SoFlow: Solution Flow Models for One-Step Generative Modeling
Authors: Tianze Luo
Haotian Yuan
Zhuang Liu
Venue: ICLR 2026 Poster (2026)
Keywords: Flow Matching Models
Consistency Models
One-step generation
Primary area: generative models

Abstract: The multi-step denoising process in diffusion and Flow Matching models causes major efficiency issues, which motivates research on few-step generation. We present Solution Flow Models (SoFlow), a framework for one-step generation from scratch. By analyzing the relationship between the velocity function and the solution function of the velocity ordinary differential equation (ODE), we propose a Flow Matching loss and a solution consistency loss to train our models. The Flow Matching loss allows our models to provide estimated velocity fields for Classifier-Free Guidance (CFG) during training, which improves generation performance. Notably, our consistency loss does not require the calculation of the Jacobian-vector product (JVP), a common requirement in recent works that is not well-optimized in deep learning frameworks like PyTorch. Experimental results indicate that, when trained from scratch using the same Diffusion Transformer (DiT) architecture and an equal number of training epochs, our models achieve better FID-50K scores than MeanFlow models on the ImageNet 256x256 dataset.

Published as a conference paper at ICLR 2026
SoFLow: SOLUTION FLOW MODELS FOR ONE-STEP GENERATIVE MODELING
Tianze Luo, Haotian Yuan, Zhuang Liu Princeton University
ABSTRACT
The multi-step denoising process in diffusion and Flow Matching models causes major efficiency issues, which motivates research on few-step generation. We present Solution Flow Models (SoFlow), framework for one-step generation from scratch. By analyzing the relationship between the velocity function and the solution function of the velocity ordinary differential equation (ODE), pro- pose a Flow Matching loss and a solution consistency loss to train our models. The Flow Matching loss allows our models to provide estimated velocity fields for Classifier-Free Free Guidance (CFG) during training, which improves generation performance. Notably, our consistency loss does not require the calculation of the Jacobian-vector product (JVP), a common requirement in recent works that is not well-optimized in deep learning frameworks like PyTorch. Experimental results indicate that, when trained from scratch using the same Diffusion Transformer (DiT) architecture and an equal number of training epochs, our models achieve better FID-50K scores than MeanFlow models on the ImageNet 256 dataset. Our code is available at github. com/ 2lab www.abbppinnttooto Fow
Size
MeanFlow
SoFlow
B/2
6.17
4.85
M/2
5.01
3.73
I/2
3.84
3.20
XI.2
3.43
2.96
(a) 1-NFE samples from our XI/2 model
(b) 1-NFE FID-50K comparison
Figure 1: Visual samples and quantitative comparison. (a) One-step samples generated by Solution Flow Models on the ImageNet 256x256 dataset. (b) With the same Diffusion Trans- former (Peebles & Xie 2023) architecture and the same number of training epochs, models (SoFlow) consistently achieve better 1-NFE FID-50K scores than MeanFlow (Geng et al. 2025) models on the ImageNet 256 x 256 dataset.
INTRODUCTION
Diffusion models (Sohl-Dickstein al.) 2015 Song & Ermon 2019 Ho et 2020 Song al,
2020) and Flow Matching models (Lipman et al. 2022 Liu et al. 20022 Albergo & Vanden-Eijnden, 202 have emerged as foundational frameworks
generative modeling. Diflusion models operate by systematically adding noise to data and then learning a denoising process to generate high-quality samples. Flow Matching offers more direct alternative, modeling the velocity fields that transport simple prior distribution to complex data distribution. Despite their power and success across various generative tasks (Esser al. 2024 Ma al. 2024 Polyak et al.) 2024), both approaches rely iterative, multi-step sampling process, which hinders their generation efficiency.
Addressing this latency becoming key area of research. Consistency Models (Song et al. 2023 Song & Dhariwal 2023 Geng et al. 2024 Lu & Song 2024) and related techniques (Kim et al. 2023 Wang et al. 2025a Frans et al., 2024 Heek et al. 2024) have gained prominence by enabling rapid, few-step generation. These methods learn direct mapping from any point on generative trajectory to a consistent, "clean" output, bypassing the need for iterative refinement. However, this paradigm introduces significant challenges. Consistency models trained from scratch often fail to

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leverage Classifier-Free Free Guidance (CFG) for enhancing sample quality, and they are further hindered by instability caused by changing optimization targets. Recent works (Peng et al.) 2025 Geng et al. 2025) address this instability by incorporating Flow Matching loss. However, this approach introduces new computational bottleneck: relies on Jacobian-vector product (JVP) calculations, which are much less optimized than forward propagation in frameworks such as PyTorch.
Data
Velocity ODE Trajectories
Noise
t,
MSE
Target
Ze + (afaa ((1)(1 t)
20
Figure 2: Illustration of solution consistency loss. The plot shows straight-line Flow Matching trajectory defined by data-noise pair (Iu, I1). where Ię is the intermediate point given by 2₁ 0g20 + 8,21 (a,, St are C'l functions with ag = 31 1, ar Bu 0). Given three time points < I < f, the mean squared error is computed between our model fo (I.. f, s) and the stop- gradderd target fe- (x- + (0/20 + (( 1),4,))
In this work, we introduce new approach for one-step generation that avoids these limitations. Instead of relying on iterative ODE solvers, we propose directly learning the solution function the velocity ODE defined by Flow Matching to generate high-quality samples, shown in Figure la Figure 4 and Figure 5 We denote this function as f( L, 8), which explicitly maps state o, at time to its evolved state x, another time To learn it with parameterized model (2x,, t, s), we first analyze the properties that enable a neural network to serve as a valid solution function.
Based on this analysis, formulate training objective comprising Flow Matching loss and solution consistency loss (see Figure 2). The resulting model, fo (T,,,,, f, s), not only accommodates the Flow Matching objective and CFG naturally but also eliminates the need for expensive JVP cal- culations during training. Our experimental results demonstrate the effectiveness of this approach, showing that our models achieve superior FID-50K scores compared to MeanFFlow models the ImageNet 256x256 dataset, using the same Diffusion Transformer (DiT) architecture and the same number of training epochs.
RELATED WORK
Diffusion, Flow Matching, and Stochastic Interpolants. Diffusion models (Sohl-Diekstein al. 2015 Song et al. 2020 Kingma et al. 2021 Karras et al. 2022) and Flow Matching (Lipman ct al. 2022 Liu et al. al, are widely adopted generative modeling frameworks. These approaches either progressively corrupt data with noise and train neural network to denoise it, or learn a veloc- ity field that governs a transformation from the data distribution to a simple prior. They have been scaled successfully for image generation (Rombach al. 2022 Saharia al.) 2022;) Podell al.) 2023) and video generation (Ho et al. 2022] Brooks et al. 2024) tasks. Stochastic interpolants (Albergo Vanden- Vanden-Eijnden, 2022 Albergo al.) (223) build upon these concepts by explicitly defining stochastic trajectories between the data and prior distributions and aligning their associated velocity fields enable effective distributional transport.
Few-step Generative Models. Reducing the number of sampling steps has become active
search direction, driven by the need for faster generation and better theoretical insights. Prior efforts fall into main streams: (i) distillation-basel approaches that compress pre-trained multi-step
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models into few-step generators (Salimans & Ho 2022: Sauer et al. 2024 Geng et al. 2023 Luo
et al. 2023b, Yin et al. 2024 Zhou et al. 2024): and (ii) from-scratch training, which learns fast
samplers without teachers, most prominently through the consistency- trriiiin family (CMs, iCT,
ECT, sCT) (Song et al. 2023 Song & Dhariwal 2023 Ging et al. 2024 Lu & Song. 2024).
Consistency Models (CMs) collapse noised inputs directly clean data, enabling one- few-step generation (Song et al.) 2023). While first applied mainly in distillation (Luo et al. 2023a Geng
ct al. 2024). later works established that they can also be trained from scratch via consistency train- ing (Song & Dhariwal 2023). Building on this, iCT (Song & Dhariwal 2023) simplifies objectives
and improves stability with robust losses and teacher-free training, while ECT (Geng et al., 2024) introduces a continuous tattoouus---meeeee ODE formulation that unifies CMs and diffusion. The recent sCT
framework (Lu & Song 2024) further stabilizes continuous-time consistency training, scaling CMs up to billion-parameter regimes. Beyond fixed start and end settings, newer approaches extend CMs to arbitrary timestep transitions, aligning better with continuous-timg dynamics.
Specifically, Shortcut Models (Frans et al. 2024) condition on noise level and step size to flexibly support both one- and few-step sampling with shared weights. MeanFlow (Geng et al. 2025) intro- duces interval-averaged velocities and analyzes the relationship between averaged and instantaneous velocities. IMM (Zhou et al. 2025) matches moments across transitions in a single-stage objective, reducing variance and avoiding muult distillation. Flow-Anchored CMs (Peng et al., 2025 reg- ularize shortcut learning with a Flow Matching anchor, improving stability and generalization On the distillation side, Align Your Flow (Sabour et al., (202)) unifies CM and FM into continuous-time flow maps effective across arbitrary step counts, further enhanced with autoguidance and adversar- ial fine-tuning, Finally, Transition Models (Wang al. (2025b) reformulate generation around exact finite-interval dynamics, unifying few- and many-step regimes and achieving monotonic quality gains as step budgets increase.
PRELIMINARY: FLOW MATCHING
We first introduce the setting of Flow Matching. Flow Matching models learn a velocity field that transforms a known prior distribution like the standard Gaussian distribution to the data distribution. More precisely, we denote the data distribution as p(2o), and the prior distribution as p(z1), which is the standard Gaussian distribution N (O, T,,) in our setting, where n represents the data dimension.
general noising process is defined as 24 ax + 3,31, where 2, E IR", , E [0, 1), a and 3r are continuously differentiable functions satisfying the boundary conditions ov 1, 0i 0, Bo O. and 31 = 1. Then, the marginal velocity field associated with the noising process is defined as
v( t) (x)=) 10 + R(11),
which is a conditional expectation given x, of the conditional velocity field. of and 8 denote the derivatives of a and 3r. According to the Flow Matching framework, given the marginal velocity field vIr, t). new samples can be generated by first sampling 21 p(x1), and then solving the initial value problem (IVP) of the following velocity ordinary differential equation (ODE) from t 1 to 0:
dx(() ((X()),). x(1) = -1- dt
To construct generative model based on this principle, straightforward approach is to approxi- mate the marginal velocity field using a neural network va, parameterized by 6. which is trained by minimizing the following mean squared objective:
(0) = Tr,,,, [Ivo(E,, 1) u v(z,, 1)
However, directly optimizing neural network with this loss is impractical since the real velocity field o(2,, t) is a conditional expectation given 2;, To overcome this challenge, a conditional variant
of the Flow Matching loss is introduced (Lipman et al. 2022 Liu et al. 2022
(G)
(6) = E, Eo .n, [ f) - (a/20 + 1121)1.
(4)
Minimizing fCFM is equivalent to minimizing the original objective Lva- This loss function is tractable since only need to sample data-noise pairs construct targets for the network.
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4 SOLUTION FLOW MODELS
4.1 FORMULATIONS
The main purpose of this work is to investigate how to train neural network that can directly solve the velocity ODE, thereby eliminating the reliance on numerical ODE solvers in Flow Matching models. We study this problem under the assumption that the velocity field v( t) E R", with Ie E R" and t E [0, 11. is continuously differentiable and globally Lipschitz continuous with respect to Tz, i.e. [[r I) eyy, l) V L. - wel12, Var, 3/t E &R", [0, 1].
According to the existence and uniqueness theorem of ODEs, these two assumptions guarantee that, given initial condition (() Is, where 2t R 1 E [0, 1] can be arbitrarily chosen, a unique ds = =(X(ss, s) exists. Since the initial condition can be varied, we denote this unique solution by notation f(,,, t) Then we immediately have two
solution (s), [0, t] to the velocity ODE dx
identities by the definition of the ODE, for any IR.", < 11s
fxr, t,t,
(5)
t,f = (f,,,,t
(6)
where (1,, E IR is the partial derivative funetion of (1, with respect to the third variable. Equivalently, the two equations can also be written as the following integral equation:
f(r,t, xxtt J. u(f u), u)du.
J,
Since f(12, f, s) maps an initial value at time i to the unique solution of the velocity ODE at time s, we denote it as the solution function in this paper. Owing to the continuous differentiability of the velocity field vxze, f), the solution function f(z,, s) is also continuously differentiable. To realize one-step generation, we need to train a model fo(z,, t. s) to approximate the ground truth solution function f s), which is determined uniquely by the velocity field v(z,,t).
Under the setting discussed above, the following two conditions are sufficient to ensure that fo(x,,t,s) f(x,, s) for all E R", 12,0 stts
To(xe,t,t) = It.
(8)
So 1, s)((E, 1) + (2So(xr,,,,)) =
1,
(9)
where f1 t, E R"*n is the Jacobian matrix function fo( t. s) with respect to the first variable, o(2,,t) t) is multiplied with it via matrix-vector product, and O2fo(x,, tx,,t,s) R2 is the partial derivative vector function of fo s) with respect to the second variable.
This can be proven by expanding the following derivative for 0 <ssist<LL
गा (oo( 11,,,) fel s), (af(x,,1,1 +025a (,),,, (10)
=v(
The expression is equal to 0 due to the conditions in Eq. and Eq. Thus, know that the model fx,,,t,s)
t, s) is indeed the true solution function tt),,
fa t, fo fe(f(z,, t),, t,, s) ff s), s, f f(z,,t,s),
(11)
where we use the property that f(x,, t) = fo(z,, t, = 2, and fo(f t, 4,,4)) s) is invariant with respect to I, allowing us to set t and = s to finish the proof.
4.2 LEARNING OBJECTIVES
We now consider how to construct efficient objectives for neural networks to learn the ground truth solution function, according to the two conditions mentioned in the previous section. To ensure the first boundary condition in Eq. 8lis satistied, we adopt the following parameterization:
fo 4, 8) ((t, b(l, Fo 4,, s),
(12)
where a(t, s) and b(t, s) are continuously differentiable scalar functions satisfying a(t, t) = 1 and b(L, t) for all & [0, 1), and Fo(x,, 8) denotes raw neural network. Following the notation
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in the last section, use (ia(t, @za(t, s), 0,b(t, s), 0zb(t, s) to denote the partial derivatives of aft, s) and b(t, s) with respeet to the first and second variables. our experiments, we choose two specific parameterizations, including the Euler parameterization and the trigonometric parameteri- zation:
fo 5)= = 2t + (s t)
(13)
fo t, = cos (2(0- t)) 2t sin (2 (0(-)))
t)) Fo
(14)
Trivially, these two parameterizations satisfy the boundary condition fa (a(x,,t, t,t) = It. Then, we construct two loss functions for training using Eq.
Flow Matching Loss. boundary condition Eq. gives fo(x,, t, Zz, Vx, E IR", t E [0. TI. Thus, we have:
Firstly, we consider special case of Eq.. 9 when 8. Recall that the
0 fL Ln, 0= afo(z1,
(efe(zn,t,)) 0rfo(zn,,tt),
(15)
01
where In E 12 is the identity matrix. Then Eq. 9 can be simplified to v(zr, aafo(xr, Under our parameterization mentioned above, we have
va t) (a Oza(t, t)x, O2b(t,
b(t,t) (zFo(x
(16)
where the complex term containing (x t) is canceled since b(t, t) = 0 by our choice of parameterization. Now we obtain a Flow Matching loss for our neural networks:
(Fm(0)= - =Br.)-)) (t, MSE)
||8za(t, f)2, + () - (a/.ro + 8, 11)1
n
(17)
where is the data dimension, 0,20 + 8111 is used to replace the intractable marginal velocity field v(x,, t) (x [a/.00 + [[11] during the training process following the standard Flow Matching framework. In addition, this velocity term can also be provided by teacher model in distillation situations, but in this paper, we focus on the from-scratch training setting.
Previous works (Geng et al., 2024 2025) have demonstrated that choosing an adaptive weighting function is beneficial for few-step generative models. Following their approach, we choose
WFM (t. MSE)
(18)
lab((,, {MSE c)
as our weighting function, where [8zb(t, t) used to balance the raw network's gradients across time, and MSE represents the original mean squared error. Here is a smoothing factor to prevent excessively small values, and p factor that determines how robust the loss is. For p O, the objective degenerates to the mean squared error. For p > O, this factor will reduce the contribution of the data points with large errors in a data batch to make the objective more robust.
The original Flow Matching framework samples t uniformly, while more recent works (Esser al. 2024 Geng et al. 2025) suggest sampling from a logit-normal distribution. We also sample t from (N (J4M, RM)), where a-)) and N represent the sigmoid function and normal distribution.
Solution Consistency Loss. We now consider how to build training target for the < t situation using Eq. 9 According to the Taylor expansion, we have an approximation equation:
fo(a 1, fola, + v(x,, () 1), = (8.Jo(z,, s)u(2,, () 02fol,,, ())) +0(1,, (19)
t 1
where E (s, t) close to t. We therefore propose solution consistency loss LsCM(B):
(t, (MSSE
(SCM(0)= E
[[ 8) - so- (It + (afx0 + (1x1)(( - t),
n
(20)
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where is the data dimension, e- means applying the stop-gradient operation to the parameters, and (a/20 + 3,21) is again used to replace the intractable o(2,, t) = E E1(z.==1 laf + 1(x1]
following the common practice. The adaptive weighting is chosen follows:
"scM (t, MSE) (t 1) b(t, s)
(21)
MSE (1-1)²
+ c)P²
where the first term ensures the gradient magnitude of the raw network (T s) s) is stable, and the second term is again used provide a more robust loss by scaling down the coefficient for data points with large mean squared errors when p > 0. Besides, we divide the mean squared error in the adaptive term by (t 1)2 since its magnitude proportional to 1)2.
As for the sampling method of t, l, during training, we first sample and s from two logit-normal distributions, o (N (14, and ([V'ss o.)), respectively. Here, 8 is clamped ensure < t- 10-0, We then determine using the following method:
1 (s 1 t) K),
(22)
where k K represent the current and total training steps, respectively. The function r(k, K) rep- resents monotonically decreasing schedule that gradually moves / towards t throughout training. To avoid numerical issues, is clamped to ensure < t - 10-4. This schedule decreases from an initial value rinit to end value rend- In our ablation studies, we exponential, cosine, linear, and constant schedules. For more implementation details, please refer to Appppediix A]
The total training loss is combination of the Flow Matching and solution consistency losses: L(O)- = (O) + (1 ASM The parameter À controls the balance between them by deter- mining the fraction of data batch dedicated computing LFv( while the remaining fraction is used for (f). We perform ablation studies to determine the optimal value of X.
4.3 CLASSIFIER- GUIDANCE
Classifier-Fres Guidance (CFG) (Ho Salimans 2022) is a standard technique in diffusion models for enhancing conditional generation. The models are trained with randomly dropped conditions to mix conditional and unconditional data. During inference, CFG is applied by linearly combining predictions from the label-conditional and unconditional models enhance generation quality.
To apply CFG to our models, we first introduce the ground-truth guided marginal velocity field:
0/g ug(= t, c) = wu(r,, t c) + (1 - w)v(z,,)) t),
(23)
where t) = 1o 3,a1] denotes the unconditional marginal velocity field, represents conditions (e.g., class labels), v( / c) Le + }[22] denotes the conditional marginal velocity field, and w is the CFG strength. To ensure our conditional model
serves as the solution function for the velocity ODE defined by this guided field (which depends purely on the data distribution and model-independeet), it should satisfy the following equations:
Solm,, 1, .t.
(24)
0 fo (,), + O2 Analogous to the unconditional setting, we define the guided Flow Matching loss (0) as
(25)
WP MSE) ||0a((t, ()xr +
Ci ar, and the guided solution consistency loss (0)
- u (2 (,(,,))))))
(26)
n
"wscM(t MSB) [ t, fo (It (e) 112
CSCM( (0) = E. t.l.8.3..e
(27)
where is the data dimension, O denotes the stop-gradient operator applied to the targets, and the adaptive weighting functions remain consistent with the unconditional case. The total guided training loss (,(() is defined as their linear combination (C) (1 (0)
However, since the unconditional velocity field is generally intractable during conditional training, (1 I, c) is not directly accessible. To address this, we train the network to concurrently predict

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We conduct our major experiments on the ImageNet 256x256 dataset (Deng et al. 2009). SoFlow models operate within the latent space of pre-trained VAE Rombach et al.] (022). which is common practice in recent works (Peebles Xie 2023 Frans et al. 2024] Geng et al. 2025). The tokenizer converts 256x256 images into 32x 32x4 latent representation. We assess generation quality using the Fréchet Inception Distance (FID) (Heusel et al. 2017), computed over a set of 50,000 generated samples. To evaluate computational efficiency, we report the number of function evaluations (NFE), with a particular focus on the single-step (I-NFE) scenario. All models presented trained from scratch. In addition to the standard time variable t, our model incorporates an additional time variable s. We provide this to the networks by feeding the positional embeddings For more implementation details, please refer to Published as a conference paper at ICLR 2026
the unconditional velocity field. Specifically, randomly replace the condition with empty label @ (with a probability of 0.1) and update the model using the unconditional Flow Matching loss [Eq. 17 and the unconditional solution consistency loss Eg. 20p. Consequently, we approximate v(2,, 1) using the model prediction Muncood = @2a(t, 1) + Ø2b(1, F8- (st, 1, ())
For data points with non-empty condition we compute the guided losses by substituting the velocity term 2 (22, t, c) in Eq. 26 and Eq. 27 with the estimator wa(a.o + f(z1) (1 w) Runcond
where is the CFG strength. Here, 0i + ffar is used to replace the intractable conditional marginal velocity field v(a,, & | c). similar to the unconditional formulation.
Notably, the term w(aaa0 8/a1) (1 urnuUncdd typically exhibits higher variance than the original term af2o + B[x1, primarily due to the scaling effect of w. This increased variance can hinder training convergence and degrade final performance. We observe that the model learns the guided velocity field via Eq. 27 when conditioned on c. Therefore, we can obtain a model-predicted guided veloeity U'guided via Oza (t, t) + Øzb(t, F (Ie, t,t,t,t, t, c), and then employ
Umix - mmwuaa/0 + 1(x1) + (1 w) Uuncondd + (1 m7 Uguided
(28)
to approximate the target (2,, t, (with expectations also taken over (Tu, T1)), where 0 < m S acts as a velocity mixing ratio. Since the stochastic term 0,20 + 3,,11 is the dominant source of variance, using a small m effectively mitigates this issue, as the model-predicted Uggidedd possesses significantly lower variance.
The inference process of our model is straightforward. Since 40 = alt, s)at + b(t, (,(r,, a neural solution function to the velocity ODE, we only need to sample 21 (0, Ix) and apply this function with t = and = 0 to obtain the clean data in a single step. Furthermore, similar to consistency models (Song et al. 2203), our model supports multi-step sampling by adding noise the predicted data and recursively applying the solution function.
5 EXPERIMENTS
5.1 SETTINGS
Vaswani al. 2017) of their difference, Appendix A
5.2
ABLATION STUDY
We conduct various ablation studies to determine the optimal hyperparameters for SoFlow models. Following the methodology of[Geng et al. (2025), we utilize the DłT-B/4 architecture in our ablation experiments, which features a "base" sized Diffusion Transformer with a patch size of 4. We train our models for 400K iterations. For performance reference, the original DiT-B/4 (Peebles & Xie 2023) achieves an FID-50K of 68.4 with 250-NFE sampling. MeanFlow (Geng et al., 2025) reports an FID-50K of 61.06 with I-NFE sampling. and the authors claim to have reproduced SYT-B/4 (Ma) et al. 2024) with an FID-50K of 58.9 using 250-NFF, sampling. The FID-50K scores mentioned here are without CFG. Our ablation studies are conducted two groups: a first group of three experiments without CFG and second group of three with CFG. Table I
presents our results,
which are analyzed as follows:
'SD-VAE: http::htttttttggggeeeeeeee http://www..ac.cta/aaiaaaaccccca
ft-mse

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((kK)
FID-50K
A
FID-50K
FID-50K
Exponential
58.57
0%
66.36
0.0
68.25
Cosine
60.17
25%
61.42
0.5
59.04
Linear
59.01
50%
59.61
58.57
Constant
59.52
75%
58.57
1.5
63.72
(a) + schedule r(k, K)
(b) Flow Matching data ratio (c) Loss weighting coefficient
Ote, Br and a(t, s), s(),s)
FID-50K
w
FID-50K
FID-50K
Linear, Euler Linear, Trigonometric Trigonometric, Euler
11.59
1.5
34,47
0.25
11.59
17.50
2.0
20.35
13,94
13.12
2.5
14,60
0.75
16.29
Trigonometric, Trigonometric
19.01
3.0
11.59
1.0
17.22
(d) Noising schedule and parameterization
(e) CFG strength
(I) Velocity mix ratio
Table 1: Ablation studies on 1-NFE ImageNet x56x256 class-conditional generation. All mod- els have 131M parameters and are trained with batch size of 256 for 400K iterations from scratch.
I + Schedule rkk, K). We compare exponential, cosine, linear, and constant schedules in our experiments (Table Tap. The FID-50K results show that the speed of the | + t transition during the training process has a relatively small influence on performance. We choose the best exponential schedule following previous works (Song & Dhariwal 2023 Geng al. 2024).
Flow Matching Data Ratio A. Although our solution consistency loss alone is sufficient to en- able one-step generation (corresponding to 0% ratio Table Ib), experimental results show that incorporating large ratio of Flow Matching loss is effective for improving performance.
Loss Weighting Coefficient p- We observe that the choice of loss metric has a significant influence on the performance of one-step generation, which aligns with previous works (Song & Dhariwal 2023 Geng et al. 2024; 2025 Dao et al. 2025). As shown in Table Ic the original mean squared settings.
loss with performs much worse than p 0.5
Noising Schedule and Parameterization. Our method is compatible with different Flow Match- ing noising schedules and solution function parameterizations. As presented in Table ld exper- iments show that the linear noising schedule It = (1 l)zn + lx along with the Euler pa- rameterization fo(2,, s) It + (s (1 performs best, compared to the trigono- metric noising schedule It = cos( f)) + sin( 1t) and its corresponding parameterization fo(2e, 1, cos( [ 1)) + sin (5 ()) (.),
CFG Strength w. Unlike diffusion models, our model enables CFG during the training stage, so the resulting 1-NFE inference is guided without additional inference-time CFG. Experiments in Table demonssttat that CFG can also greatly improve the generation quality for our model.
Velocity Mix Ratio m. Experiments Table Tf) show that relatively small is beneficial for performance. This can be explained by the fact that the CFG guidance strength w amplifies the variance in the velocity term, and smaller m can suppress this variance by partially replacing the velocity term with the model's prediction of the guided velocity field.
5.3 COMPARISON WITH OTHER WORKS
ImageNet 256 256 Results. We begin by analyzing the 1-NFE FID-50K results of our model
across various model sizes (numbers of parameters), shown in Figure Our model's 1-NFE performance gradually improves with an increase model parameters, which is consistent with
previous observations Diffusion Transformers (Peebles & Xie 2023 Ma at. 2024 Geng al. 2025).
Next, compare model's one-step generation performance against previous models, with the
results summarized in Table To make relatively fair comparisons, we train our model by setting the batch size to 256 and the training epochs 240, consistent with the MeanFlow models (Geng al.
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method
epochs
params
NFE
FIDJ
Generative Adversarial Networks
BigGAN (Brock et al.
2018
1 112M
6.95
StyleGAN-XL Sauer
2022
166M
2.30
GigaGAN (Kang et al. 2023)
569M
3,45
Masked and Autoregressive Models
Mask-GIT CChag al. 2022)
555
227M
8
6.18
Mag ViT-v2 et al. 2023
1080
307M
64
1.78
LlamaGen-XX XL Sun al. 2024
300
775M
576
2.62
VAR Tian et al. (0244)
350
2.0B
1.80
MAR Li et al.
2024
800
943M
64
1.55
RandAR-XL Pang et al. 2025
300
775M
256
2.22
Multi-step Diffusion Models
LDM-4-G [Rombach et al.
2022
170
395M
250x2
3.60
MDTv2 (ao et al. 2023
700
676M
250x2
1.63
DIT-XI/2 (Peebles & Xie
2023
1400
675M
250x2
2.27
SiT-XL/2 [a et aL 22024
1400
675M
250x2
2.06
FlowDCN-XL/2 XL/2 Wang et al.
2024)
400
675M
250x2
2.00
SiT-REPA- XL/2 (Yu et al. 2024)
800
675M
250x2
1.42
Few-step Difffusion Models
iCT-XL/21 (Song & Dhariwal 2023
675M
1/2
/
Shortcut-XLL XL/2 (Frans et al. (0224)
250
675M
1/4
10.60 7.80
IMM-XL/2 (Zhou ct 2025
3840
675M
1x2/2x2
7.77 3.99
MeanFlow-B/2 (Geng et al. 2025
240
131M
6.17
MeanFlow-M/2 Geng al. 2024
240
308M
5.01
McanFlow-L/2 Geng el al. 2025
240
459M
3.84
MeanFlow-XLL/2
(Geng 2025
240
676M
1/2
/
SoFlow-B/2
240
131M
1/2
4.85
4.24
SoFlow-M/2
240
308M
1/2
3.73 3.42
SoFlow-L/2
240
459M
1/2
3.20
2.90
SoFlow-X1./2
240
676M
1/2
2.96 2.66
Table 2: FID-50K results for class-conditional generation on ImageNet 256x256. x2 denotes an NFE of per sampling step incurred by CFG. Entries in the format "1 / 2" indicate that the corresponding FID scores are reported for 1-NFE and 2-NFE sampling, respectively. Results as reported in Zhou et al. (2025).
2025). Experimental results show that our model consistently outperforms MeanFlow models across all evaluated model sizes when trained from scratch. Specifically, for smaller models with DiT-B/2 and DiT-M/2 architectures, our model demonstrates significant improvements, achieving FID-50K scores of 4.85 and 3.73, respectively. Our model also achieves superior FID-50K values for larger architectures, namely DiT-L/2 and DiT-XL/2, reaching 3.20 and 2.96, respectively. Notably, our models employ CFG during training, which enables generation with exactly 1-NFE during inference, similar to McanFlow models. Furthermore, our XL/2 model achieves a strong 2-NFE FID-50K score of 2.66, surpassing the 2.93 achieved by MeanFlow-XI./2.
In terms of computational efficiency, since our method obviates the need for JVP computation, benefits from lower GPU memory usage and faster training speeds compared MeanFlow models. Please refer to Appendix D for detailed comparison.

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ImageNet 256x256
8/2
M/2
1-NFE FID-50K
L/2
7
6
5
80
120
160
200
240
# Training Epochs
Figure 3: FID-50K performance of our models. The plot shows the FID-50K performance of our models with varying numbers of parameters, all trained from scratch on the ImageNet 256x256 dataset. We apply CFG during training and report the FID-50K scores of generated images using I-NFE sampling process. The results consistently demonstrate that the performance of our models improves the model size increases.
CIFAR-10 Results. In Table 3 report conditional generation results on the CIFAR-I0 (Krizhevsky et al., 2009) dataset, where performance is measured by the FID-5OK metric with 1-NFE sam- pling. For our model, ture (Ronneberger et al. 2015) developed from Song al. (2020), aligning with prior works. Our method applied directly to the pixel space, with resolution of 32x32. For more implementation details, please refer to Appeeddix A Our method achieves compet- itive performance compared to prior approaches on this dataset.
method
NFE
FID
iCT (Sogg Dhariwal
2023
2.83
ECT GGng ct al 2024
1
3.60
sCT
Lu & Song
2024
2.97
adopt the U-Net architec-
IMM (Zhou
2025
3.20
MeanFlow (Gieng ct al. 20255
2.92
SoFlow
2.86
Table 3: FID-50K results for unconditional generation on CIFAR-10.
CONCLUSION
We have presented SoFlow, a simple yet effective framework for one-step generative modeling, Our approach directly learns the solution function the velocity ODE, enabling single-step sampling without iterative solvers. By leveraging bi-time formulation and hybrid training objective com- bining a Flow Matching loss and a solution consistency loss, SoFlow naturally supports CFG during training and avoids JVP calculations that are not well-optimized in deep learning frameworks like PyTorch. Our method demonstrates competitive performance on the class-conditional ImageNet 256 x 256 generation task, outperforming MeanFlow models when trained from scratch under the same settings.
ACKNOWLEDGMENTS
We thank Kaiming He for helpful discussions. We gratefully acknowledge the use the Neuronic GPU cluster maintained by the Department of Computer Science at Princeton University. This work was substantially performed using Princeton Research Computing resources, consortium led by the Princcton Institute for Computational Science and Engincering (PICSciE) and Research Computing at Princeton University. We also thank Princeton Language and Intelligence (PLI) for H100 GPU support for this work."
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REFERENCES
Michael S Albergo, Nicholas M Boffi, and Eric Vanden-Eijnden. Stochastic interpolants: A unifying framework for flows and diffusions. arXiv preprint WwXiv:2303308797, 2023.
Michael Samuel Albergo and Eric Vanden-Eiinden. Building normalizing flows with stochastic interpolants. In ICI.R, 2022.
Andrew Brock, Jeff Donahue, and Karen Simonyan. Large scale gan training for high fidelity natural image synthesis. In ICLR, 2018.
Tim Brooks, Bill Peebles, Connor Holmes, Will DePue, Yufei Guo, Li Jing, David Schnurr, Joe Taylor, Troy Luhman, Eric Luhman, et al. Video generation models as world simulators. OpenAl Blog, 1(8):1, 2024.
Huiwen Chang, Han Zhang, Lu Jiang, Ce Liu, and William Freeman. Maskgit: Masked generative image transformer. In CVPR, 2022.
Quan Dao, Khanh Doan, Di Liu, Trung Le, and Dimitris N Metaxas. Improved training technique for latent consistency models. In ICLR, 2025.
Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In CVPR, 2009.
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 ICML, 2024.
Kevin Frans, Danijar Hafner, Sergey Levine, and Pieter Abbeel. One step diffusion via shortcut models. In ICLR, 2024.
Shanghua Gao, Pan Zhou, Ming-Ming Cheng, and Shuicheng Yan. Mdtv2: Masked diffusion trans- former is a strong image synthesizer. arXiv preprint arXiv:2303. 14389, 2023.
Zhengyang Geng, Ashwini Pokle, and Zico Kolter. One-step diffusion distillation via deep equi- librium models. NeurlPS, 2023.
Zhengyang Geng, Ashwini Pokle, Weijian Luo, Justin Lin, and J Zico Kolter. Consistency models made easy. In ICLR, 2024.
Zhengyang Geng, Mingyang Deng, Xingjian Bai, Zico Kolter, and Kaiming He. Mean flows for one-step generative modeling. In NeurlPS, 2025.
Jonathan Heek, Emiel Hoogeboom, and Tim Salimans. Multistep consistency models. arXiv preprint arXiv:2403.06807 2024.
Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter.
Gans trained by a two time-scale update rule converge to a local nash equilibrium. In NeurIPS, 2017.
Jonathan Ho and Tim Salimans. T.Xiv:2207.12598, 2022.
Classifier-free diffusion guidance.
arXiv preprint
Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models, In NeurIPS, 2020.
Jonathan Ho, Tim Salimans, Alexey Gritsenko, William Chan, Mohammad Norouzi, and David J Fleet. Video diffusion models. NeurIPS, 2022.
Minguk Kang, Jun- Yan Zhu, Richard Zhang, Jaesik Park, Eli Shechtman, Sylvain Paris, and Taesung Park. Scaling up gans for text-to-image synthesis. CVPR, 2023.
Tero Karras, Miika Aittala, Timo Aila, and Samuli Laine. Elucidating the design space of diffusion- based generative models. In NeuriPS, 2022.

---

Published as a conference paper at ICLR 2026
Dongjun Kim, Chich-Hsin Lai, Wei- Wei-Hsiang Liao, Naoki Murata, Yuhta Takida, Toshimitsu Uesaka, Yutong He, Yuki Mitsufuji, and Stefano Ermon. Consistency trajectory models: Learning proba- bility flow ode trajectory of diffusion. In ICLR, 2023.
Diederik Kingma, Tim Salimans, Ben Poole, and Jonathan Ho. Variational diffusion models. In NeurlPS, 2021.
Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. Technical Report, 2009.
Tianhong Li, Yonglong Tian, He Li, Mingyang Deng. and Kaiming He. Autoregressive image generation without vector quantization. In NeurlPS, 2024.
Yaron Lipman, Ricky TQ Chen, Heli Ben-Hamu, Maximilian Nickel, and Matthew Le. Flow match- ing for generative modeling. In ICLR, 2022.
Liyuan Liu, Haoming Jiang, Pengcheng He, Weizhu Chen, Xiaodong Liu, Jianfeng Gao, and Jiawei Han. On the variance of the adaptive learning rate and beyond. In ICLR, 2019.
Xingehao Liu, Chengyue Gong, al. Flow straight and fast: Learning to generate and transfer data with rectified flow. In ICLR, 2022.
Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization. In ICLR, 2017.
Cheng Lu and Yang Song. Simplifying. stabilizing and scaling continuous-time consistency models. In ICLR, 2024.
Simian Luo, Yiqin Tan, Longbo Huang, Jian Li, and Hang Zhao. Latent consistency models: Synthe- sizing high-resolution images with few-step inference. arXiv preprint arXiv:2310.04370 2023a.
Weijian Luo, Tianyang Hu, Shifeng Zhang, Jiacheng Sun, Zhenguo Li, and Zhihua Zhang. Diff- instruct: A universal approach for transferring knowledge from pre-trained diffusion models. In NeurlPS, 2023b.
Nanye Ma, Mark Goldstein, Michael S Albergo, Nicholas M Boffi, Eric Vanden-Fijnden, and Sain- ing Xie. Sit: Exploring flow and diffusion-based generative models with scalable interpolant transformers. In ECCV, 2024.
Ziqi Pang, Tianyuan Zhang, Fujun Luan, Yunze Man, Hao Tan, Kai Zhang, William T Freeman, and Yu-Xiong Wang. Randar: Decoder-only autoregressive visual generation in random orders. In CVPR, 2025.
William Peebles and Saining Xie. Scalable diffusion models with transformers. In ICCV, 2023.
Yansong Peng, Kai Zhu, Yu Liu, Pingyu Wu, Hebei Li, Xiaoyan Sun, and Feng Wu. Flow-anchored consistency models. arXiv preprint arXiv:2507.03733, 2025.
Dustin Podell, Zion English, Kyle Lacey, Andreas Blattmann, Tim Dockhorn, Jonas Müller, Joe Penna, and Robin Rombach. Sdxl: Improving latent diffusion models for high-resolution image synthesis. In ICLR, 2023.
Adam Polyak, Amit Zohar, Andrew Brown, Andros Tjandra, Animesh Sinha, Ann Lee, Apoorv Vyas, Bowen Shi, Chih Yao Ma, Ching- Yao Chuang, et al. Movie gen: A cast of media founda- tion models. arXiv preprint arXiv:2410.132 2024.
Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Bjöm Ommer. High- resolution image synthesis with latent diffusion models. In CVPR, 2022.
Olaf Ronneherger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomedi- cal image segmentation. In MICCAI, 2015.
Amirmojtaba Sabour, Sanja Fidler, and Karsten Kreis. Align your flow: Scaling continuous time flow map distillation. In NewrIPS, 2025.
12

---

Published as a conference paper at ICLR 2026
Chitwan Saharia, William Chan, Saurabh Saxena, Lala Li, Jay Whang, Emily L Denton, Kamyar Ghasemipour, Raphael Gontijo Lopes, Burcu Karagol Ayan, Tim Salimans, et al. Photorealistic text-to-image diffusion models with deep language understanding. In NeurIPS, 2022.
Tim Salimans and Jonathan Ho. Progressive distillation for fast sampling of diffusion models. In ICLR, 2022.
Axel Sauer, Katja Schwarz, and Andreas Geiger. datasets. In SIGGRAPH, 2022.
Stylegan-xl: Scaling stylegan to large diverse
Axel Sauer, Dominik Lorenz, Andreas Blattmann, and Robin Rombach. Adversarial diffusion dis- tillation. In ECCV, 2024.
Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsupervised learning using nonequilibrium thermodynamics. In ICML, 2015.
Yang Song and Prafulla Dhariwal. Improved techniques for training consistency models. In ICLR, 2023.
Yang Song and Stefano Ermon. Generative modeling by estimating gradients of the data distribution. In NeurlPS, 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 ICLR, 2020.
Yang Song, Prafulla Dhariwal, Mark Chen, and Ilya Sutskever. Consistency models. In ICML, 2023.
Peize Sun, Yi Jiang, Shoufa Chen, Shilong Zhang. Bingyue Peng, Ping Luo, and Zehuan Yuan. Autoregressive model beats diffusion: Llama for scalable image generation. arXiv preprint 0rXiv:2406.065255 2024.
Keyu Tian, Yi Jiang. Zehuan Yuan, Bingyue Peng. and Liwei Wang. Visual autoregressive modeling: Scalable image generation via next-scale prediction. In NeurIPS, 2024.
Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention all you need. NeurlPS, 2017.
Fu-Yun Yun Wang, Zhengyang Geng, and Hongsheng Li. Stable consistency tuning: Understanding and improving consistency models. In ICLR 2025 Workshop Deep Generative Model in Machine Learning: Theory, Principle and Efficacy, 2025a.
Shuai Wang, Zexian Li, Tianhui Song, Xubin Li, Tiezheng Ge, Bo Zheng, and Limin Wang. Explor- ing den-like architecture for fast image generation with arbitrary resolution. In NeurIPS, 2024.
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:2509.044 04394, 2025b.
Tianwei Yin, Michaël Gharbi, Richard Zhang, Eli Shechtman, Fredo Durand, William Freeman, and Taesung Park. One-step diffusion with distribution matching distillation. In CVPR, 2024.
Lijun Yu, Jose Lezama, Nitesh Bharadwaj Gundavarapu, Luca Versari, Kihyuk Sohn, David Minnen, Yong Cheng, Agrim Gupta, Xiuye Gu, Alexander G Hauptmann, et al. Language model beats diffusion-tokenizerr is key visual generation. In ICLR, 2023.
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. ICLR, 2024.
Linqi Zhou, Stefano Ermon, and Jiaming Song. Inductive moment matching. In ICML, 2025.
Mingyuan Zhou, Huangjie Zheng, Zhendong Wang, Mingzhang Yin, and Hai Huang. Score identity distillation: Exponentially fast distillation of pretrained diffusion models for one-step generation. In ICML, 2024.
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APPENDIX
A IMPLEMENTATION DETAILS
We first provide a detailed description of the different scheduling strategies used to determine the intermediate time variable during training. Recall that I computed from and & as follows:
f (s t) r(k: K),
where k K represent the current and total training steps, and the function r(k, K) is monotonically decreasing function that controls the progression of from an initial value near t towards t itself as training advances. Below specify the four schedules compared in our ablation studies:
Exponential Schedule:
Pend Pinit
rkk = Pinit x
Cosine Schedule:
1
(T
))
k K
r(k, K) = road + (Fimit - Fend) x
cos cos
2
Linear Schedule:
k
r(k, K)
Pinit (rend Tinal
K"
Constant Schedule:
r(k, K) = Tead
In all cases, is clamped satisfy < 1 ensure numerical stability; similarly, we also clamp to satisfy < -- 10-4 The initial value Tinit and end value Tend are hyperparameters controlling the starting and final relative position between and We now detail the training and architectural configurations for our models on two benchmark datasets. All experiments are run on NVIDIA H100 GPUs.
ImageNet 256 256. We employ the AdamW optimizer (Loshchilov & Hutter 2017) with stant learning rate of x 10-4 and betas set to (0.9, 0.99), without learning rate decay or weight decay. Following standard practice, Average (EMA) with a decay rate of 0.9999. For time sampling, the logit-normal distribution pa- rameters set μFM -0.2, OPM 1.0; μr 0.2, ar 0.8; μt, 1.0, a, 0.8. The Flow Matching data ratio is 75% and the adaptive loss weight coefficient = 1.0. The schedule param- eters are Tint
evaluate model performance using Exponential Moving
and Fend 500 We use a linear noising schedule with Euler parameterization. The CFG strength w is set to 2.5, 2.25, 2.0, and 2.0 for the B/2, M/2, L/2, and XL/2 models, re- spectively, while the velocity mix ratio m is set to 0.25 for all models. Following common practice, we gradually decay the CFG strength to 1.0 in high-noise regions, setting the decay threshold at t > 0.8 for ablation studies and > 0.75 for the main experiments. Specifically, the decay speed is
determined by the function exp 30 ) where L' min - 1 10-6) for t ddeaay-
Here, /deay denotes the threshold time and 10-6 is used to ensure numerical stability. We employ this specific smoothing function rather than a direct decay to ensure that the guided velocity field remains continuously differentiable. Finally, architectural details provided in Table
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CIFAR-10. Training uses batch size of 1024 for 800K iterations, consistent with MeanFlow. We adopt the RAdam (Liu et al. 2019 optimizer with a learning rate x 10 following Song & Dhariwal (2023): Geng al. (2024). We evaluate model performance using EMA with decay
rate of 0.9999. Time sampling parameters are: μFM -0.9, 9PM = 1.6; f4 -0.9, oz = 1.6;
(ss 4.0, 0 1.6. The Flow Matching data ratio is 0% with p 0.75. Schedule parameters Timit and Fend are set to 1.0 and 32ud respectively. Linear noising schedule and Euler parameterization are used. Our data augmentation setup follows Karras et (2022), where vertical flipping and rotation augmentations are disabled.
Architectures
SoFlow-B/4
NoFlow-B/2
SoFlow-M/2
SoFlow-1/2
SoFlow-XL.2
Parameters (M)
131
131
308
459
676
FLOPs (R/ G)
5.6
23.0
54.0
81.0
119.0
Depth
12
12
16
24
28
Hidden dimension
768
768
1024
1024
1152
Attention heads Patch size
12
16
16
16
1x4
2x2
2x2
2x2
2x2
Training epochs
240
240
240
240
Table 4: Configurations on the ImageNet 256x256 dataset. We detail the specifications for our models, which are based on the Diffusion Transformer (Peebles Xie 2023) architecture. The configurations scale from 131M parameter model (B/4) to a 676M parameter model (XL/2) to evaluate performance across different capacities.
B MORE VISUAL SAMPLES
We present additional visual results generated by our models for the ImageNet 256x256 and CIFAR- 10 datasets
Figure 4 and Figure 5 respectively.
Figure 4: Curated conditional 1 I-NFE samples by our XL/2 model on the ImageNet x56x256 dataset.
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Figure 5: Uncurated 1-NFE unconditional samples by our U-Net model on the CIFAR-10 dataset.
C PSEUDO-CODE FOR GUIDED TRAINING PROCESS
To facilitate a deeper understanding of the implementation, we detail the pseudo-code for the guided training process in Algorithm
Algorithm Train Solution Flow Models with CFG
Input: model fo t, s, c), Flow Matching data ratio CFG strength w, velocity mix ratio m
Sample a data batch: no, C ~v Pdata c) Split the data batch for two losses by X: (x. 225CM) = zo, (cFM CSCM) =c
Sample tpM. tscM, IsCM, SSCM according to (olsubsection 4.2
Compute 2.FM, ot .... ..-SCM 2z 2,SM using standard Flow Matching framework Compute FM and s,SCM with Eq. 28 with w and m
Randomly replace mix and SCM with e.FM and uSCM with CFG drop rate 0.1
mis
Replace JM and SCM with the empty label 6 correspondingly
Compute (0) according to [q. by replacing the ( c) term with FM
Compute (f) according to Eg. 27 by replacing the vy ( 1, term with SCM
mix
Update fo via gradient descent according to (6) = AC (0) + (1 A) (0)
D TRAINING EFFICIENCY
is important to note that GPU memory consumption and training speed rely heavily on implemen- tation details, hardware, and frameworks. To ensure a fair comparison, evaluate the training costs of MeanFlow and SoFlow models using an identical PyTorch codebase, differing only by replacing solution consistency loss with the MeanFlow loss.
The results are presented in Table 5 Currently, PyTorch's memory-efficient attention implemen- tation supports standard forward and backward passes but lacks support for the Jacobian- vector product (JVP) computation required by McanFlow. Consequently, MeanFlow must rely on the stan- dard "math" attention backend. In contrast, solution consistency loss is fully compatible with
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memory-efficient attention. As result, model reduces peak GPU memory usage by approxi- mately 31% and increases training speed by 23% compared to MeanFlow, demonstrating significant gains in computational efficiency.
MeanFlow (Math Attn.)
SoFlow (Math Attn.)
SoFlow (Efficient Attn.)
GPU Memory Training Speed
51.45 GB
38.95 GB
35.44 GB
2.39 iters/sec
2.8 iters/sec
2.94 iters/sec
Table 5: Training Speed and GPU peak memory usage comparison.
E THEORETICAL ANALYSIS
In this section, we provide additional theoretical analysis to demonstrate that our model can effec- tively learn the global ground truth solution function based our local consistency objectives.
Specifically, we analyze the error between our model fo (2t, t, s) and the ground truth f(z,, t, s). Using the fundamental theorem of calculus, the error be expressed as:
f s) fo( = J.
(29)
This equality holds because the boundary terms satisfy fo([ L, 8) fo L, s) and [o( 1, 8), 8) ((,,,, Applying the chain rule, the integrand term can be expanded as:
0 म. (,, = (),
(f (a t,),
(30)
Since f(z,, t, s) is the ground truth solution function of the velocity ODE defined by v(z,, t), have 0,5 1, 1, = vff(r,, 1, 1). We define the residual term Ro 1, 8) as:
Ro t, 8) = fe t, s)u(z,, t) + f2 xt, t, s),
(31)
where we refer to Ro( 1, s)1 as the partial differential equation (PDE) error. Substituting this into Eq. 30, we simplify the expression to:
a di
(fo, ()).
(32)
Combining this with Eq. 29
obtain:
((1,1,)) so(o))8)= J:" (F1,
(33)
During the training process, model minimizes the difference loss Il (Vt, t, fo- (xx + v(2,, (1 - 4,, with an adaptive scaling function. To derive an upper bound for the error between our model's prediction and the ground truth, assume that after training, the loss term [lfo(z,, t, s) - fo (æt + o(2,, () t), I, l|2 is uniformly bounded by emax - Z/. where Cmax is constant and It 1| represents the magnitude of the error.
Assuming fe t, s) and v(z,. t) are twice-comme cicc-continuuusll differentiable with bounded second-ords derivatives, we can apply the Taylor expansion bound the discrepancy:
fo- (It + v(z,, (1 - 1), Ro(x,, t, s) D)1 H(t 11)
(34)
This implies the remainder is uniformly bounded by H(t - 1)2 given the bounded second-order derivatives. Using the triangle inequality, we combine these bounds:
Ro t, s)112 S Cmax + H - Cmax + H tlr(k,
(35)
where rkk, K) = | is the decreasing schedule function determining discussed in our paper, and k, K denote the current and total training steps, respectively. For simplicity, we denote this
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upper bound as 6, ie., || t, s)ll I 6. Finally, substituting this back into Eq. 33 arrive at the global error bound:
(Ie, - fola,, 8)]12 S S. R| (f , 1,0,, ()) S - t/6.
(36)
As the training loss upper bound Cmax and the schedule function value r(k: K) decrease sufficiently during training, our model fo (x),,t,)) effectively approximates the solution function f t, s) with low error.
Furthermore, we provide theoretical analysis to show that our objective not only enables our model to learn the ground truth solution function also implicitly minimizes the ODE error [[0gfo(r,,t,s) v v(fo(1z, s),), s) To derive a bound for the ODE error, make three mild assumptions. First, we assume that after training, the residual satisfies (x, S for all I I 1, where ond, we assume that v1 t) is continuously differentiable and Lipschitz continuous with respect to Ty, ie., ||u( t) v(ye, t)lla I Lul|æ; Bell2- Third, uously differentiable and its partial derivative with respect to is Lipschitz continuous; that is, 10gRo(xr, (,, 81) 0g Ro (3 82)ll2 L|s1 s2/, where L is the Lipschitz constant. We first prove the following lemma:
E R and 0
6max + H|s t/r(k: K) as discussed above. Sec-
assume that Rø (, is contin-
Lemma.
Under the stated assumptions, Rg s) is uniformly bounded by:
||0gRa t, s)1 - 2V
(37)
Proof. For brevity, we denote Ra Ra(,,, t, s) as g(s) R". We aim to bound [gg( given that [|g())ll2 S 5 and llg((s,) g (s2)ll I L|s1 s2). According to the fundamental theorem of calculus, we have:
g(s + h) g(s) S'' g'ss Th)/dr.
(38)
We can rewrite the term ('))) as follows:
g((s) g (s) h g(s +1) - g(8) S/¹ g'ss Thihdr
(39)
= g(s - h) - g - - (g((s - (s()))drr.
Taking the Euclidean norm both sides and applying the triangle inequality yields:
[hlll' (s)lla < (lg(s + h)]l2 + ([(())l2 + 11, (g'(s + th) - g (s))
(40)
Using the assumption [gg())|2 < 5 and the Lipschitz continuity of g'(s), we can bound the terms using the triangle inequality for integrals:
[klll' (s))2 v 5 J. llg((s rh) o (x)]l2/bdd dr
(41)
$ 28 +
L|Th|[h|dd dr
(42)
Jo
28 Lh2
dr
(43)
28 Lh?
(44)
Note that this inequality holds for any h + 0. We consider the case where h 0 to form upper bound. Dividing by [h], we obtain:
25
Lh
(s)(
<
+
Wh. > 0.
(45)
h
2
18

---

Published as a conference paper at ICLR 2026
To find the tightest bound, minimize the night-hand side with respect to h. The minimum occurs when the derivative with respect to h is zero (since the second derivative positive), yielding:
28
-
=
L
46 -
h 2/ 8
(46)
h
Substituting this optimal h back into the incquality gives:
28
L
ll (s) S
2 T.
VLO vLS = 25.
(47)
2/8/L
2
This concludes the proof of the lemma.
Next, taking the partial derivatives with respect to 8 in Eq. 33 we have:
(af s) s) asfo(z,,t, t, (o( s) 8, + f. t,
(48)
Using this equation, we can write the ODE residual vector follows:
0sfo(2e,4, (())
= = t, ((fo(21,1,8) Ro ( ': 0 2, (49)
= =(f((z,,t)) 8), - uff((z,, R (,,,, 8), L:
We first use the Lipschitz continuity of (x t), along with Eq. 36
(, s) v(fo(zr, s), 11 S L. (f s) folm,, t, s) S Lols t/ō.
(50)
Since have uniform bound [[ 1, || < 6, it follows that:
|| (Ze, s), s)1(2 S 6.
(51)
Finally, for the integral term, using the triangle inequality and the lemma, we have:
Vr. 0g (o,o
< J.a
1,2), ()),,,)))222)) 1|2d
2
(52)
2v Lödl
2|s LT.
Combining these three terms, we obtain:
[a
(r, (Mx, 1, 8) v(forr, 4, 8), a)1 S 2 O(V8).
(53)
We have theoretically proved that the ODE error t,) ((f)) t, s,,,, s) l12 of our model fe f, s) is bounded by the PDE error Ro , f, s)]12. where 5 Emax + s K). During
training, the training erгor emax minimized and the schedule function r(k, K) also decreases small value. Thus, we conclude that our model implicitly minimizes the ODE error during training with theoretical guarantee.
LARGE LANGUAGE MODEL USAGE
We only adopt large language models to polish the writing.
19


Reviews:
[{"review_id": "mVNExk98ed", "reviewer": "Reviewer_eADR", "summary": "The paper derive a loss for learning the flow map $f(x_t, t,s)=x_s$ following the supervision of flow matching. The authors closely follow the derivation of Align Your Flow [1] and MeanFlow [2]. The main contribution of the paper is showing that applying the linear approximation of a small step (i.e., $t\\rightarrow s$) as in equation 13 is not needed and the flow map can be learn with a finite difference approximation instead as in equation 14.\n\n[1] Sabour, Amirmojtaba, Sanja Fidler, and Karsten Kreis. \"Align Your Flow: Scaling Continuous-Time Flow Map Distillation.\" arXiv preprint arXiv:2506.14603 (2025).\n\n[2] Geng, Zhengyang, et al. \"Mean flows for one-step generative modeling.\" arXiv preprint arXiv:2505.13447 (2025).", "questions": "Could the authors provide empirical comparison for training costs of SoFlow vs. MeanFlow ?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The paper is well written (with slightly confusing notation).\n\n2. The proposed method is backed by theory.\n\n3. The proposed method achieve state-of-the-results (though comparable with baselines).\n\n4. Proposed method suggest that in practice using JVP in [1,2] is not needed, potentially saving training cost.", "weaknesses": "1. the proposed method is of low novelty:\n\n    a. The presented derivation is very close to Align Your Flow [1].\n\n    b. The proposed preconditioning (referred as \"Euler parametrization\" ) of the flow map $f_{\\theta}(x_t,t,s) = x_t + (s-t)F_{\\theta}(x_t,t,s)$ was already suggested by [1] and essentially results in making the actual network parameters $F_{\\theta}$ learn the mean field as in [1].\n\n    c. Equation 13 was already used by [1], and composed with the Euler parametrization returns the mean flow objective [2].\n\n    d. Approximating $v_t(x_t,t)=\\mathbb{E}[\\alpha_t' x_0 + \\beta_t' x_1]$ with $\\alpha_t' x_0 + \\beta_t' x_1$ inside the flow map as in equation 14 is justified only for small $t-s$ where marginalization trick [3] can be applied as done in [2].\n\n2. The statement that the proposed method \"consistently outperforms Meanflow\" (line 425) is somewhat limited given that in Table 2 for DiT-L and DiT-XL the difference between SoFlow and MeanFlow is 0.12 and 0.08 FID (resp.). \n\n3. Though is seems the two method, SoFlow and MeanFlow, achieve comparable results, since SoFlow do not use JVP it can potentially reduce training cost which can be significant in large scale training. However, no empirical comparison or analysis of this is found in the paper.\n\n4. No empirical validation of the method on the multistep setting. In particular the range 2-4 steps since previous work [1,2] reported state-of-the-art results in this range.\n\n[3] Lipman, Yaron, et al. \"Flow matching guide and code.\" arXiv preprint arXiv:2412.06264 (2024).", "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": "tdII5V64b9", "reviewer": "Reviewer_izTu", "summary": "This manuscript proposes a variant of consistency models, which can be trained from scratch without JVP and achieve better 1-NFE FID than MeanFlow on ImageNet 256x256. The method involves a combination of flow matching and consistency training loss with carefully designed scheduling, loss weighting, and CFG handling. The results are promising, showing that JVP is not the only way to achieve comparable results to MeanFlow on ImageNet.", "questions": "A few questions about design choices:\n- Is a randomly sampled $s$ really necessary? For 1-NFE sampling, theoretically $s$ can just be zero. I wonder if there is a performance gain when using a random $s$ compared to setting $s=0$.\n- Regarding the velocity mix ratio, $m=0$ is not tested in the ablation studies. Theoretically $m=0$ could work as long as there is a FM loss, basically the FM loss learns the velocity from data, and then the consistency loss at $m=0$ reduces to consistency distillation from the learned velocity (basically it's online self distillation).", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The ImageNet 1-NFE results of the proposed method is very strong, beating MeanFlow without using the inefficient JVP operation. This could imply a very positive message to the community: JVP may not be really necessary for strong performance in consistency training/distillation. Rather, a well-designed training objective with careful CFG handling could also achieve something very competitive. \n- All technical details (theories, design choices, ablation results) look very sound and the presentation quality is good. Notably, the introduction of velocity mix ratio is an interesting design choice, which clearly makes sense in terms of reducing variance and accelerating convergence.", "weaknesses": "- While I'm very positive on based on the evaluation results, this manuscript does not really introduce novel fundamental approaches that clearly distinguish it from prior art. For example:\n\n  - The model architecture is similar to prior CM extensions that learns mappings from arbitrary t to s (e.g., CTM, MeanFlow)\n  - Combining FM loss with consistency loss is also seen in prior work (e.g., CTM)\n  - The CFG handling method can be broadly seen as an online guidance distillation approach (where the unconditional velocity is modeled by an unconditioned network prediction [1, 2]). MeanFlow also uses similar CFG handling (which is very important to FID).\n  - For the consistency loss, the velocity mixing design is essentially mixing the consistency distillation objective (with $v_\\text{guided}$ as the teacher, essentially it's online self distillation) with the consistency training objective (using real data).\n  \n  While this work does a very good job integrating all these techniques together, the limited novelty makes it difficult for a clear accept in my opinion.\n\n- No multi-step results are presented in the paper. Multi-step generation is closer to applications of large-scale models. MeanFlow demonstrated very strong 2-NFE results. Beating it on 2-NFE could have further strengthened this work.\n\n[1] Chen et al. Visual Generation Without Guidance. ICML 2025\n\n[2] Tang et al. Diffusion Models without Classifier-free Guidance", "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": "3pP1L45qH2", "reviewer": "Reviewer_aKZH", "summary": "This paper proposed the SoFlow, a one-step generation paradigm for flow-matching diffusion models (FM). The main motivation for this paper is to directly approximate the unique solution of the ODE of the FM. The experimental results show that SoFlow could successfully achieve the one-step generation.", "questions": "1. How much influence does using the expectation of the velocity to replacing the velocity term? Is there any way to quantify the bias after replacing it with expectation?\n\nTo summarize, I believe this paper is novel. Approximating the ODE solution directly is an interesting approach. The experimental results prove its validity. But the writing should be clearer. The primary concern for me is the writing of this paper, which I believe does not meet the standards of ICLR. During the rebuttal, I recommend that the author improve the quality of their writing. Therefore, I rate it as marginally above the acceptance threshold, temporarily.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 1, "contribution": 4, "strengths": "1. This paper proposes a novel method for one-step generation. This paper demonstrates how to approximate the unique solution of the ODE for the FM using sound theoretical proofs. Then, they build a learn target to train the model. In this way, one-step generation could be regarded as the solution from T to 0, thereby being directly solved via SoFlow. To the best of my knowledge, this method is the first one to achieve this and will truly benefit the development of FM. \n\n2. The experimental results clearly show that SoFlow works well.", "weaknesses": "1. I have to say that the writing of this paper should undergo a significant revision. Firstly, in Eq. 1, there is no explanation for $\\alpha^{'}_t$ and $b^{'}_t$. Eq 5 should at least be split into two equations, not integrated into one. I highly recommend that the author add the integrated formulation of a unique solution. In this way, the reader could clearly know why $f(x_t, t, t)$ contains two time-related variables, one for the start and one for the end. Meanwhile, it may be better to expand Eq. 6 - Eq. 12 in the Appendix. This section is one of the most crucial parts of this paper and warrants a more detailed explanation. Meanwhile, this could simplify the audience's budget. Then, why is there no equation number in lines 192 and 202 lines.  The equation in 192 lines should also explain how it is derived. This is also important and directly related to your loss. Then, $\\alpha$ and $b$ are redefined twice in 123 lines and again in Eq. 10, which may mislead the reader. In the end, the CFG section should reorder the logic. Currently, it is challenging for the reader to understand the rationale behind introducing a condition. The solution of the ODE itself does not contain the condition $c$. Meanwhile, the dataset itself cannot recognize the condition. This will lead the reader to question the validity of the CFG section.\n\n2. I think the author should add a section to illustrate how to achieve distillation. Well, training from scratch is necessary, but distillation is also essential and can further enhance the contribution of this paper (can refer to the consistency models).\n\n3. Experimental results in the Table. 1 has some inconsistency in the Table. 2. For example, the noise scheduler part shows the best FID is 59.97. But the true best FID is 3.35.", "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": "nmefpvbDUb", "reviewer": "Reviewer_NydX", "summary": "### **Motivation**\n- The paper is motivated by the computational inefficiency of recent Consistency Models, which still rely on local supervision using Jacobian–Vector Products (JVPs) to enforce consistency of velocity fields at intermediate timesteps.  \n- Such methods remain local approximations of ODE dynamics — they match instantaneous velocities rather than learning the global solution function that defines the full trajectory.  \n- To overcome this, the authors propose directly modeling the ODE solution itself with a neural network, aiming to bypass both the high JVP cost and the reliance on multi-step numerical integration or teacher distillation.\n\n\n### **Methods**\n- Approximates the true ODE solution using a neural network \\( f_\\theta(x_t, t, s) \\) under sufficient conditions ensuring equivalence.  \n- Introduces **Flow Matching Loss** (for boundary alignment at \\( s=t \\)) and **Solution Consistency Loss** (for temporal consistency across \\( s<t \\)).  \n- Removes the need for JVP computation and integrates Classifier-Free Guidance (CFG) directly into training for guided one-step generation.\n\n### **Contributions**\n- Presents a solution-function–based training approach as an alternative to velocity-field supervision in existing flow and consistency models.  \n- Describes a training-time CFG procedure (including variance-reducing velocity mixing) compatible with one-step inference. \n- Reports improved FID-50K on ImageNet 256×256 (3.35 vs. 3.43 for MeanFlow) using the same DiT-based backbone and cifar-10", "questions": "Please refer to the “Weaknesses” section for detailed points.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- **[S1] Clear motivation and formulation:** The paper articulates a coherent motivation for moving from locally supervised velocity-field models toward a solution-function approach, addressing computational inefficiency in consistency-based methods.\n\n- **[S2] Consistent empirical improvements:** SoFlow demonstrates FID gains across all MeanFlow model sizes on ImageNet-256×256, while maintaining stable one-step generation, indicating practical viability of the proposed approach", "weaknesses": "**[W1] Lack of theoretical grounding in global solution learning** \n \nSoFlow claims to learn the global ODE solution operator (Eq. (6)) directly, but its training relies only on local consistency objectives (eq. (14)). While this condition theoretically implies $ f_\\theta = f $, the actual supervision—based on first-order Taylor approximations and stochastic sampling of $(t, l, s)$—remains inherently local, creating a gap between the theoretical formulation and practical optimization.\n\nTraditional Flow Matching constructs a global velocity field via CFM, where local supervision is provably consistent with the true global dynamics. Similarly, Consistency Models enforce locally consistent trajectories using Jacobian-based constraints to form a globally coherent velocity field. In contrast, SoFlow extends this idea to the solution trajectory itself, but without a comparable theoretical foundation.\n\nFrom a functional approximation perspective, to the best of my knowledge, there is no established theoretical evidence that a neural parametrization of the ODE solution trajectory can recover global behavior solely through such locally defined objectives. As this forms the paper’s main conceptual contribution, providing additional theoretical analysis or empirical validation would help substantiate the claim.\n\n**[W2] Ambiguity in guidance supervision**\n\nUnlike MeanFlow, which learns a local velocity field\n$v_\\theta(x, t) \\approx \\mathbb{E}_{p(x_0, x_1|x_t)}[\\alpha_t' x_0 + \\beta_t' x_1]$ and guarantees global trajectory consistency via explicit numerical integration,\n\nSoFlow directly parameterizes a solution operator\n$f_\\theta(x_t, t, s) \\approx x_t + \\int_t^s v(f(x_t, t, u), u)du.$\n\nThis global mapping is nonlocal—it depends on the entire integral path of $v$—and therefore the training problem is fundamentally different in nature.\n\nThe proposed solution consistency loss (Eq. 14) is based on a first-order Taylor approximation:\n$f_\\theta(x_t, t, s) - f_\\theta(x_t + v(x_t, t)(l - t), l, s) \\propto \\partial_1 f_\\theta(x_t, t, s) v(x_t, t) + \\partial_2 f_\\theta(x_t, t, s),$\nwhich supervises local temporal consistency between nearby timepoints $(t, l, s)$.\nHowever, this formulation does not explicitly enforce\n$\\partial_s f_\\theta(x_t, t, s) = v(f_\\theta(x_t, t, s), s),$\nthe defining property that would make $f_\\theta$ an integral curve of the true ODE.\nThus, even if Eq. (6) provides sufficient conditions for $f_\\theta = f$,\nthose conditions are not directly constrained in practice, leaving a theoretical gap between local training objectives and the desired global solution behavior.\n\nMoreover, under Classifier-Free Guidance (CFG), SoFlow supervises a guided trajectory\n$f_\\theta(x_t, t, s; c, w),$\nimplicitly assuming that\n$\\partial_s f_\\theta(x_t, t, s; c, w) =v_g\\big(f_\\theta(x_t, t, s; c, w), s \\mid c\\big), \\quad v_g(x, t \\mid c) = wv(x, t \\mid c) + (1-w)v(x, t),$\nwhere the unconditional component $v(x, t)$ is itself an approximation.\nIn MeanFlow, this guided field $v_g$ remains a local and well-defined quantity.\nIn contrast, SoFlow assumes that the integrated, extrapolated trajectory is itself consistent with some valid conditional ODE solution—an assumption that is neither enforced by the objective nor theoretically guaranteed.\n\nFrom a mathematical standpoint, the learning problem shifts from a well-posed local field approximation\nto an ill-posed operator regression problem over nonlinear solution trajectories.\nThe paper would benefit from clarifying whether the learned $f_\\theta$\nis empirically observed to satisfy\n$\n\\partial_s f_\\theta(x_t, t, s; c, w)\n\\approx\nv_g(f_\\theta(x_t, t, s; c, w), s \\mid c),\n$\nand whether such trajectories can be interpreted as genuine integral curves of a consistent guided flow field,\nrather than heuristic extrapolations fitted by the loss.\n\n\n**[W3] Limited evaluation of model performance and validity:**  \n\n  The paper reports only FID scores, which primarily reflect perceptual quality but not distributional coverage or faithfulness to the learned dynamics. To properly assess whether the proposed solution operator $f_\\theta$  approximates the true ODE solution and preserves the target data distribution, additional quantitative evaluations would be valuable — for instance:  \n  - Inception Score (IS) or precision/recall to measure coverage and diversity.\n  - Deviation or residual norms (e.g., $\\| \\partial_s f_\\theta - v(f_\\theta, s)\\|$) to measure how closely the learned operator satisfies the underlying ODE.  \n\n Such analyses would better demonstrate whether SoFlow learns a genuinely faithful solution operator rather than merely optimizing perceptual similarity.\n\n**[W4] Explanation for scheduling and mixing effects:**\n \n  While Table 1(a)–(f) includes results for the $k$ scheduling (Eq. 16) and the velocity mix ratio $ m $, the paper provides only empirical trends without clarifying the underlying mechanism.  \n  For instance, the paper states that a smaller $ m $ improves performance by reducing the randomness of the velocity term, but it is unclear whether this effect arises from variance reduction, regularization, or an implicit bias toward model-predicted guidance.  \n  Similarly, the scheduling function $ r(k, K) $ in Eq. (16) is varied (exponential, cosine, linear), yet the text does not explain how the schedule theoretically affects optimization stability or convergence. \n \nCould the authors elaborate on why these heuristics influence performance — and whether the model remains stable and consistent under different schedules or $ m $ values?\n\n**[W5] Comparsion table:**  \n  While the proposed method focuses on one-step generation and demonstrates consistent improvements over MeanFlow across model sizes, the experimental comparison remains somewhat narrow.  \n  For a fairer assessment of one-step efficiency and quality, it would be beneficial to include either (i) 2-step results of recent consistency-based models or (ii) one-step flow distillation models such as [1~5].\n  Including these comparisons would allow readers to more accurately contextualize SoFlow’s efficiency–quality trade-off against both direct training and distilled few-step baselines.\n\n[1] sCT, Simplifying, Stabilizing, and Scaling Continuous-Time Consistency Models., arxiv, 2024\\\n[2] CTM, Consistency Trajectory Models: Learning Probability Flow ODE Trajectory of Diffusion, ICLR, 2024\\\n[3] IMM, Inductive Moment Matching, ICML 2025.\\\n[4] Align Your Flow: Scaling Continuous-Time Flow Map Distillation, arxiv, 2025\\\n[5] Progressive Distillation, Progressive Distillation for Fast Sampling of Diffusion Models, Neurips, 2023", "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)