflict, it more efficient exclusively optimize the trajectory flow matching objective for faster initial convergence.
Next, investigate the effect of the transition duration. With the midpoint (k, + ke )/2 fixed 200K iterations, we vary the total transition iterations from to 400K. Our results indicate that longer, smoother transition leads to better generation quality. This highlights the importance of gradually reducing the bias of the training objective by smoothly transitioning between trajectory flow matching and MeanFlow.
Flow matching ratio. In (b), we compare our &-Flow framework with the McanFlow base- line across various flow matching ratios (%r t). Our results show that a-Flow consistently out- performs MeanFlow for all evaluated ratios, confirming the effectiveness of our proposed method. Thhe EMA version in https://hugiingg
ft-mse

---

FID
MounFlow DIT-XU2 MounFlow DIT-XL22
1.00
Cunstantee
3.2
Flow DIT-XL/2 -o- Flow XTT (CS)
valuu
Signoody
DST-XI/2+
DXT-XI-2+
Ragaoid (4--0o
3.0
Signoldd uee. caras
2.6-
Signord aa
Bagauiid a4--200
Bignnidyy.ex
1.00
Intermediate Timestop for NFE=2
0.75
0.50
0.25
8.000
23%
2006
266 Training lieration.
1838
2015
2008
2008
4005
Figure 5: Comparing ODE vs consistency sam pling for MeanFlow and a-Flow models.
Figure 6: Visualization of consistency step ratio schedule.
A key finding is that a-Flow achieves its best 1-NFE performance relatively low flow matching ratio. Specifically, it reaches the best FID of 40.0 at 25 % of = and the best FDD of 781.0 at 50 % of r f, while MeanFlow requires higher ratio of 75% achieve its best FID of 43.1 and FDD of 819.2. This aligns with our motivation: by pretraining on trajectory flow matching, a-Flow is less reliant on the flow matching objective and can focus more on the overall MeanFlow objective, leading to superior one-step generation quality.
Furthermore, we observe that for a-Flow, the flow matching ratio presents clear trade-off between 1-NFE and 2-NFE performance. For instance, the 75% ratio yields worse NFE=1 but better NFE=2 generation results compared the 50%-ratio version. This indicates that higher proportion of flow matching improves the model's s ability to generate images in slightly higher number of steps.
Sampling. As shown in Figure 5 we compare ODE sampling (solid line) and consistency sam- pling (dotted line) for 2-NFE generation across different intermediate sampling timesteps, using MeanFlow-XLL2. a-Flow-XL/2, and yields better generation performance for both a- F-ooo-XLL22 and a-Flow-XL/2+, achieving the best FID scores of 2.09 at timestep 0.4 and 2.28 at timestep 0.45, respectively. contrast, ODE sampling performs better for HeenFFlowXXI.22 which attains its best FID of 2.39 at timestep 0.35. In Table we select intermediate sampling timesteps that balance FID and FDD; see Table 3 for details.
The results show that consistency sampling
5.3 EXPERIMENTS ON KINETICS-70N 17 256
We evaluate the generalization capa- bility of a-Flow on video genera- tion. As shown in the Figure 7 Flow consistently outperforms Mean- Flow across all metrics and settings. When training without CFG, a-Flow achieved improvements of 2.5 (NFE FID), 21.5 (NFE 1 FDD), 21.5 (NFE FVD), 1.4 (NFE FID), 11.9 (NFE 2 FDD), and 13.5 (NFE 2 FVD); When training with CFG: a-
NFE
NFE 2
Method
FID
FDD
FVD
FID FDD FVD
McanFlow-B/2
53.2 1031.9 619.5 49.5 995.3 580.8
a-Flow-B/2
50.7 1010.4 598.0 48.1 983.2 567.3
CicanPlow-B/2-fff
32,3
779.0 298.1
26.6 719.8 249.6
H-Floo-B/2-cff
29.6 709.8 276.2 24.9 657.4 217.3
Figure 7: Kinetics-700 17 x 256 experiments.
Flow achieved improvements of 2.7 (NFE FID). 69.2 (NFE I FDD), 21.9 (NFE I FVD), 1.7 (NFE 2 FID), 62.4 (NFE FDD), and 32.3 (NFE 2 FVD). These experimental results successfully demonstrate the effectiveness and robust generalization of the a-Flow framework across more di- verse datasets and modalities.
CONCLUSION
Our work provided a principled analysis of the MeanFlow framework, analyzing its objective and establishing the necessity of flow matching supervision during training. Motivated by this under- standing, we proposed the a-Flow objective as generalization of MeanFlow loss, allowing us to train consistently stronger few-step image generation models from scratch.
10

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7 REPRODUCIBILITY STATEMENT
We are committed to ensuring the reproducibility of our results. To this end, we include all the necessary implementation details in Appendix G ensuring that our methodology can be faithfully reproduced. We will publicly release our source training, inference, and evaluation code, as well as the pre-trained checkpoints for ImageNet-IK 256².
REFERENCES
Michael S Albergo and Eric Vanden- Eijnden. Building normalizing flows with stochastic inter- polants. arXiv preprint wrXiv:2209.1557// 2022.
David Berthelot, Arnaud Autef. Jierui Lin, Dian Ang Yap. Shuangfei Zhai, Siyuan Hu, Daniel Zheng, Walter Talbott, and Eric Gu. Tract: Denoising diffusion models with transitive closure time-distillation. arXiv preprint arXiv:2303.04248 2023.
Nicholas M Boffi, Michael Albergo, and Eric Vanden Eijjden,, How build consistency model: Learning flow maps via self-distillation. arXiv preprint arXiv:2505. 18825, 2025a.
Nicholas Matthew Boffi, Michael Samuel Albergo, and Eric Vanden-Eijnden. Flow map matching with stochastic interpolants: A mathematical framework for consistency models. Transactions on Machine Learning Research, 2025b.
Tim Brooks, Bill Peebles, Connor Holmes, Will DePue, Yufci Guo, Li Jing, David Schnurr, Joe Taylor, Troy Luhman, Eric Luhman, Clarence Ng. Ricky Wang, and Aditya Ramesh. Video generation models as world simulators. 2024. URL httppt:tts///penaaa wtw//reeearc......eeeeeeeeo -s-ss-aswwoos wwrr-sswwwwd---ooooo-oo--------- ------a-ooo-----mllllaoooo
video-generation--nee
Joao Carreira, Eric Noland, Chloe Hillier, and Andrew Zisserman. A short note on the kinetics-700 human action dataset. arXiv preprint arXiv: ptt.:/907999977 069.7, 2019.
Minshuo Chen, Kaixuan Huang. Tuo Zhao, and Mengdi Wang. Score approximation, estimation and distribution recovery of diffusion models on oowdimensionnl data. International Conference Machine Learning, pp. 4672-4712. PMLR, 2023.
Sitan Chen, Sinho Chewi, Jerry Li, Yuanzhi Li, Adil Salim, and Anru R Zhang. Sampling easy as learning the score: theory for diffusion models with minimal data assumptions. arXiv preprint arXiv:2209. 12215, 2022.
Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hi- erarchical image database. In 2009 IEEE conference on computer vision and pattern recognition, pp. 248-255. Ieee, 2009.
Prafulla Dhariwal and Alexander Nichol. Diffusion models beat gans on image synthesis. Advances in neural information processing systems, 34:8780-8794 2021.
Patrick Esser, Sumith Kulal, Andreas Blattmann, Rahim Entezari, Jonas Müller, Harry Saini, Yam Levi, Dominik Lorenz, Axel Sauer, Frederic Boesel, et al. Scaling rectified flow transformers for high-resolution image synthesis. In Forty-first international conference on machine learning. 2024.
Kevin Frans, Danijar Hafner, Sergey Levine, and Pieter Abbeel. One step diffusion via shortcut models. In The Thirteenth International Conference on Learning Representations, 2025. URL
http::///peeeeeeee..ee. net www.:neet//ooototooooo
forum? 200.mmm2111001016666..... zB6LnXc5)
Zhengyang Geng, Mingyang Deng, Xingjian Bai, Zico Kolter, and Kaiming He. Mean flows for one-step generative modeling. arXiv preprint arXiv:2505.1 00.25555.3444 /3447, 2025a.
Zhengyang Geng, Ashwini Pokle, Weijian Luo, Justin Lin, and J Zico Kolter. Consistency models made easy. In The Thirteenth International Conference on Learning Representations, 2025b. URL htt.ps:://ppnreeiie. net. http://wwweeeeeeeeoooooooooo/o//

---

Xiangming Gu, Chao Du, Tianyu Pang, Chongxuan Li, Min Lin, and Ye Wang. On memorization in diffusion models. arXiv preprint arXiv:2310.02664 2023.
Yi Guo, Wei Wang, Zhihang Yuan, Rong Cao, Kuan Chen, Zhengyang Chen, Yuanyuan Huo, Yang Zhang, Yuping Wang, Shouda Liu, et al. Splitmeanflow: Interval splitting consistency in few-step generative modeling. arXiv preprint arXiv:2507.1688. 2025.
Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. Gans trained by a two time-scale update rule converge local nash equilibrium. Advances in neural information processing systems, 30, 2017.
Jonathan Ho and Tim Salimans. vrXiv:2207.12590 2022.
Classifier-free
diffuusion guidance.
arXiv preprint
Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models, Advances in neural information processing systems, 33:6840-665 6851, 2020.
Tero Karras, Miika Aittala, Timo Aila, and Samuli Laine. Elucidating the design space of diffusion- based generative models. Advances in neural information processing systems, 55:26565-26577 2022.
Tero Karras, Miia Aittala, Jaakko Lehtinen, Janne Hellsten, Timo Aila, and Samuli Laine. Analyz- ing and improving the training dynamics of diffusion models. In Proceedings of the IEEE/CVF Conference on Computer Vision and Puttern Recognition, pp. 24174-24184, 2024.
Dongjun Kim, Chieh- Lai, Wei- Wei-Hsiang Liao, Naoki Murata, Yuhta Takida, Toshimitsu Uesaka, Yutong He, Yuki Mitsufuji, and Stefano Ermon. Consistency trajectory models: Learning prob- ability flow ODE trajectory of diffusion. In The Twelfth International Conference on Learning Representations, 2024. URL httph:tt/////tttcttt /ttt:s::/operrrevieeeee wtteiiiwwwwweeeeeeeeeeeeeeeeeeee wwtp//eet/t/tteeeeeeeeeeeettteet for
Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization, arXiv preprint arXiv:1412.69800 2014.
Tuomas Kynkäänniemi, Tero Karras, Miik Aittala, Timo Aila, and Jaakko Lehtinen. The role of imagenet classes in fr), 'echet inception distance. arXiv preprint arXiv:2203. 06026, 2022.
Yaron Lipman, Ricky T. Q. Chen, Heli Ben-Hamu, Maximilian Nickel, and Matthew Le. Flow matching for generative modeling. In The Eleventh International Conference Learning Repre- sentations, 2023. URL / review.www....e wwt.tiiwwwweeeeeeeeeee..e...eeee htte://foooumn1ooecoceo
Xingchao Liu, Chengyue Gong, and qiang liu. Flow straight and fast: Learning to generate and transfer data with rectified flow. The Eleventh International Conference on Learning Repre- sentations, 2023. httt:tttt////peeeeette net. wwwpee/wwwwiw..eeeooooooo.eee... XVViiriiwiiiixiI
Cheng Lu and Yang Song. Simplifying, stabilizing and scaling time consistency models. In The Thirleenth International Conference on Learning Representations, 2025. URL https:
//operreiiew. net./ wwt.tf..ooo..eeeee..eeee formm/orrrm// oett://forrmmm/.TT..yy.5y5 SugyJx
Martin Marek, Sanae Lotfi, Aditya Somasundaram, Andrew Gordon Wilson, and Micah Goldblum. Small batch size training for language models: When vanilla sgd works, and why gradient accu- mulation is wasteful. arXiv preprint arXiv:2507.07101, 2025.
Sam McCandlish, Jared Kaplan, Dario Amodei, and OpenAl Dota Team. An empirical model of large-batch training. arXiv preprini 11220:28/2.06162, 2018.
Maxime Oquab, Timothée Darcet, Théo Moutakanni, Huy Vo, Mare Szafraniec, Vasil Khalidov, Pierre Fernandez, Daniel Haziza, Francisco Massa, Alaaeldin El-Nouby, et al. Dinov2: Learning robust visual features without supervision. arXiv preprint arXin:2304.077193, 2023.
William Peebles and Saining Xie. Scalable diffusion models with transformers. In Proceedings of the IEEE/CVF international conference on computer vision, pp. 4195-4205, 2023.
Yansong Peng, Kai Zhu, Yu Liu, Pingyu Wu, Hebei Li, Xiaoyan Sun, and Feng Wu. Flow-anchored consistency models. arXiv preprint 20x.i::557770373888 2025.
12

---

Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al. Learning transferable visual models from natural language supervision. In International conference machine learning, pp- 8748-8763. PmLR, 2021.
Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Bjöm Ommer. High- resolution image synthesis with latent diffusion models. In Proceedings of the IEEE/CVF confer- ence on computer vision and pattern recognition, pp. 10684-10695, 2022.
Tim Salimans and Jonathan Ho. Progressive distillation for fast sampling of diffusion models. arXiv preprint arXiv:2202.005// 2022.
Axel Sauer, Dominik Lorenz, Andreas Blattmann, and Robin Rombach. Adversarial diffusion dis- tillation. In European Conference on Computer Vision, pp. 87-103. Springer, 2024.
Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsupervised learning using nonequilibrium thermodynamics. In International conference on machine learn- ing. pp. 2256-2265, pmlr, 2015.
Jiaming Song, Chenlin Meng, and Stefano Ermon. Denoising diffusion implicit models. In Interna- tional Conference on Learning Representations, 2021a. URL https:///penrreview wwt.rreeieeeeeeeeeeeeeee.
Forum?id=Stlgiarce
Yang Song and Prafulla Dhariwal. Improved techniques for training consistency models. In
The Twelfth International Conference on Learning Representations, 2024. URL https://
openreview wwtpeeewwwwwweeeeeeeeeeee forum Ieforrmmid-iie WN-
Yang Song and Stefano Ermon. Generative modeling by estimating gradients of the data distribution.
Advances in neural information processing systems, 32, 2019.
Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. In Interna- tional Conference on Learning Representations, 2021b. URL htttp:://pperreview..er net/
forum?id=PxTIG12202
Yang Song, Prafulla Dhariwal, Mark Chen, and Ilya Sutskever. Consistency models. In Andreas Krause, Emma Brunskill, Kyunghyun Cho, Barbara Engelhardt, Sivan Sabato, and Jonathan Scar- lett (eds.), Proceedings of the 40th International Conference on Machine Learning, volume 202 of Proceedings of Machine Learning Research, pp.
PMLR, 23-29 Jul 2023.
Peng Sun, Yi Jiang, and Tao Lin. Unified continuous generative models. arXiv:2505.07447. 2025.
arXiv preprint
Thomas Unterthiner, Sjoerd van Steenkiste, Karol Kurach, Raphaël Marinier, Marcin Michalski, and Sylvain Gelly. FVD: A metric for video generation, 2019. UtL/Itttps://ppenrereee.
Fu- Yun Wang. Zhaoyang Huang, Alexander Bergman, Dazhong Shen, Peng Gao, Michael Lingel- bach, Keqiang Sun, Weikang Bian, Guanglu Song, Yu Liu, et al. Phased consistency models. Advances in neural information processing systems, 37:83951-84009, 2024.
Zidong Wang. Yiyuan Zhang, Xiaoyu Yue, Xiangyu Yue, Yangguang Li, Wanli Ouyang, and Lei Bai. Transition models: Rethinking the generative learning objective. 2025a.
Zidong Wang, Yiyuan Zhang, Xiaoyu Yue, Xiangyu Yue, Yangguang Li, Wanli Ouyang, and Lei Bai. Transition models: Rethinking the generative learning objective. arXiv preprint arXiv:2509.04390 2025b.
Ling Yang. Zixiang Zhang, Zhilong Zhang, Xingchao Liu, Minkai Xu, Wentao Zhang, Chenlin Meng, Stefano Ermon, and Bin Cui. Consistency flow matching: Defining straight flows with velocity consistency. arXiv preprint arXiv:2407.02394 2024.
13

---

Tianwei Yin, Michaël Gharbi, Taesung Park, Richard Zhang, Eli Shechtman, Fredo Durand, and Bill Freeman. Improved distribution matching distillation for fast image synthesis. Advances in neural information processing systems, 37:47455-47444 2024a.
Tianwei Yin, Michaël Gharbi, Richard Zhang, Eli Shechtman, Fredo Durand, William T Freeman, and Taesung Park. One-step diffusion with distribution matching distillation. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 6613-6623 2024b.
Sihyun Yu, Sangkyung Kwak, Huiwon Jang. Jongheon Jeong. Jonathan Huang, Jinwoo Shin, and Saining Xie. Representation alignment for generation: Training diffusion transformers is easier than you think. In The Thirleenth International Conference on Learning Representations, 2025.
URL
net forum? wetprruummd-DDSSSGG2222
Huijie Zhang, Jinfan Zhou, Yifu Lu, Minzhe Guo, Peng Wang, Liyue Shen, and Qing Qu The emergence of reproducibility and consistency in diffusion models. In Forty-firsl International Conference on Machine Learning, 2024. URL
net ?
Hsl10g2kc0
Linqi Zhou, Stefano Ermon, and Jiaming Song. Inductive moment matching. In Forty-second International Conference on Machine Leurning, 2025. URL http:::///peeeeeviewe.nntt
forum?id-pwiit 7yUb.
Mingyuan Zhou, Huangjie Zheng, Zhendong Wang. Mingzhang Yin, and Hai Huang. Score identity distillation: Exponentially fast distillation pretrained diffusion models for one-step generation. In Forty-first International Conference Machine Learning, 2024.
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A RELATED WORK
Diffusion Models. Diffusion models have become a dominant paradigm in generative modeling
for vision domains (Sohl-Dickstein et al. 2015 Song & Ermon 2019 [Ho et al., |022 Song al. b Dhariwal & Nichol| (021). The classical diffusion framework defines a forward noising process and corresponding reverse process that model learns approximate. Early works such as DDPM (Ho et al.) 2020) and score-based generative modeling (Song & Ermon 2019) demonstrated high-quality image generation. later extended continuous-time SDEs and ODEs (Song et al. 2021b). (Dhariwal & Nichol (02T) further improved sample fidelity with larger architectures and classifier guidance. More recently, the community has explored flow-based parameterizations that directly learn continuous velocity fields (Liu et al.) [023] Lipman et al. [203]] /Albergo & Vanden- Eijnden 2022). These flow matching approaches simplity training, unify score- and likelihood- based models, and are used in large-scale systems such as Stable Diffusion 3 (Esser et al. 2024).
Few-step Diffusion. Despite their quality, diffusion models are computationally expensive due to iterative sampling. A large body of work accelerates sampling to few steps even one step. Distillation- based approaches include progressive distillation (Salimans & Ho (022)
(0022)BBele Berthelot et al.)
2023), and often incorporate adversarial objectives (Yin et al. 2224b Zhou al. [0244] [auuer ct al. 2024). UCGM (Sun et al. 2025) develops a unified training scheme for multi-step and few- step diffusion-based methods.
A closer research direction (which our method follows as well) includes the methods which are trained from scratch and support few- and even one-step generation by design. Consistency Mod- els (CMs) (Song et al., 2023) learn to map noisy inputs directly to clean data by enforcing self- consistency. Extensions improve stability and scalability (Song & Dhariwal, 2024 Lu & Song) 2025| Geng et al. 2025b). Trajectory-basse
methods learn the dynamics of the entire denoising pro- cess, enabling arbitrary jumps along the diffusion path. PCM (Wang et al.) 2024) scale consistency distillation to large scale models and optimize with preselected time intervals. Shortcut diffusion models (Frans et al., 2025) learn direct mappings with shortcut constraints. MeanFlow (Geng al.) 2025a) predicts time averaged velocities with continuous consistency, while Guo et al. (2025) ex- plore this idea for discrete consistency. While concurrent work Guo et al. (2025) proposed a similar loss function to our a-Flow loss (Equation (8)) our paper offers theoretical and empirical contri- butions. Theoretically. we provide a deeper analysis comparing the a-Flow loss and the MeanFlow loss. Empirically, we introduce novel techniques specifically designed to improve the performance of the a-Flow loss, and we successfully demonstrate its efficacy and the benefits of curriculum learn- ing on the large-scale ImageNett dataset. Hybrid approaches combine consistency and flow match- ing: Consistency-FN (Yang et al. 2024) enforces velocity self-consistency, FACM (Peng et al.) 2025) anchors consistency to flow objectives, and IMM (Zhou al. 2025) matches the output distri- butions via moment matching instead of exact outputs. Consistency Trajectory Models (CTM) (Kim) al. 2024) generalize consistency training to support transitions between any two timesteps, com- bining one-step generation with progressive refinement. [Boffi et al. (202b)) introduces Flow Match- ing Mapping (FMM), a unified framework that extends CMs, CTM, and progressive distillation. In particular, it demonstrates that existing methods can be interpreted within a common Euler and progressive perspective, while also proposing a novel class of Lagrangian methods. Building on this foundation, (Boffi et al. 2025al presents systematic algorithmic framework and demonstrates the superior performance of the Lagrangian approach. Our work, on the other hand, provides a distinct and detailed unification focused specifically the Euler method. Within this framework, we unify Flow Matching, Shortcut models, and MeanFlow. Transition Models (TiM) (Wang et al.] 2025a) derive an exact continuous-time dynamics equation for arbitrary-step transitions. These methods achieve one- to few-step sampling with steadily improving fidelity.
work/Guo
B LIMITATIONS
Our &-Flow loss enables high-quality training of discrete MeanFlow models without requiring JVP computation. However, in practice, the continuous objective (i.e., setting 0) remains important, likely due to the variinae trade-off inherent in the consistency objective (Song
et al. 2023 Song & Dhariwal 2024).
15

---

We occasionally observed unstable training in large-scale models with guidance integration, both for the vamilla MeanFlow model and our a-Flow variant. Thus, our framework should not be viewed as silver bullet for addressing the well-known instability issues of consistency mod- els Gieng et al. (025h).
The a-Flow objective uses pure flow matching supervision up to k2 iterations, after which the consistency objective is applied. Before this point, the model's few-step performance weak, which can make progress harder to monitor.
Our gradient analysis provides actionable insights but remains empirical; it does not fully explain, from a theoretical perspective, why flow matching is so critical for consistency.
Although we motivate larger batch sizes for fine-tuning by the high variance of the consistency loss, the observed improvements (see Table 4) may instead reflect that small batches are more sensitive to hyperparameters (Marek et al. 2025), and that beyond certain size, batch-size scaling exhibits diminishing returns (McCandlish et al. 2018).
C FAILED EXPERIMENTS
We also wish to share with the community several experiments that did not succeed during the course of this project. Some of these directions were likely underexplored on our side, while others may represent genuine dead-ends. Nevertheless, we believe documenting them may serve useful reference for future work.
devoted several weeks to exploring decomposed training of the MeanFlow objective with individually tuned weighting functions for each term, drawing inspiration from EDM Karras et al. (2022) to map out the design space. Unfortunately, every configuration we attempted produced worse results than the default adaptive loss heuristic, which
particularly frustrating outcome.
Consistency sampling (see Figure did provide the improvements had anticipated. In- terestingly, the optimal midpoint consistently emerged at = 0.5, which coincides with the default MeanFlow setting. We suspect this effect is related to the training distribution, which has mode slightly lower 0.5. Following the original work, we employed logit-normal distribution with location parameter -0.4.
We experimented with LoRA fine-tuning and introduced separate prediction heads for vanilla velocity and mean velocity. Neither approach yielded promising results.
We conducted roughly 50 ablations the train-time noise schedule for vanilla MeanFlow models. None resulted in noticeably better performance, even when factorizing the joint distribution p(t. into prl)p(r(t) and exploring alternative supervision distributions for flow matching in parallel.
We investigated additional representation alignment losses Yu et al. (2025) with the aim of ac- celerating convergence in MeanFlow models. However, the observed gains were insufficient to justify the added complexity of the training framework.
We also experimented with different EMA schedules, but these attempts did not lead to meaningful improvements.
D PROOFS OF THINGS
D.1 LOSS DECOMPOSITION
Proof. The MeanFlow loss is given by:
Lxp (0) Er,ris uo((,))),)) (t duo- (Z,, r, dt
(9)
16

---

(unpacking the norm and regrouping terms yields)
deep (21, r,t)
Be, [1, r, 1)
Uo [2 Ben
2. (1 r)) ug (2 r,
(10)
dt
(om(()
Cx(
duo (21,
ddug (ze, T; 1) 12
-2 (t (r) (z,, t|x)
+ r)2
(11)
dt
Does not depend on
D.2 La LOSS UNIFICATION
The proof for flow matching and shorteut models is straightforward, We will only show the proof for the third bullet point. For brevity, let's At s and a
Proof of theorem
7
At
t -
At uo (z)
(0) Er,r,
10-
At
--r
- At
1-r u1- (Zi- A1)
(4)
t -
At
A-At-a
Eir,,
uo (2t,
Uz
At
--
---
(12)
duo (Zi, r,t) Al O (A21) )
ug
(2 ()
df
t uo( ug- (21, At
E
At
duo (z,, ug (21,1,() + (A41) ){²
Uz 1 (I-
dl
where (i) uses the Taylor expansion over ug- (21- 1 Al):
ug- (z,-At, - At) = ug (Zt,T,1) dug (Zi, T, Al A + O
dt
dug (z r,t) A (A2t). Thus,
and (ii) uses the fact that
dt
lim (0) a-0
= lim
(oLa(0)
At-0
L At
Al Vo uo (2),) (uo(2));1)) ug- (2),,,) r-
lim Bi,r,
2
At-0
dl uo- +) + ) At Vo (o0, (-
duo (Z(,Tit)
Ur (t
r))
t
Iim ELnz
2
At-+0
At
duo (,t)
V: (6-r" r)
dt uo- (x)
AL ( Vouo(Zerr()) (0 (2,,11) vr + (1 r) dug (z {)) ).
-
At
-Be
2
dl
V VNLAx (0),
(13)
17

---

Proof of equivalence with consistency model. By setting 8.
ve, and uo 0,t)
Al
(z+ fo(zt, 1)) /t, At = 1 and a
we have:
t
At
L (0) = R,
ug 2z, r,t)
2/=
t --
t
r
ug- (zi An, At)
t ビール
r
Zt
fz(
Al
E1,
.U:
(14)
At
t
t
1
Al 21 At
So- (21- At)
t
t- At
[[ t) fo (ZI- t - At)1
((is) LET, (0),
Bt.z,
t&t
where (i) plug in the reparameterization and 0, (ii) uses the fact that 2, 2t at Al-v Thus tr-
Since the discrete CT uses timestep partition to determine and At, the (iii) holds for special timestep partition when At
Ca 0) could be reparameterized to Len, (0) with a loss weighting function t
given fixed From Theorem 2 in Lipman et al. 2023). because ug (,,, is
independent of 0, we have:
dug- (21, r. 1) dt [
CHP (0) Bi,r,z
uo uo( + (t (t)
(15)
dug (z1, r, f)
v(ze,t) -(((,,) ()) r) us [u((21
+C,
dt
with a constant independent of Thus,
V (
(uo(z,,T,)) - v(ze, f) + (t r) dus (z1, r,t) t)
-Binn
2 V us (2,,T,))
dl
( Et.zh
fo(z),t)
2t - So((o.))t) - t)
2. V
t
(16)
dfe- (z(,t)
(z) - So (Ze, t))
u(zz,t)=
dt
dfo (2t, V Lor. (0),
= -B,
2
Vo fo(
dt
where (i) plug in the reparameterization and
0, and use the fact that:
dug (z,, 0,t)
1
t (() dffe (z),
(zi for (z)
(17)
dt
t2
Thus VOLNE could reparameterized to Ler. (0) with a loss weighting function 7
E MORE PROOFS
E. ANALYSIS OF OPTIMAL SOLUTION SPACE BETWEEN TRAJECTORY FLOW MATCHING LTEM TRAJECTORY CONSISTENCY LTc
Assumption 1. Assume that the uz(zi, r,t) has infinite model capacity, and approximate any continuous function to an arbitrary level of accuracy based on the Universal Approximation Theo- rem. And 2a
Rdd draw from random distribution p(a:).
18

---

Assumption has been widely used in previous works Karras et al. (2022): Gu et al. (2023 Zhang
et al. (2024) to analysis the optimal solution of diffusion training loss, which simplified specific network architecture constraint without loss of generality. The trajectory flow matching loss defined in Equation (6) could be re-written as:
LIEM (uo)
E [|leo(21; r, - well2]
(18)
1,
[ 1, Jad
[
21 [ N (z): t)x, f21) (2,,r,,
-t - t
112 (((pa))d dz, dt dr.
BB
Crer(Mo,),,rss))
(19)
Since Assumption implies that uz(2,, r, approximate any continuous function, min- imize LIFM (Ug) by independently minimizing the inner integral (TFx(Ug. f, z,) for each fixed r,),
Given Zt,T, 4, L (,,, ) is convex optimization problem with respect to uo(z,, v, 1) and the optimal solution is uniquely identified by setting the gradient with respect to ug (2g,r, ()) to O.
V 1) (TEO(UO (,,, 0,
(20)
+ -vo(ee()))) / N (ze: - -t)x,t²1)
- Zt =
x
p(x))dx = 0,
(21)
= / N (1-20, " (2e, r,
21 - x
:p(x))dx = O,
(22)
t
Exep(æ) [ (z 1),,)),,1 (21) ax
2t
u 1H (zz, 1)-
(23)
E Bm<p(a) [ W(z:: (1 - t)x, T2
This shows that the loss LrwM has only one optimal solution. However, the situation is different for the Trajectory consistency Lrc defined in Equation (6):
dus (,,,
LrC = t,r,ze
(t ((t-r) r)- ug (zt, r,t)
(24)
dt
For fixed 2t, rt,, the loss Lro (Zt, r, t) is linear func- tion w.r.t uo(2,, so it does not have a lower bound. Any function (z, which satisfies the condition u (Zz, r, dug- (x....)) oo minimizes the loss. This confirms that the unconstrained Lre has a large, easily optimizable solution space.
Loss Lrp
dt
-60
Crucially, this finding does not contradict the difficulty of optimizing Ler, noted in prior work Song et al. (2023) Lu & Songl (205). In those papers, the optimization prob- lem was made difficull by the explicit boundary condi- tion fe (2, 0) 2o. which constrains the velocity field as up 21, 0, (zz ff ()) /t En.
-80
100
200
300
400
500
Training Iteration
Figure Optimization of Lrc.
E.2 ANALYSIS OF OPTIMAL SOLUTION BETWEEN TRAJECTORY FLOW MATCHING LIe TRAJECTORY CONSISTENCY Lrc
Following the same derivation above, minimizing CNF is equivalent to minimizing CHe 2z, r,t) for every fixed 2,, r, under Assumption | Lwp ( r,))
where:
=/3 N (ze; (1 t)xx,t21 (2T) (100(2,, r,t) - U2 + (t r)
dug- (2x, r.))
(25)
11 p(x)da.
dt
19

---

Since the term ug- is independent of us. Lnp 2t, r. remains convex optimization problem over ug. Consequently, for all 2e, r, i, the optimum satisfies:
V up( (oo Cur( Ze, T,, O,
(26)
dug M (t,1711
N (zri - (1)x)x,(11) uionee (Z1, 2t x Rd
+ (1 r)
(z) = 0,
t.
dl
(27)
dug du (Z.:
Ug 1, + (1 - r)
uà, Nen.
(28)
dl
1
U (2,, 1
ue TEN (z,, Zz, l)dl.
(29)
As detailed in Geng et al. (025a).. the constant term in the last equation is zero. Furthermore, to minimize the (0) loss with Us Ur, the same assumption in Assumption I makes it equivalent to minimizing La Ze, r, t) for every fixed ze, r, t, where
r,, =a / N (21:(1 x) PT) (2,,,
(a vz + a) ug- (ze, r, 8)) 112 p(z)dx.
Applying the same zero-ge zero-gradee prr--gradeen condition, the optimal solution of up (z1, r, t) satisfies:
U (Z,,,,) - (1 - -a) (0. 8) = a u0
Thus, when 1:
u (2. = ug (2),t),
(30)
when a < 1:
ug (2,, r,t) = a up IFN Zz, t) + (1 a) ue (z., r,
(31)
= a. ug IPM Zz, t) + (1 aja 1o 8) + (1 a)² u00. (,,,
(32)
(33)
(x) u TEw (Zis 4x);
(34)
k=0
where 44 =r - (1-a)*(2 (t-)), r). Let d = (1-a))* thus Ad d - = a(1 - a)*, so:
U T,t, Σ
(Ax )b+1) (0 (tr)) + delt - r)),
(35)
k=0
(Th -1-4)) U6 Tk)
(36)
k=0
where Tr. = + Aa - r). This is left Riemann sum, and as - 0, 2. +
and:
1 f ug now (z, E)dt = u6xx (2t,
lim (zt, a
(37)
t t-r
20

---

Assumption 2.
The optimal velocity function uggTen(21, and the parametric velocity function (,,, satisfy the following properties:
Lipschitz Continuity of Optimal Velocity: There exists constant L1 O such that for all 11,22 E [0, 1] and corresponding Ztg, Ztz
10o.ren(2 (1) Ug 1PM Etg 12)1l2 S Lilti ////
(38)
Lipschitz Continuity of Time Derivative: There exists a constant To such that for all t1 [r, (where r [0,1] 1)) and corresponding Ze, Z1g
du, duo (Zt, t2) dt
I L2ft1 - t2l-
(39)
dt
Boundedness of Gradient and Time Derivative: There exist constants C1, C2 0 such that for all r, E [0, 1] with r S t, and for the corresponding zt.
duo Zz, T,
II (0 C1 and
S C2
(40)
dt
The Lipschitz continuity of the score function is a common assumption widely adopted in the the- oretical analysis of score functions in diffusion models Chen al. (2022) 2023). Here, assume the Lipschitz continuity of the optimal velocity function ug rru (Zt, () and the derivative of ug over
t, du.(
de The boundedness of Vø uo r, f)]l2 holds in reality as we using gradient clip over each element in V uz(2,, to stablize training. Thus, under finite a:
|1 (2,, ugg r2,,r,t)|22
(Nr Tk) ug V.r (2r, 7)) dr
k=0 T8+t
(41)
T2
| (2n, Tk) (m(T (2)
T6+1
(42)
Tk
L1 (Tk - -T
(43)
1
Ty
(Th T2+1)2
(44)
t-r r
2
Li
(1 a) 2k
(45)
k=0
L1
(46)
2 - a
| () (2,,
r, f)22) decreases with and lim | (z), ue (z,, t)]l2
O; hence the two optimal solution coincide - 0, which align with our previous proof.
E.3 NON-SYMPTOTIC DISTANCE BETWEEN GRADIENTS FROM a-FLOW AND MEANFLOW
From Equation g when U.. ei the gradient of a-Flow loss is:
21

---

(0).
E
[a
Vo (01,,,1;1) (uo(2,,1,1) - (a vr + (1 a) ug (2s, T])))]
t,r,z
E
V p (z r,t) (uo(z), vv)]
Gr
E
(2o(2), Vo (z ((00 - ug (2))
a
(1- a)) Vo uz r, 1 {" dug (2 dr
VOLTF (0) + E
2,1,2t
dl
(47)
And the gradient of MeanFlow loss could be written as:
V (0) = Br,r, vouu(zn (uo(z,, - - (t-r) -r) dug- (Zz, r, ()
df
(48)
(t (t-r). Vo ro,,,,,) duig- Az1 r, L)
VoLTEN(O))
+ E tinise
dt
Thus, the distance between VeCa(O) and V V (0) is:
(f) - VHLH((0)) (0)122
(2 (2 r, 1) - /¹² (1 (x-
dup (T),
dup (2z,
dr
dt
dl
1 ,
(2,,
dus- (Z),,,)
dus (zr, T, T)
V uo 1, 00112
E
dr
dr
t.r,z
dl
dl
dl
SG E
Lz r)dr + a(l - r2
tirize
a
/ L2 r)2 Cz m)
=0 C,E,r
(49)
Because 1,r E 0,, 1]. Ber (2 of distribution of r and f.. Thus the upper bound of VoL. (0) VoLue ||, is linearly depend on a, and will vanish to when a 0, align with our Theorem that VoL 0 VoLa-oof (0)
(2(1-r) ) will be finite constant no matter the choice
E.4 VARIANCE OF a-FLOW GRADIENT
Let X Vouo(zz,),)) (uo(Ze,T,() v2) and Y Vjuo(zer),)) fe dugg (zr. dr.
22

---

V[VoCa(0)]
y [a- u... (1 (u)
( (a v, + (1 - a) (Z., s)])]
V [X +(1 [))]
2
[ + +11 a)
V r
V GM
2
VV 1) +11 V
E
|
f.rizt
1 pt dus (1
27
v VoL 0) + at)
uo r, 1)2
dr
dl
2
VoLTFN + Ci C2 V [tt [(t-
Thus, the upper bound of the variance VeCnr(O) increases as a approaches O.
F ANALYSIS DETAILS
The detailed implementation of DiT-B/2 provided in Table where adopt the DiT-B/2-non-cfg setting. For loss evaluation, at each checkpoint we use batch size of 128 and run 1000 iterations to compute the mean loss along with its 5% and 95% percentiles, which are reported in the figure. To measure the cosine similarity between different losses, we calculate V LTF,, VL and Vre on the same batch and then compute their pairwise cosine similarities, which is defined as:
a
cosine similarity (a, |al| |b]
given two vectors a, This procedure is also repeated over 1000 iterations obtain the mean similarity and its 5% and 95% percentiles, as shown in the figure.
G IMPLEMENTATION DETAILS
Implementation details for models trained on ImageNet- IK 256 are shown in Table For Kinetics- 700 17 2562, we employ DiT-B/2 architecture trained for 600K iterations, with and without CFG. For CFG training, we set guidance scale w 1.0 and classifier weight = 0.7.
H ADDITIONAL EXPERIMENTS
H.1 ABLATION STUDY OVER BATCH SIZE
Training diffusiion/fowwbaseee models can be challenging due to the high variance of their gradients.
Past research Zhou al. (2025) Karras et al. (2022] 2024) often used large batch sizes (1024 or 4096) to mitigate this issue. In this section, implementation details in Table 3 for an additional 60 epochs using a large batch size.
fine-tune McanFlow-XL/2 model (with
As shown in Table 4 batch size of 512 achieved the best I-NFE FID of 3.05 and FDD of 164.3. A batch size of 1024, however, yielded the best FDD of 93.4. Overall, a batch size of 1024 performed McanFlow-XLL
well across all metrics, so designate this configuration applicd to fine-tune the ShanFlow-LLP model, leading to the McanFlow-Xil XL/2+ results in Table Our proposed Fllow-X11/24 model outperforms MeanFlow-XI./2+ in several key metrics: 1-NFE
The same setting is
23

---

Table 3: Configurations on ImageNet 256x256. B/2-non-cfg is our ablation and analysis model in the main text.
Configs
DiT-B/2-non-cfg
DIT-B/2
DIT-XL/2
DiT-XL12+
Network Architectures
Params (M)
131
131
676
676
FLOPs (G)
23.1
23.1
119.0
119.0
Depth
12
12
28
28
Hidden dim
768
768
1152
1152
Heads
12
Patch size
2x2
2 2
2 x 2
2 x 2
Training hyperparameters
Training steps
400K
1.2M
1.2M
1.2M
Batch size for training
256
256
256
256
Fine-tuning steps
75K
Batch size fine-tuning
1024
Dropout
0.0
Optimizer Ir schedule
Adam Kingma & Ba 2014
constant
Ir
0.0001 (0.9, 0.95)
Adam (81,
Weight decay
0,0
EMA half-life
6931
Gradient clipping
16
Autoencoder used
-d-vaaeft-ema
a-Flow hyperporameters
Ratio =
Table
(b)
25%
50%
50%
(r, sampler
logitnorm(-D0 (0.4, 1.0)
Ü,,t
Table 5 a
Whether to EMA for ug-
Table
No
No
No
Adaptive weight
Table
(b)
w xx/ (I
+
Schedule of a
25
25
k-
600K
600K
Table (b)
ke
1.2M
1M
1M
n
Table
(c)
x 10-3
x 10
x 10-3
CFG truining
w
1.0
0.2
0.2
1.0 [0.0, 1.0] No
0.92
0.92
CFG triggered is in Whether EMA for CFG
|0.0, 0.75] No
(0.0,0.75) No
2-NFE Sampling
Method
ODE
ODE
consistency 0.55
consistency
Intermediate timestep
FID (2.58 3.06), I-NFE FDD (148.4 165.7) and 2-NFE FID (2.15 2.16), only in 2-NFE FDD (96.8 vs. 93.4). These results demonstrate the overall effectiveness of our a-Flow method. Notably, the results in Table are obtained using labels sampled from the ImageNet dataset distribution, whereas the results in Table [ use randomly generated labels. In general, sampling labels from the ImageNet distribution leads lower FID scores compared to using random labels.
H.2 ABLATION STUDY OVER a-FLOW DESIGN SPACE
This section contains an ablation study on a-Flow, specifically for a E 1). We use DiT- B/2-non-cfg model (see Table 3) that is pre-trained on flow matching for 200k iterations and then fine-tuned on a-Flow for another 200k iterations, Across all experiments, a remains a constant, and the ratio of L is 25 %.
24

---

NFE
NFB
Batch Size
Algorithm 3 a-Flow: Sampling
FID
FDD
FID
FDD
t1 t2 timest.aps randn,like (x)
CN sequeenc of
256
3.13
167.2
2.31
97.1
512
3.05
164.3
2.21
95.2
zange (N):
1024
3.06
165.7
2.16 93.4
I a conal starcy.samp ing:
2048
3.29
169.6
2.10 96.6
2 = z tn (z, r=0, t=tn)
4096
3.13
168.9 2.16
95.1
tm randn like (x1
elir ODD.samp1 ingz
Table 4: Ablation study over the fine-tuning batch size using the data distribution over class labels.
(tn tm) (2 v-tm, E-tn)
0,,t
Ug
FID
FDD
Loss weight
FID
FDD
14g
EMA
188.1
1761.6
964.6
Non-EMA
319.0
4009.9
1/
(1
1
0.5
55.0
918.5
EMA
202.8
1832.3
w
1/
+
52.2
883.6
02
Non-EMA
59.2
964.6
(I
49.7
845.2
Reformulate the training objective.
(b) Adaptive loss.
FID
FDD
Method Shortcut Model
FID
FDD
59.8
1017.3
10
49.7
845.2
i (( f/ar)
59.2
964.6
x
46.2
860.8
Adaptive loss
49.7
845.2
2 x 10
50.3
833.0
0.005
45.6
857.8
x 10-3
57.2
863.7
McanFlow
43.3
822.3
(c) Consistency step ratio.
(d) Overall ablation study.
Table 5: Ablation study a-Flow.
Training objective. Here, we set a 10 Table shows that the model only converge when Ü..t was set to and without using EMA for ug- This is key difference from Shortcut Models Frans et al. (2025), which set U. = Ug We suspect their objective only works when a is larger (e.g., 0.5).
Geng al. (202aa) uses an adaptive weight: w = 1/ (1 +c) = 1/ + c). 12h. could deniv limo-vo Ca aCCr. When is close O, approximate LAF as La/a. This gives us a new adaptive weight, w = 1/ (C,/a++ c) a/ (Co + = a/ (I) A12 + e) as both e and a is very small. As shown Table (()), this new weight performs better empirically, especially compared to the original MeanFlow adaptive weight.
Adaptive loss. From Equation
Consistency step ratio. Ablating the in Table 5 consistency step ratio. This value was then used as the clamping value for our schedule.
(c) reveals that 5 10-3 to be the optimal
Table shows that by combining these improvements, our discrete a-Flow approach significantly reduces the performance gap between Shortcut models and the McanFlow model.
H.3 OPTIMAL RATIO OF r = 1 UNDER DIFFERENT NFE
In this subsection, extend the experiment results in Table 2 (b) to NFE 5. As shown in Table 6 the optimal ratio of r = required to achieve the best generation quality for NFE is also 75%. We hypothesize this is because Flow Matching inherently possesses degree of few-step generation capability. Therefore, when mixing the training of standard Flow Matching and a-Flow, varying the ratio introduces trade-off between single-step and few-step generation.
25

---

Model
NFE
NFE
5FE
%
Schedule
FID FDD
FID
FDD
FID
HDD
Constanto.0
46.0 879.6 44.3 867.7 44.0 843.8
0%
Sigmoid 400K
40.4 822.5 38.9 811.8 38.2 779.0
Constanto.0
44.4 844.1 42.1 836.3 41.3 817.8
25%
40.0 785.4 37.1 782.9 36.9 770.0
Constanto.0
43.9 844.1 42.1 836.3 38.4 783.3
50%
40.2 781.0 37.1 775.0 35.5 743.5
Constanto.o
43.1 819.2
38.5
787.6
36.0 752.1
75%
Skg
42.2 810.5 36.2 754.7 33.7 716.1
Table Optimal Kt = r under different NFE.
NFE
NFE 2
Model
FID
FDD
FID
FDD
FMoK->200K + MeanFlow2oox
42.3 798.1 37.6 772.4
TFMok -200 MeanFLoWanDK---2
42.3 821.8
37.2 780.2
Sigmoida
40.2 781.0
37.1 775.0
REP AOK 400K Mean HeanFLow.ook-eeom
32.0 667.4
25.5 594.3
REP 400K + bngg
30.7 657.1 25.3 601.5
Table 7: Finetuning from pretrained flow models.
H.4 FINETUNING FROM PRETRAINED FLOW MODELS
In this subsection. we study the performance of a-Flow by finetuning from pretrained flow mod- els. Specifically, We have conducted experiments comparing a-Flow with standard MeanFlow fine- tuning when starting from pre-trained Flow Matching (FM), Trajectory Flow Matching (TFM), and FM + Representation Alignment Yu et al. (2025) (REPA) models. We use DiT-B/2-non-cfg configs Table 3 for backbone models. Both FM, TFM and FM + REPA are pretrained by ourselves. Ex- periment results shown Table 7 A specific configuration for a-Flow deserves mention: for the experiment labeled REPAOK-440K + the pre-trained model was provided at the 400K iterations, set the initial value of to 0.5 instead of 1.0 at the beginning of the a-Flow fine-tuning phase.
we adjusted the initial a parameter. Since
Under direct epoch-to-epoch comparison, starting from the pre-trained model, a-Flow sistently outperforms MeanFlow for one-step generation (NFE 1) and achieves comparable perfor- mance for two-step generation. Specifically, under NFE 1, a-Flow (FID 40.2, FDD 781.0) demon- strates significant gains: it surpasses the FM + MeanFlow baseline by 2.1 FID and 17.1 FDD, and surpasses the TFM MeanFlow baseline by 2.1 FID and 40.8 FDD. Furthermore, when initial- ized with the same FM+ REPA model, F-Flow (FID 30.7, FDD 657.1) still outperforms MeanFlow by 1.3 FID and 10.3 FDD one-step generation. Since the a-Flow method inherently consists of three sequential stages: TFM pre-training, a-Flow annealing. and MeanFlow fine-tuning, these re- sults directly demonstrate the effectiveness of the a-Flow annealing stage. This stage particularly beneficial for improving the quality of one-step sampling.
H.S DISTILLATION
In this subsection, conduct the distillation experiment. Specifically, distilled the a-Flow model (using the DiT-B/2 architecture) with a DiT-B/2-REPA teacher model, which we trained for 400K iterations.
The experimental results, summarized in Table clearly demonstrate that distillation improves the performance of a-Flow for both NFE and NEE 2 settings. The distilled a-Flow model, using the Sigmoid - schedule (the first line in Table above), outperformed the a-Flow trained from seratch with the Sigmoidok-x400k schedule (the last line of the Table 2 (a)). For NFE 1, the
26

---

NFE 1
NFE 2
Schedule
FID
FDD
FID
FDD
36.6 756.9
33.2
701.3
Sigmoid - 150K
37.1
749.2
33.4
702.9
10gmo11d100K-2000
37.8
757.5
33.5
704.7
Sigmoid 250K
36.7
743.2
34,1
706.9
Sigmoiduk
36.9
738.9
33.6
704.3
Table 8: Distillation of a-Flow-B/2.
MeanFlow
Variance of gradient nomm
MeanFlow
0.04
d-Fllow
0.3
a-Flow
0.03
Variance
0.2
0.02
0.1
0.00
0.0
100K
200K
300K
100K
200K
300K
400K
(a) Variance of loss
(b) Variance of gradient norm
Figure 9: Stability comparison of a-Flow and MeanFlow. The green dotted line indicates the transi- tion from TFM pretraining a-Flow annealing, while the blue dotted line represents the subsequent transition from a-Flow annealing to MeanFlow training.
distilled model achieved improvements of 3.4 (FID) and 28.5 (FDD). For NFE 2, the distilled model showed improvements of 3.9 (FID) and 81.6 (FDD).
It is important to note that the optimal distillation schedule shifted to Si sigmoid This change expected, as the teacher model (DiT-B/2-REPA) was already pretrained. This pretrained knowl- edge allows the distillation process to reducing the initial training stage.
H.6 STABILITY ANALYSIS BETWEEN MEANFLOW AND F-FLOW
In this subsection, we analyze the training dynamics of MeanFlow and a-Flow by evaluating the variance of the MeanFlow loss and the gradient norms throughout the training process. Experiment results are shown in Figure 9 where the variance for each loss and gradient norm is estimated over 500 iterations.
As shown in Figure the variance of the loss and the gradient norm for a-Flow are lower than those for MeanFlow during the TFM pretraining and a-Flow annealing stages (to the left of the blue dot line). This indicates a more stable training process for a-Flow in these early stages. This stability aligns with the statement that a-Flow annealing effectively transitions the training objective from "high-bias, law-variance" state to the necessary "low-bias bias, high-variance" state.
Despite these early stability advantages, the a-Flow formulation ultimately trains the full MeanFlow loss in its final stage (as + 1). Consequently, a-Flow cannot fully circumvent the inherent instability introduced by the MeanFlow objective.
H.7 MORE GUIDANCE METHOD
Beyond standard Classifier-Free Guidance (CFG) training, we explored an alternative guidance method called u-target guidance. This method is based on optimizing the following loss function:
Lu.go (0) = E Uuo 2,, t|c) (1 Wa.ge) Utaget (2,, r, - Wait uo r. t|Ø) | (50)
t.r.
where Wa.gt is the guidance strength. The specific definition of the target vector, Uraget (2, r, t), depends the underlying loss formulation being used:
27

---

NFE
NFE
Guidance method
FID
FDD
FID
FDD
u-target guidance
25.1 645.1
9.5 392.7 445.8 7.7 365.2
CFG guidance
10.0
Table 9: Ablation of guidance methods.
For MeanFlow loss, Uraget (ee r) = Uz - - -7) dug (x),
de
For a-Flow loss, Mtaget Et, = (a Dz.t + +11 a) (zu, 8)).
The fundamental difference between u-target guidance and CFG training lies the unconditional prediction component. The u-target guidance uses un [z, (z, r, t|Ø) for the unconditional prediction conditioned, instead of unconditional prediction (z 1, (/23) employed in CFG training.
We compared u-target guidance with CFG guidance using MeanFlow-BB Fl2 architecture, training on ImageNet-1K 1K 2562 for 400K iterations. We set Wa.gt 2.5. Experiment results are shown in Table The u-target guidance performs significantly worse than CFG guidance when using NFE and is still slightly worse even at NFE 2. Thus in the main paper we use the CFG guidance.
H.8 DISCUSSION OF THE GRADIENT CONFLICT
In this subsection, discuss the gradient conflict between VOCTAN and VeLTo in Figure 3a In general, for the minimizer x² of some arbitrary, regular function f(x) g(x) + h(r), one can trivially show the relation Vzg(x*)= -Va((=") from the necessary condition Vzf(x*) 0.
However, our empirical analysis in Figure 3a shows this gradient relationship from the early start of the optimization process, where the gradients of LrAM and Lrc are already strongly negatively correlated (specifically, < -.9).. Crucially, this correlation becomes weaker (around -0.7) as training progresses.
This behavior is opposite to what would be expected if the negative correlation simply resulted from convergence to stationary point. Instead, the decreasing correlation indicates that the two loss components exert conflicting optimization signals early in training, and gradually become more aligned as the model parameters adapt.
This suggests that the observed gradient dissimilarity reflects different underlying mechanism than the trivial relation implied by v=f(=*) = O. This mechanism could be better explained by how far apart the corresponding minimizers arg min g(x) and arg min h(x) are. This evolution likely reflects the interaction between two loss components whose minima are initially far apart: early in training. their gradients point toward distinct optima, leading strong opposition, whereas as the model parameters evolve and the shared representation (and with the optimizer's preconditioning reweighting the directions) begins to satisfy both objectives, the corresponding descent directions become partially aligned.
LLM USAGE I
disclose the usage of Large Language Models in this As requested by the ICLR 2026 policy
section. LLMs were primarily used in two capacities:
Coding assistance for experiments. LLMs provided code auto-completion functionality to ease the process of implementing and analyzing the experiments.
Writing assistance for paper writing. We used LLMS to assist with grammar and phrasing validation while working the submission.
28

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J RANDOM VS BALANCED CLASSES FOR FID COMPUTATION
We treat EDM series (Karras et al. 2022 2024) as the standard in FID (Heusel et al. 2017) eval- uations, which use a randomly sampled class label (from 0 999) for each sample in constructing 50,000 synthetic examples with the model. We found a curious way to decrease the FID values by up to 10% by using "balanced" class sampling: instead of using 50,000 independently sampled random classes, one can generate 50 samples for each of 1000 classes. This greatly improves FID results, but not FDD (i.e., Fréchet Distance the DINOv2 (Oquab et al. (023) feature space) or FCD (Kynkäänniemi et al. 2022 (i.e., Fréchet Distance in the CLIP-LL-based (Radford et al. 2021 feature space).
Since it is not standard practice in the community, we only report it separately from the random class sampling results and with the appropriate notice. But we emphasize that it might be a more reasonable way to evaluate FID since it reduces the variance (we are less likely to sample an unlucky set of classes). We provide the results for it in Table I0
NFE
NFE
Method
Class sampling
Params Epochs
FID
FDD
FCD
FID
FDD
FCD
McanFlow-XI./2
Random U 1000
676M
240
3.47
185.8
3.39
2.46
108.7
2.40
a-Flow-XI./2 (ours)
Random 100000
676M
240
2.95
164.6
3.14
2.32
105.7
2.42
a-Flow-XL/2+ (ours)
Random U 1000
676M
240+60
2.58
148.4
3.07
2.15
96.8
2.31
McanFlow-XII -1./2"
Balanced
676M
240
3.33
182.8
3.34
2.26
106.1
2.36
a-Flow-XLL Miw- (ours)
Balanced
676M
240
2.81
162.4
3.10
2.16
103.2
2.37
u-Flow-XL/2+ (ours)
Balanced
676M
240+60
2.44
147.2
3.04 1.95 94.6
2.30
Table 10: Balanced vs random class sampling for FID, FDD and FCD.
It is curious observe that while greatly improves FID results, FDD and FCD are barely affected. We believe that this constitutes one more reason for the community to switch from FID to more robust metries which correlate better with human perception, like FDD and FCD.
29

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K ADDITIONAL EXPLORATION OF THE MEANFLOW LOSS
0.8
nitoo OS
088
Averago CostPEry BECc)
Ear malo 75%
ratio 75%6
0.4
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Average ena Eyon
0.4
-04
B00K
200K Training Iteration
300K
40OK
100K
2008
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(V LEw', LTc)
(V LIFX, V Vee)
Cow
Avergg Coo Eura V
Eyp rato 75%
Avenge L VCye)
75%
0.4
0.4
0.4
0.4
300K
200K Training lteration
10CK
40CK
100K
7000K
120K
420K
Training lieration
cos (V LYPA, V Lue)
(d) cos (V LFH, VLF)
Crz milio
App rallo U/%
Average Co V.rc)
Cow
0.4
0.4
Avenge con(V
0.0
0.0
-0.8
-08
400K
105K
2008
300K
Training
Training
(e) ous (VLr,, VLFF)
(l) cos (V Lro, LTPM)
Figure 10:
Average cosine similarities between the gradients of different losses (LrEn, Lew, LCT., LND) for DiT-B/2 McanFlow model trained with 0% and 75% of flow matching.
30

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L
ADDITIONAL VISUALIZATIONS
MeanFlow- CeonFlow-DiI-XLA DDI-XL/22
K-Flow-DiT-XLJY a-Flow- DIT-X12
Figure 11: Uncurated samples (seeds 1-16) for Class 15 (robin) for NFE=1.
MeanFlow- D12-
a-Flow-D e-Flow-DiT-XLA DiT-XL/2
a-Flow- D +
Figure 12: Uncurated samples (seeds 1-16) for Class 15 (robin) for NFE=2.
31

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MeanFlow- DDI--L/22
a-Flow- D/2-XL/2
a-Flow-Dii DiT--L/22 +
Figure 13: Uncurated samples (seeds 1-16) for Class 29 (axolotl) for NFF=1.
MeanFlow- DDI--LL22
a-Flow-Dizz DiT-XL/2 + a-Flow- DiT-XLL
Figure 14: Uncurated samples (seeds 1-16) for Class 29 (axolotl) for NFE=2.
32

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MeanFlow- MeanFlow-Dit-XLA DiT-XL/2
a-Flow DIT-XL/2
a-Flow- -iT--L2- +
Figure 15: Uncurated samples (seeds 1-16) for Class 33 (loggerhead) for NFB=1.
MeanFlow- MeanFlow-Dil-XLA DDI-XL/2
a-Flow- DLT-
a-Flow-DiTs - +
Figure 16: Uncurated samples (seeds 1-16) for Class 33 (loggerhead) for NFE=2.
33

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MeanFlow- DiI-XL/2
a-Flow DiT-XL/2
a-Flow- DiT-XL/2 +
Figure 17: Uncurated samples (seeds 1-16) for Class 88 (macaw) for NFE=1.
MeanFlow DDI-XLL2
a- DL
a-Flow-Dize DDT--I/2-
Figure 18: Uncurated samples (seeds 1-16) for Class 88 (macaw) for NFE=2.
34

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MeanFlow- DDI-XL/2
Q-Flow Di2-XL/2
a-Flow- -iT--LL
Figure 19: Uncurated samples (seeds 1-16) for Class 89 (cockatoo) for NFF=1.
MeanFlow- DiI--LL22
Q-Flow DL2-
a-Flow- DiT--X// +
Figure 20: Uncurated samples (seeds 1-16) for Class 89 (cockatoo) for NFE=2,
35

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MeanFlow- DiI-XL/2
a-Flow DiT-XL/2
a- Dii--Di--//2 +
Figure 21: Uncurated samples (seeds 1-16) for Class 127 (white stork) for NFE=1.
MeanFlow- DDII-LL22
a-Flow-Dizz DiT--X/22 + a-Flow- DLT-XL/2
Figure 22: Uncurated samples (seeds 1-16) for Class 127 (white stork) for NFE=2,
36

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MeanFlow- DDI-XL/22
a-Flow- DL2-XL/2
a- -iT--L-- +
Figure 23: Uncurated samples (seeds 1-16) for Class 279 (arctic fox) for NFE=1.
MeanFlow- DDI--L/22
a-Flow-Diie DiT-XX/2+ + a-Flow- DI2-XL/2
Figure 24: Uncurated samples (seeds 1-16) for Class 279 (aretic fox) for NFE=2,
37

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MeanFlow- DDI--L/2
a-Flow DiT-XL/2
a- -IN---I--//24
Figure 25: Uncurated samples (seeds 1-16) for Class 980 (volcano) for NFF=1.
MeanFlow- DiI-XL/2
a-Flow- DD
a-Flow-Dize DiT-XL/1
Figure 26: Uncurated samples (seeds 1-16) for Class 980 (volcano) for NFE=2,
38

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MeanFlow- DDI--LL2
a-Flow DiT-XL/2
a-Flow- G-Flow- -Iw-DiT-XL/2+
Figure 27: Uncurated samples (seeds 1-16) for Class 975 (lakeside) for NFE=1.
MeanFlow DiT-XLL2
a-Flow- DIT-XL/2
C-Flow-JJ G-Flow-DIT-- DDW---/I--/2
Figure 28: Uncurated samples (seeds 1-16) for Class 975 (lakeside) for NFE=2.
39


Reviews:
[{"review_id": "cswsiCMxSy", "reviewer": "Reviewer_Tpep", "summary": "This paper analyzes the MeanFlow framework for few-step generative modeling and introduces a generalized family of objectives called $\\alpha$-Flow. The authors show that the MeanFlow loss decomposes into two competing components—trajectory flow matching and trajectory consistency—whose gradients are strongly negatively correlated, leading to slow convergence. To address this, they propose $\\alpha$-Flow, which unifies MeanFlow, Shortcut Models, and flow matching under a single formulation. By gradually annealing the training objective from trajectory flow matching to MeanFlow (a curriculum learning approach), $\\alpha$-Flow disentangles the conflicting objectives, improving optimization efficiency. When trained from scratch on class-conditional ImageNet-1K with DiT backbones, $\\alpha$-Flow achieves consistent gains over MeanFlow, reaching new state-of-the-art FID scores for 1- and 2-step generation using standard architectures.", "questions": "The authors state that \"the 75% ratio yields worse NFE=1 but better NFE=2 generation results compared to the 50%-ratio version”. Extrapolating this observation, I wonder whether the optimal ratio grows with the target inference NFE. Have the authors checked this? More generally, have the authors run evaluations on larger NFE?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "- The paper introduces a comprehensive framework to train few-step generative models by unifying and streamlining existing approaches. This, combined with the experimental ablations, provides a clear recipe for practitioners.\n\n- The experimental section is complete, and provides a lot of insights.\n\n- Section C (Failed experiments) includes approaches tried by the authors that did not succeed. This is a great idea that should be more common.", "weaknesses": "- The authors fail to cite some related works like https://arxiv.org/abs/2406.07507\n\n- The fact that the gradients of L_{TFM} and L_{TC_c} are so negatively correlated is not surprising since adding the two losses and completing the square yields a constant value at the optimal vector field. Meanwhile, no such relationship exists between L_{FM’} and L_{TC_c}. Hence, it is reasonable that the correlation between the gradients of these two losses fluctuates around zero. From this perspective, there is nothing mysterious about the empirical analysis in Section 2. In fact, the negative correlation must hold between the gradients of any decomposition of a least squares loss.\n\n- In Section 4.1, it would be good to devote a paragraph to clarifying/proving how the MeanFlow loss is related to the \\alpha-Flow loss. The connection is stated in Theorem 1, but this statement is not obvious at a first glance. While proven in the appendix, having a sketch in the main text would help the reader.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "dcB2teg1Ph", "reviewer": "Reviewer_S4Q1", "summary": "The authors propose to train a MeanFlow model along a curriculum that transitions from flow matching to MeanFlow optimization. This curriculum is inspired by two observations: the MeanFlow objective can be decomposed into a flow matching loss and a consistency loss, and the two losses have conflicting gradients during training. By optimizing the flow matching loss (with a narrow solution space) first and then transitioning to the easier MeanFlow objective (with a larger solution space), the authors claim one can achieve better convergence. The curriculum is implemented by optimizing the $\\alpha$-Flow objective that interpolates between flow matching (at $\\alpha=1$) and MeanFlow (at $\\alpha=0$). Various annealing strategies for $\\alpha$ are explored in the paper, and $\\alpha$-Flow in general outperforms vanilla MeanFlow.", "questions": "- **[Q1] Why is there a large gap between FIDs in Table 1 and Table 2?** Did the authors use smaller number of training epochs for ablations?\n\n- **[Q2] How does $\\alpha$-Flow training from scratch compare to MeanFlow training initialized from a pre-trained diffusion / flow model?** I am curious how MeanFlow initialized with e.g., REPA [1] or REPA-E [2] DM would perform compared to $\\alpha$-Flow. I believe the embedding layer for the second time parameter can easily be implemented by copying embedding modules for the original time parameter in pre-trained DMs.\n\n[1] Representation Alignment for Generation: Training Diffusion Transformers Is Easier Than You Think, ICLR, 2025\n\n[2] REPA-E: Unlocking VAE for End-to-End Tuning with Latent Diffusion Transformers, 2025", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- **[S1] This paper is novel, as it presents a new perspective on the slow convergence of MeanFlow.** By decomposing the MeanFlow objective into a sum of flow matching and consistency training losses and showing that the two have conflicting gradients, this paper elucidates a possible source of instability and slow convergence in MeanFlow training.\n\n- **[S2] This paper is significant, as it provides a simple and practical recipe for stable training of MeanFlow.** By pinpointing a source of instability in MeanFlow training, the authors are able to develop a simple curriculum learning strategy for stabilizing convergence of MeanFlow. The authors provide strong evidence (on ImageNet 256$\\times$256) that the curriculum indeed yields better MeanFlow models. In particular, experiments show one can use a lower flow matching ratio with $\\alpha$-Flow, implying that the proposed curriculum effectively mitigates the tension between flow matching and consistency training.", "weaknesses": "- **[W1] Experiments in Section 5 only indirectly support the claim that $\\alpha$-Flow mitigates the conflict between trajectory flow matching (TFM) and trajectory consistency (TC) objective.** An experiment such as an analogue of Figure 2 for $\\alpha$-Flow training could directly support the hypothesis that the proposed curriculum resolves the tension between TFM and TC.\n\n- **[W2] The paper is missing a simple but crucial baseline: training from a pre-trained diffusion model.** In particular, with diffusion models (DMs) trained via representation alignment offering much better generative performance, initialization from such DMs may be a simpler alternative to $\\alpha$-Flow without the added complexity of tuning the alpha decay schedule.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Mzfp93AmRG", "reviewer": "Reviewer_gT18", "summary": "This paper provides a detailed analysis of the MeanFlow objective for few-step generative modeling. Motivated by the observation that MeanFlow dedicates the majority of its training to border-case supervision (i.e., fitting to vanilla flow matching with $r=t$), the authors conduct in-depth investigation into the MeanFlow objective. Specifically, they show that the MeanFlow loss naturally decomposes into two components — trajectory flow matching ($L_{TFM}$) and trajectory consistency ($L_{TC_c}$) — whose gradients are strongly negatively correlated, leading to optimization conflicts. The authors further discover that such issue can be partially mitigated by introducing an auxiliary flow matching loss ($L_{FM'}$). \n\nIntended for a training recipe with higher efficiency, the authors propose $\\alpha-$Flow, a unified framework that interpolates between flow matching and MeanFlow via a curriculum parameter $\\alpha$. The framework includes several prior objectives (Flow Matching, Shortcut Model, Consistency Training) as special cases, and progressively transitions from $L_{TFM}$ to MeanFlow through an $\\alpha-$schedule. This framework achieves improved performance on ImageNet $256\\times256$ compared to various previous baselines under comparable compute budgets.", "questions": "1. **Distillation applicability**\n\nCould the proposed $\\alpha-$Flow framework be extended to a *distillation* setting, where the conditional velocity $v_t$ is replaced by a teacher model’s prediction? If so, would this substitution affect any design choices?\n\n2. **Counterintuitive claim on $L_{TC_c}$'s optimization**\n\nIt is well known that the JVP term in the consistency objectives introduces strong training instability and is notoriously difficult to optimize. But in the paper, the authors argue that, without a boundary condition, $L_{TC_c}$ is easier to optimize given its unconstraint solution space. Could the authors elaborate on this seemingly contradiction? Is this claim mainly referring to that it's easy to optimize $L_{TC_c}$ to 0 but not to the solution space intersecting with $L_{TFM}$?\n\n3. **Adaptive weighting strategy**\n\nFor $\\alpha = 1$, the derived adaptive weighting is applied on the vanilla flow matching objective, which could potentially break the L2 loss required by the flow matching training recipe; yet, for $\\alpha \\to 0$, no adaptive weighting is not applied to the more challenging JVP-based objective. Although this empirically yields positive results, the design choice seems counterintuitive. Could the authors offer further intuition or justification for why this adaptive weighting behaves favorably in practice?\n\n4. **Fine-tuning the model**\n\nThe trajectory flow matching phase in $\\alpha-$Flow. effectively serves as a warm-up for stabilizing the model from random initialization. Have the authors experimented with starting from a pretrained FM model instead?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. **Insightful empirical analysis**\n\nThe decomposition of the MeanFlow objective and the empirical analysis of gradient correlations in Section 3 provide a novel perspective on the training dynamics of the trajectory-based generative modeling. These analysis not only provide clear motivations for the proposal of $\\alpha-$Flow, but also effectively deepens the understanding of optimization behavior in few-step generative models, which is timely and valuable given the growing research focus on this field.\n\n2. **Comprehensive and clear ablation studies**\n\nFor each design choice introduced in the proposed $\\alpha-$Flow framework, the authors conduct extensive ablation experiments accompanied by clear explanations that justify their settings. These analyses substantially improve the clarity of presentation, strengthen the empirical grounding of their claims, and enhance the reproducibility of the overall training recipe.\n\n3. **Improved empirical results**\n\nAcross all comparable settings, $\\alpha-$Flow demonstrate superior performance over multiple baselines, including MeanFlow, FACM, and Shortcut Models. The improvements are consistent across different model scales and sampling steps, underscoring the effectiveness and robustness of the proposed framework.\n    \n4. **Detailed limitations and failed experiment discussions**\n\nIt is commendable that the authors explicitly discuss both the limitations of their framework and the failed experimental attempts (Appendix B–C). Such transparency is rare and valuable: it not only helps future researchers avoid redundant directions but also provides practical insights into the challenges of optimizing few-step generative models at scale. This openness meaningfully contributes to the community’s collective understanding and progress.", "weaknesses": "1. **Stability of $L_{TC_c}$**\n\nThe proposed method does not directly address the well-known stability issue of the $L_{TC_c}$, which the authors also acknowledge in Appendix B. This instability has been repeatedly identified as the primary bottleneck in scaling consistency-based methods to large-scale applications such as text-to-image or text-to-video generation (Lu & Song, 2024; Chen et al., 2025; Zheng et al., 2025). As a result, while the paper offers valuable insights into the gradient conflict within the MeanFlow objective and presents improved training heuristics, the overall scope remains somewhat limited. That said, since the authors explicitly note that “our framework should not be viewed as a silver bullet for addressing instability,” it would nonetheless strengthen the paper to include an empirical comparison of training stability between α-Flow and vanilla MeanFlow (e.g., visualizations of training losses, gradient norms, or JVP magnitudes as $\\alpha \\to 0$).\n\n2. **Comparisons with contemporaneous works**\n\nIn Section 4.1, the authors demonstrate that $\\alpha-$Flow subsumes Consistency Training (CT) under certain network parameterizations. However, the paper does not include corresponding CT reproduction results in Table 1. Including such a baseline would substantiate the claim that $\\alpha-$Flow not only unifies but also improves upon prior consistency-based methods.\n\nMoreover, the proposed framework shares conceptual similarities with SplitMeanFlow (Guo et al., 2025). While the authors briefly mention this work in the appendix, a more direct discussion or empirical comparison—highlighting differences in optimization behavior, efficiency, or loss design—would enhance the paper’s soundness and contextual positioning within the recent literature.\n\n3. **Supporting evidence of \"$L_{TC_c}$ is easier to optimize\"**\n\nThe paper claims that $L_{TC_c}$ has a “larger solution manifold” and is therefore easier to optimize, but this assertion appears counterintuitive, as the JVP term in consistency objectives are generally known to suffer from great instability during training. A more rigorous justification (or some empirical results) would make the argument more convincing and clarify the optimization dynamics of $L_{TC_c}$.\n\nLu, Cheng, and Yang Song. \"Simplifying, stabilizing and scaling continuous-time consistency models.\" arXiv preprint arXiv:2410.11081 (2024).\n\nChen, Junsong, Shuchen Xue, Yuyang Zhao, Jincheng Yu, Sayak Paul, Junyu Chen, Han Cai, Song Han, and Enze Xie. \"Sana-sprint: One-step diffusion with continuous-time consistency distillation.\" arXiv preprint arXiv:2503.09641 (2025).\n\nZheng, Kaiwen, Yuji Wang, Qianli Ma, Huayu Chen, Jintao Zhang, Yogesh Balaji, Jianfei Chen, Ming-Yu Liu, Jun Zhu, and Qinsheng Zhang. \"Large Scale Diffusion Distillation via Score-Regularized Continuous-Time Consistency.\" arXiv preprint arXiv:2510.08431 (2025).\n\nGuo, Yi, Wei Wang, Zhihang Yuan, Rong Cao, Kuan Chen, Zhengyang Chen, Yuanyuan Huo et al. \"Splitmeanflow: Interval splitting consistency in few-step generative modeling.\" arXiv preprint arXiv:2507.16884 (2025).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "SczP2ckRpp", "reviewer": "Reviewer_9SDg", "summary": "This paper provides a theoretical and empirical analysis of MeanFlow, a recently proposed \nframework for few-step generative modeling. The authors decompose the MeanFlow objective \ninto trajectory flow matching and trajectory consistency components, \nrevealing that their gradients are strongly negatively correlated during training, causing \noptimization conflict. Motivated by this insight, they propose α-Flow, a unified family of \nobjectives.The paper is well-motivated and clearly written.", "questions": "1. **Why does curriculum learning specifically help?** Can you show that gradient conflict \n   decreases during α-annealing? Is it warm-starting or conflict reduction?\n\n2. **How does α-Flow perform on other datasets and resolutions?** ImageNet-256 is the only \n   evaluation—generalization unclear.\n\n3. **What if you train MeanFlow with just α-Flow's pretraining phase (L_TFM for 150K steps), \n   then fine-tune on L_MF?** Does curriculum truly outperform simple pre-training?\n\n4. **Why is the manifold interpretation important?** The paper claims L_TCc has a large manifold, \n   but provides no measurement or proof. Does manifold size actually affect optimization?\n\n5. **Can you provide theoretical guarantees for α-Flow?** Does curriculum learning provably \n   reduce optimization difficulty or bias?\n\n6. **How does α-Flow perform with other guidance methods** (not just classifier-free guidance)? \n   Why does guidance cause instability?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. **Clear problem motivation and analysis**: The paper provides valuable insights into *why* \n   MeanFlow works by decomposing its objective into interpretable components. The gradient \n   analysis in Figure 2 effectively demonstrates the negative correlation between L_TFM and \n   L_TCc gradients.\n\n\n2. **Well-motivated curriculum learning strategy**: The three-phase training approach (trajectory \n   flow matching pretraining → α-Flow transition → MeanFlow fine-tuning) is well-justified by \n   the observed gradient conflicts and is intuitive.\n\n\n3. **Thorough ablation studies**: Table 2 and Table 5 provide detailed ablations on curriculum \n   schedule, flow matching ratio, training objectives, adaptive loss weights, consistency step \n   ratio, and more. This demonstrates systematic investigation.", "weaknesses": "**α-Flow is an incremental generalization**\n- The core idea is straightforward: interpolate between L_TFM (α=1) and MeanFlow (α→0) via \n  curriculum learning.\n- While Theorem 1 unifies existing methods, this is largely a mathematical observation rather \n  than a conceptual innovation.\n- The curriculum learning strategy is intuitive but not fundamentally novel—using easier \n  objectives first is a well-established practice.\n\n**Gradient analysis is empirical, not theoretical**\n- Figure 2 shows empirical gradient correlations but provides no theoretical explanation for \n  *why* the gradients conflict.\n- The paper hypothesizes: \"L_TCc has a very large solution manifold compared to L_TFM's \n  narrow manifold\" (Section 3.2) but provides no proof or rigorous analysis.\n- No formal characterization of when/why conflicting gradients occur in optimization of \n  multiple objectives.\n\n**Curriculum learning for generative models is not new**\n- Curriculum strategies in diffusion/flow models have been explored:\n  - Progressive distillation (Salimans & Ho, 2022) uses curriculum-like approaches\n  - EDM (Karras et al., 2022) employs careful scheduling\n  - Consistency model training already uses staged objectives\n- The novelty of applying curriculum to MeanFlow is limited.\n\n**Main contribution is improved MeanFlow, not a fundamental advance**\n- α-Flow is essentially MeanFlow with curriculum learning + reduced flow matching ratio\n\n\n**Theorem 1 proof is incomplete**\n- The proof in Appendix D.2 shows α-Flow reduces to existing objectives at specific α values \n  but doesn't establish theoretical properties (convergence, optimality, bias-variance).\n- For MeanFlow: proof shows ∇_θ L_MF(θ) = ∇_θ L_α→0(θ) as α→0, but this is just asymptotic \n  equivalence, not equivalence at finite α.\n- Missing: formal analysis of how well finite-α approximates α=0.\n\n**Manifold hypothesis (Section 3.2) is unsupported**\n- Hypothesis: \"L_TCc has a large solution manifold; L_TFM has a narrow manifold.\"\n- No measurement of manifold size, rank, or structure.\n\n**Notation could be clearer**\n- L_TFM vs. L_FM′: distinction subtle and easy to confuse\n- L_TCc notation for consistency loss is non-standard (why \"Cc\"?)\n- Multiple uses of subscript/superscript make equations hard to parse\n\n\n**Missing experimental details**\n- How is \"gradient similarity\" computed exactly? (Appendix E brief on this)\n- How many iterations used for averaging in Figure 2?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)