FACM: Flow-Anchored Consistency Models
Authors: Yansong Peng
Kai Zhu
Yu Liu
Pingyu Wu
Hebei Li
Xiaoyan Sun
Feng Wu
Venue: ICLR 2026 Poster (2026)
Keywords: Image Generation
Consistency Model
Primary area: generative models

Abstract: Continuous-time Consistency Models (CMs) promise efficient few-step generation but face significant challenges with training instability. We argue this instability stems from a fundamental conflict: Training the network exclusively on a shortcut objective leads to the catastrophic forgetting of the instantaneous velocity field that defines the flow. Our solution is to explicitly anchor the model in the underlying flow, ensuring high trajectory fidelity during training. We introduce the Flow-Anchored Consistency Model (FACM), where a Flow Matching (FM) task serves as a dynamic anchor for the primary CM shortcut objective. Key to this Flow-Anchoring approach is a novel expanded time interval strategy that unifies optimization for a single model while decoupling the two tasks to ensure stable, architecturally-agnostic training. By distilling a pre-trained LightningDiT model, our method achieves a state-of-the-art FID of 1.32 with two steps (NFE=2) and 1.70 with just one step (NFE=1) on ImageNet 256$\times$256. To address the challenge of scalability, we develop a memory-efficient Chain-JVP that resolves key incompatibilities with FSDP. This method allows us to scale FACM training on a 14B parameter model (Wan 2.2), accelerating its Text-to-Image inference from 2$\times$40 to 2-8 steps. Our code and pretrained models:
https://github.com/ali-vilab/FACM.

Published as a conference paper at ICLR 2026
FACM: FLOW-ANCHORED CONSISTENCY MODELS
Yansong Pengl.2-, Kai Zhu!!?, Yu Liu', Pingyu Wul.2, Hebei Li', Xiaoyan Sun 3. Feng Wu
'University of Science and Technology of China ?Tongyi Lab IInttittee of Artificial Intelligence, Hefei Comprehensive National Science Center pengyansong ustc..c... edu.cr.
THE WORLD
Figure 1: FACM scales effectively to high-resolution text-to-image synthesis with a 14B parameter model (top) and achieves state-of-the-art few-step generation ImageNet 256 (bottom).
ABSTRACT
Continuous-tim Consistency Models (CMs) promise efficient few-step generation but face significant challenges with training instability. We argue this instabil- ity stems from fundamental conflict: Training the network exclusively on a shortcut objective leads to the catastrophic forgetting of the instantaneous veloc- ity field that defines the flow. Our solution is to explicitly anchor the model the underlying flow, ensuring high trajectory fidelity during training. We intro- duce the Flch-Anchored Consistency Model (FACM), where a Flow Matching (FM) task serves dynamic anchor for the primary CM shorteut objective. Key to this Flow-Anchoring approach is a novel expanded time interval strategy that unifies optimization for single model while decoupling the two tasks ensure stable, architecturally-agnosit. training. By distilling pre-trained Light- ningDiT model, our method achieves state-of-the-art FID of 1.32 with two steps (NFE=2) and 1.70 with just one step (NFE=1) on ImageNet 256x256. To address the challenge of scalability, we develop a memory-efficiae
Chain- JVP that resolves key incompatibilities with FSDP. This method allows us to scale FACM training on a 14B parameter model (Wan 2.2), accelerating its Text- to-Image inference from 2x to 2-8 steps. Our code and pretrained models: https: / wiigh..t......... com/ ali vi wivlb//lAAC
Work done during their internships at Tongyi Lab. Corresponding author

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Published as a conference paper at ICLR 2026
INTRODUCTION
As generative models scale to unprecedented sizes and applications demand real-time synthesis, the need for efficient, few-step samplers has become paramount. Consistency Models (CMs) have emerged as a promising paradigm for few-step generation (Song et al. 2023). Early successful works were largely based on discrete-tinn tecerete-timeeee formulations (Song et al. 2023 [Song & Dhariwal [203) Luo et al.) 2023), which are inherently prone to discretization errors. While their continuous- time counterparts can circumvent these errors, they have been historically hindered by severe training instability. Recent approaches, notably sCM (Lu & Song 2024), have made significant strides in stabilizing continuous- time training through a combination of regularization techniques and architectural modifications. Concurrently, Flow Mapping methods (Geng et al., 2025; Sabour et al., 2025 Wang et al. (025)
exemplify another line of research that has aimed to stabilize training. By reformulating the shorteut objective itself, these methods either model the "average velocity" to arbitrary endpoints, or introduce additional self-consistency constraints between multi-timesteps. Although these methods provide stable few-step sampling, they fail to address the root cause of instability. Their reliance single, over-coupled objective to learn the flow and shortcut simultaneously prevents explicit task decoupling and compromises perfect trajectory fidelity.
This paper addresses the root cause of instability in the continuous CM objective from a novel perspective. We posit that the standard continuous CM objective, while powerful for learning direct "shortcut" across a probability flow, is inherently unstable when trained in isolation. This is because the approach implicitly assumes the model has robust understanding of the underlying flow. However, training exclusively on the shortcut objective can induce catastrophic forgetting of this flow, leading to training collapse. Our key insight is that stability can be achieved by explicitly anchoring the model in the very flow it is shortcutting.
The most direct way to achieve this Flow-Anchoring is re-introduce the explicit training of the instantaneous velocity field that defines the flow. We propose that an objective based on Flow Match- ing (FM) (Lipman et al. 2022) can act as crucial anchor, enabling the primary shorteut objective to be trained effectively. Based on this principle, we introduce the Flow-Anchored Consistency Model (FACM), which employs a simple yet effective training strategy combining two distinct objectives:
Flow-Anchoring Objective that learns the flow's velocity field to provide stability.
Shortcut Objective that learns the efficient one-step consistency mapping.
Our architecturally-agness method is stabilized by innovative expanded time interval strategy that decouples these objectives into distinct domains, while forming a continuous target that unifies the optimization for single model, supporting high-fidelity and stable training. By distilling a pre- trained LightningDiT model, our approach sets new state-of-the-an FID scores of 1.70 (NFE=1) and 1.32 (NFE=2) on the ImageNet 256x256 benchmark. To enable scalability, we solve a key memory bottleneck caused by the Jacobian- Vecctian--Vectr Product (JVP), which is incompatible with modern training techniques like Fully Sharded Data Parallel (FSDP). We introduce a memory-efficient Chain-JVP that computes derivatives sequentially by module, avoiding prohibitive memory spikes. This allows us to train 14B parameter model and accelerate its inference from 2 40 to just 2-8 steps.
2 BACKGROUND
Diffusion and Flow Matching, Generative models aim to transform a prior distribution po (e.g..
(0, I)) to a data distribution P1- A dominant approach Diffusion Models (Ho et al. 2020 Song
ct al. 2020 Karras et al. 2022). which learn to reverse a predefined noising process. Flow Matching
(FM) (Lipman et al. 2022 Liu et al. 2022 Albergo & Vanden-Eijnden 2022 Albergo et al. 2023
Kingma& & Gao 2024 offers a more direct framework to learn the probability flow ODE by regressing its output against target velocity dæ,/ = vg (*,,,,) t). A common approach uses the OT-FM path
2; (1 - t)aso fæ between a noise sample æn and data sample æ1, which has constant
conditional velocity of 01 - Zo. This leads to the practical FM objective:
(f) = Ey,, |[
- (x) -20)1
Consistency Models. Consistency Models (CMs) (Song et al. 2023) are trained to map any point 24 on an ODE trajectory directly to its endpoint æi in a single evaluation. While early successful
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works were largely based discrete-ting -imettt-emmeeee formulations that are prone discretization errors (Song Dhariwal 2023 Geng et al. 2024 Luo et al. 2023 Zheng et al. 2024), our work focuses on the continuous- time formulation. This approach requires the total derivative of the model's output to be zero: dfo (tr
With the standard parameterization fo t) æt + (1 t) Fa (2 8) and the boundary condition So (x1, 1) 21, this implies the network Fg must satisfy:
dt
Fo -1
dt
Here, represents the conditional velocity æ1 To from the underlying flow. In the distillation paradigm, this velocity is provided by pre-trained FM teacher. This objective, relying on Jacobian- vector product (IVP) for the derivative term, is notoriously unstable to train (Lu Song. 2024).
Recently, Flow Mapping methods ZZouu et al., 2025 Geng et al., 2025 (Sabour et al. 2025 Wang et al., 2025] Guo al. (225)) have extended consistency models with a unified objective, but they do not address the root cause of instability and compromise perfect trajectory fidelity.
FLOW-ANCHORED CONSISTENCY MODELS (FACM)
This section first analyzes the core instability of continuous time Consistency Models (CMs), iden- tifying the "missing anchor" as the root cause. We then present our solution, the Flow-Anchored Consistency Model (FACM), detailing its mixed-objective training strategy. Our analysis reframes the challenge of training continuous-time Consistency Models. We argue that the instability is not an inherent flaw of the shortcut objective itself, but a consequence training on isolation, which causes the model to lose its anchor in the flow's underlying velocity field.
3.1 REVISIT THE SHORTCUT TARGET OF CONSISTENCY MODELS
To understand the mechanics of the generative shorteut, first re-examine the consistency model's learning objective. The goal of a consistency function fo (z,, t) is to map any point æ; on an ODE trajectory to its endpoint æ1. Using the OT-FM parameterization fo( t) æ₁ + (1 t) F& (2x t), the ideal shortcut fa 1) = æ can only be achieved if the network Fg learns to predict a very specific quantity:
2 xt
+ - Fo-
F)
This term has clear physical interpretation: is the average velocity required travel from point æ; to the endpoint æ1 in the remaining time 1 - t. We denote this quantity as v(æ,, t). Thus, the task of learning the one-step shortcut is equivalent to training Fş to predict this average velocity.
Now, we investigate the properties that this average velocity field must satisfy. From its definition Eq. 3 we have (1 - -1)o(a,, () æ t.. Differentiating both sides with respect using the product rule gives:
dx
-0 1) + do( (I, dt
di ((1 - o(x,, ())
(x (,.,,,) ().
(4)
dt
Rearranging the terms, we arrive key differential identity that the true average velocity field must satisfy:
v( t) = v( t) + (1 - d
(5)
dt
This identity is formally identical the continuous-ting CM learning objective (Eq. 2 and the Mcanflow identity (r 1). This confirms that the CM objective directly forces the network Fg to leam the properties of an average velocity field, thus enabling the one-step generation shorteut.
3.2 THE SOURCE OF INSTABILITY: LOSING THE FLOW ANCHOR
While Eq. correctly identifies the target, its practical implementation via the training objective = e + (1 t) 4F.- (wx.f) is notoriously unstable. The core of this instability lies in the target's self-referential nature. This dependency creates two fundamental, intertwined problems:
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Missing Instantaneous Velocity Field Supervision. The target T explicitly depends on the instanta- neous velocity v. However, the CM objective only enforces a loss on the final prediction Fg. There is no explicit mechanism ensure that the model's learned dynamics remain faithful to the underlying instantaneous velocity field v (xx, 1). The model is being asked to learn the integral of a function (average velocity) without being explicitly taught the function itself (instantaneous velocity).
Self-Referential Derivative Estimation. This lack of direct supervision on v makes the derivative term, dF,
highly unstable. The total derivative, expanded via the chain rule, is:
d.Fe (#,, (Va, )0 FFg Øl
(6)
dt
The network is optimized to estimate its own derivative to satisfy the consistency identity in Eq. Ideally, this process should facilitate a smooth transition, evolving the model from predicting the instantaneous velocity field to average velocity field that satisfies this identity. However, without stable anchor in the underlying flow, the model's output Fg quickly begins to drift. This drift has a critical consequence: the derivative term in the identity grows dominate the ground-truth velocity v, effectively diluting its supervisory signal. At this point, satisfying the identity no longer converges to the boundary condition. The training target thus becomes noisy and erratic, creating a vicious cycle that rapidly amplifies errors and ultimately leads to training collapse.
These two issues stem from the same fundamental problem: the CM objective is ungrounded. lacks stable foundation in the very flow it is supposed to shortcut. The antidote is to re-introduce the explicit supervision the instantaneous velocity field via a Flow Matching objective. This provides a stable anchor for the model's internal dynamics, ensuring that the model's 's gradient field is well-behaved, which directly stabilizes the derivative term in the CM objective and allows the primary shorteut objective be learned effectively. We term this principle Now-Anchoring.
3.3 THE FACM TRAINING STRATEGY
Based on our analysis, we introduce the Flow-Anchored Consistency Model (FACM). Instead of requiring specialized architectures, FACM employs simple and effective training strategy that mixes two complementary objectives: one for stability (the anchor) and one for efficiency (the accelerator).
3.3.1 THE FACM OBJECTIVE: AN ANCHOR AND AN ACCELERATOR
The FACM training approach harnesses the stability of Flow-Anchoring (the FM task) and the efficiency of direct shortcut learning (the CM task) within single training loop. The overall training loss. LEACM, is a sum of two complementary objectives:
CFACM = LFM + LcM
(7)
To enable the model distinguish between the tasks, each objective uses distinct conditioning signal, CFM and сcM, which we detail in Section 3.3.2]
Flow Matching (FM) Loss (The Anchor). This loss component anchors the model by regressing its output towards the instantancous velocity v. The target is constructed with a base velocity Vbssc and an optional classifier-free guidance (CFG) (Ho & Salimans 2022) term:
Ubase + (Ucond Uuncond)
(8)
where w is the guidance scale. The definitions of these components vary by training paradigm. For from-scratch training, the base is the conditional velocity, Vbass I Zo. and the guidance term is derived from the online model Fe itself. In distillation, the model is initialized with weights from a pre-trained FM model. A non-trainable copy of these weights, denoted as the "teacher" Fs- provides all velocity components for the target, with Vause = Vuncond = F2 W), making the formula equivalent standard CFG. Without CFG (w = 1), the target simply defaults to Beond. The FM loss then combines an L2 term with a cosine similarity term Lene Can b) = (a b)/ [|/[[]a||2|]b]]]:
LFM (0) E (o,, CFM) 2112 + Leos (Fo (PM), v)]
Consistency Model (CM) Loss (The Accelerator). This component acts as an accelerator, training the model to learn the generative shorteut. We interpret the consistency condition (Eq. 2
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Average Velocity (Accelerator)
Instuatanevus Velocity (Anchor)
ODE Flo
Expanded Time Interval
Auxiliary Time Condition
(t.1)
Fou (
Now Menory Gav = =(-
Cpor (C.t)
notse
noise
data
2-8=2
I 0
D
Figure 2: Two implementation strategies for the mixed cijectooeeetivv function in FACM. (A) Expanded Time Interval (default): The time domain conceptually doubled, showing the same ODE flow on two intervals. The CM task is performed on & E [0. 1]. To perform the FM task at point on the flow, the model is conditioned on сиM = 2 t, which maps the time to the alternate interval [1 to distinguish the two tasks. (B) Auxiliary Time Condition: An additional time condition is introduced the model. When r = 1, the model learns the CM task (average velocity from to 1, orange); when it learns the FM task (instantaneous velocity at I, blue).
fixed-point problem, F, = (Fa).. where the operator is T(F) & v + (1 t) ar The training objective is designed to solve this problem stably and iteratively. First, we compute the consistency residual g the stop-gradient model Fg- (Fa- = sg((F)))
dFg (x), (CM)
g Fo- (It, cCM) ((Fo)) Fo- (CM) - 0
(10)
This residual is then clamped to the range |-1, 1] to prevent extreme gradients. A perturbed target then formed as:
U Fo- (az,, cCM) at)) g.
(11)
Substituting the definition of reveals the target's 's structure as a relaxation step for the lixed-point iteration:
Uha (1 - (1) + ((())((F))).
(12)
This formulation provides a stable, interpolated learning target between the current model's output and the ideal consistency target. The final CM loss component modulated by weighting functions (x() and (() (detailed in Appendix A.3 and A.4 c)):
a norm L2 loss, Lnorss and is
(CM(O) (0) E [t()) Tenoen Fo (ax, (CM), Var)
(13)
The combination of the interpolated target Var from the CM loss and the stabilizing flow anchor from the FM loss enables effective training. is important note that our specific choices for weighting and loss functions are designed to accelerate convergence, not as prerequisites for stability, which is already guaranteed by the Flow-Anchoring principle.
3.3.2 IMPLEMENTATION OF THE MIXED OBJECTIVE
A key design question is how to encode the distinct conditioning signals, CFM for the FM loss and CCM for the CM loss, that tell the model which velocity to predict. While this conditioning can include various information like class labels, for clarity in this section, focus only the based components. We explore an effective strategy for this (Figure
Expanded Time Interval. We innovatively propose leveraging expanded time domain distin- guish between the two tasks, a strategy that requires no architectural modifications. The primary CM task operates on the interval E [0, 1). using the time directly the condition: CCM perform the FM task at the same point (defined by t), we signal this by mapping t to the alternate interval [1, 2). This is done by setting the conditioning input to CFM 2 t, which makes the two conditions decoupled, symmetric, and easily distinguishable. This mapping also ensures continuity the boundary t 1, as the CM learning objective from Eq. 2 naturally converges to the FM objective's 's target at the boundary:
To
lim u+ -
F =.
(14)
dt
This ensures smooth transition between the learning regimes.
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Algorithm 1 FACM Training
Require: Online model Fø, pretrained teacher Fs, metrics LFM, LcM
1: Sample zo; X1,t
2: Define CCM, OrM based on & (see Sec 3.3.2
3: at +- (1- 1) + 10
4: v + Fo CFM)
For training from scratch, use æ1 2g instead
5: FIm + Fa (FM)
6: FCM. V.Fo t- JVP (Fa, (Ie, CcM). (v,
>> Simultaneous forward pass and JVP
7: τ + u + (1 Fo)
8: Vaar + (1 - a()) sg( FCM) + a(l) -
> Compute relaxation target
9: LTaa LFM (FMM v) + LCM(FCM Bur))
Auxiliary Condition with Second Timestamp. Alternatively, another intuitive approach is to introduce a second time variable, r, to the model, making its full conditioning a tuple of rt, We then define CCM (1, 1) and сум (1, l). This means the model signature is effectively Fo I, r). When = 1, the model is trained on the CM task (predicting average velocity from t to 1). When r t. the model trained on the FM task (predicting instantaneous velocity at t. or from to t). We can provide this auxiliary condition to the model through a zero-initialized time embedder, which does not alter its original structure or initial output.
As shown in our ablations (Table 3), while both methods effectively stabilize training, the Expanded Time Interval strategy consistently yields the best performance. attribute this to its use of highly distinct time domains ((0, 1] vs. [1, 2], which provide clearer, more separable conditioning signals for the two tasks compared to the subtler differences in the Auxiliary Time Condition (c.g.. (t. 1) vs. (t, t)). For clarity, if t 0 represents the data distribution (i.e., 2y two + (1 11--t)a t)x1), ), the conditions for the two strategies would be vs. -t and (t, 0) vs. (t, t). respectively.
3.3.3 TRAINING ALGORITIM AND SCALABLE CHAIN-JVV IMPLEMENTATION
With the objective functions and conditioning signals defined, we present the complete FACM training strategy in Algorithm | A key component of this algorithm is the computation of the total derivative F, in the CM loss (Line 7), performed using a vacobian-vector product (JVP).
The JVP computation, however, presents critical bottlenecks when using modern acceleration tech- niques. While its incompatibility with components like Flash Attention (Dao] 2024) can resolved using methods from sCM (Lu & Song 2024), more fundamental memory bottleneck emerges from its conflict with Fully Sharded Data Parallel (FSDP) (Zhao et al. 2023). Standard JVP im- plementations require the model's 's full parameters
be materialized all-gat the operation in an FSDP setup. This reconstructs the entire parameter set on each GPU, causing prohibitive memory spike that makes training models with over ten billion (10B) parameters impossible. To overcome this, we leverage the chain rule. For a network composed of modules Fg fL f1, the JVP can be computed sequentially:
the device, forcing an
Jz,(z) v== J (= (-... (J)z(Z)) (J -()))-)))
(15)
where = =f((z=1) is the intermediate output. Our approach computes the JVP for each module sequentially, embedding this operation within the FSDP logic. Its speed consistent with standalone JVP pass, adding only standard FSDP overhead. This ensures that only one module's parameters are materialized at time. Consequently, peak memory depends on the largest module, not the entire model. and the resulting memory savings grow with the model's parameter count.
In summary, the principle of Flow-Anchoring offers a robust and fundamental solution. While other methods achieve stability, they do so with certain limitations. For instance, sCM (Lu & Song 2024) requires architectural modifications to its normalization layers, limiting its adaptability to large, pre-trained models. Other approaches like McanFlow (Geng et al. (2025). while clever, present a trade-off: by treating the instantaneous velocity as merely an edge case (r = t) of the primary average velocity objective, the learning tasks become over-coupled. As result, the supervisory

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Table I: Few-step generation on CIFAR-10 and ImageNet 256x256. "x2" indicates that CFG doubles the NFE per step. Our method sets new state-of-the the-tth-rttt on both datasets.
Unconditional CIEAR-10
Method
NFE
FID (L)
ImageNet x56
Muli-NFE Baselines
Method
Params
NFE
FID (U)
DPM Solver++ (Lu at. 2022
2,91
Multi-NFE Baselines
BDM Karras el al.
2.01
SIT-XL/2 (M et al. 2024
G75M
250x2
2,06
Fe Methods (NFF=!)
DiT-XL/Z (Peebles
2023
675M
250x2
2.27
iCT Song & Dhariwal 2023
2.83
REPA Yu et al. 2025
675M
250x2
1.42
cCT [Geng cf al. 2024
3.60
LightningDII YYao et al. 2025
675M
250
1.35
sCM (SC17 Son2
2024
2.85
Few-NFF Methods (NFF=I)
IMM Zhou
2025
3,20
iCT (Song & Dhariwal 2023
675M
34.24
MeanfJow Geng el
2025
2,92
Shoricut Frans et al. 2025
675M
10.60
FACM (Ours)
2.69
McanFlow (Geng al. 2025
676M
3.43
Few-NFE Methods (NFE=2)
FACM (Ours)
675M
1.70
TRACT (Berthelot et al.
2023
3.32
CD (LPIPST [Song
2025
2.93
Few-NFE Methods (NFE=2)
iCT-deep Song & Dhariwal
Z023
2.24
iCT (Song & Dhariwal 2023
675M
2
20.30
ECT (Geng el al. 2024
2.11
IMM Zhou at 2025
675M
1x2
7.77
(sC1) Song
2.06
MeanFlow (Cieng el al. 2025
676M
2.20
IMM Zhou at. 2025
1,98
FACM (Ours)
675M
2
1.32
FACM (Uur))
1.87
signal for the underlying flow is often diluted, which we have observed can lead to training collapses and underfitting. In contrast, FACM provides more direct and principled solution. Through our innovative expanded time interval strategy, the anchoring and shortcut tasks are functionally decoupled into distinct domains. This ensures flow anchor receives a clear, undiluted supervisory signal at all times, forcing the model to maintain a stable and high-fidelity representation of the flow. This robust theoretical foundation, combined with our scalable Chain-JVP implementation, makes FACM not only stable but also highly practical for training models at unprecedented scale.
EXPERIMENTS
4.1
EXPERIMENTAL SETUP
We empirically validate FACM on image generation benchmarks, including CIFAR-10 Krizhevsky Hinton 2009) and ImageNet 256x250 (Deng al., 2009). We evaluate models based Fréchet Inception Distance (FID) (Heusel et al. 2017) and the Number of Function Evaluations (NFE). FACM can be trained from scratch by distilling pre-trained model. Our default experimental setup involves a two-stage process. We first pre-train a FM model, incorporating our mixed-objective conditioning as detailed in Appendix (a) accelerate the subsequent distillation. We then distill this teacher using the FACM strategy. For few-step inference, we follow the standard multi-step sampling procedure for CMs as described in Appendix [A.4 (b). Further details on our experimental settings are provided in Appendix AA.4] To demonstrate scalability, we also distill 14B parameter model (Wan 2.2) on the text-to-image (T2I) task, achieving high-fidelity generation in just 2-8 steps.
(oppedix[A.4](00
4.2
MAIN RESULTS
4.2.1
COMPARISON WITH STATE-OF-THEL
As shown in Table FACM achieves sfate-of ofate-o--the-aee the-hhha results on both CIFAR-10 10 and ImageNet
256x Specifically, our method achieves FIDs of 1.70 (NFE=1) and 1.32 (NFE=2) on ImageNet 256x250 by training LightningDiT model space, and 2.69 (NFE=1) and 1.87 (NFE=2) on CIFAR-10 by training DDPM++ model (Ho et al. 2020) in pixel-space, significantly outperforming previous methods on both benchmarks. RRemarkably, our few-step model even surpasses some multi-step baselines that require hundreds of function evaluations.
4.3
A ABLATION STUDY ON THE TRAINING STRATEGY
We conduct ablation studies to validate our claims regarding the training strategy. We test on the ImageNet 256x dataset by distilling pre-trained LightningDiT model. The results provide
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Table 2: FID scores (NFE-2) on ImageNet 256x256 for different few-step methods applied to various backbone architectures. indicates our reproduction.
Backbone
Baseline (NFE=250x2)
sCM
MeanFlow
FACM (Ours)
SiT-XL/2 REPA
2.06
2.83
2.27
2.07
1.42
2.25
1.88
1.52
DiT-XL/2
2.27
2.91
2.62
2.31
LightningDiT
1.35
1.94
1.74
1.32
Table 3: Ablation stabilization strategies. All methods are distilled from the same LightiingDT
teacher. t: Our reproduction. *: For sCM, more epochs yield worse results.
Method
Params
FM epochs
CM epochs
FID (NFE=1, 1) Stable
sCM (w/o pixel norm.)
675M
800
sCM (w/ pixel norm.)!
676M
600
30'
3.04
McanFlow
676M
800
200
2.75
FACM (Auxiliary Condition)
676M
800
200
1.97
/
FACM (Expanded Interval)
675M
800
200
1.81
V
Training from scratch methods
McanFlow 1
676M
1120
2.65
FACM (Expanded Interval)
675M
800
2.27
strong evidence for our central claim: the presence of the FM objective the critical stabilizing anchor.
Different Architectures. To demonstrate the architectural agnosticism of our approach, apply architectures, including SiT-XL/2, REPA, DiT-XL/2, and LightningDTT. All methods are distilled from their respective multi-step FM models. As shown in Table 2 FACM consistently achieves the lowest FID scores across all tested backhones. This highlights that Hlow-Anchoring is fundamental principle for stabilizing consistency training that is not limited to a specific model design.
FACM, SCM, and McanFlow to range of
Stabilization Strategy. To ensure fair comparison, we distill sCM, MeanFlow, and FACM from an identical LightningDiT teacher (reproduction details in Appendix (c)). As shown in 3 FACM achieves superior results due to its principled approach to stability without requiring architectural changes. In contrast, sCM's stability is limited, depending architectural modifications (pixel normalization) and sensitive hyperparameter tuning. MeanFlow achieves robustness but at the cost of over-coupled objective (u(2, t) v(z,t) that hinders optimization by diluting the essential path modeling task. FACM's explicit task separation proves more effective, as it allows the model stably learn the shortcut while remaining anchored to the teacher's flow.
Sensitivity to Teacher Model Quality. As shown in Figure FACM's performance monotonically improves with teacher quality. This demonstrates that by explicitly anchoring the teacher's complex flow, our method can consistently benefit from stronger teachers. This confirms FACM acts as high-fidelity trajectory compression rather than a lossy compromise on the pre-trained flow.
Ablation on Key Components. As shown in Table introducing Flow-Anchoring with our Expanded Time Interval decouples the FM and CM tasks, yielding faster convergence. Fidelity is further improved as shorteut interpolation (a) and beta weighting (5) ensure smooth transition to FM supervision as 1, while residual clamping suppresses gradient spikes. Together, these components stably guide the learning dynamic via the FM anchor, leading to significantly better trajectory fidelity,
Sensitivity FM Loss Weight. The FM loss is prerequisite for stability, but the minimum required weight λFM depends on the model's initialization. Our investigation reveals nuanced picture that strongly supports a simple default choice (c.g., ÀFM 1.0). As summarized in Table (1eft vs. right), the non-finee con-finetuned setting requires at least XPM 2 0.1 avoid collapse, whereas the finetuned setting remains stable with ApM as low as 10 These results lead to a key conclusion: while a non-zero FM weight is essential, FACM is highly robust to the specific weight across several orders of magnitude once stability is achieved. This robustness, which stems from our decoupled design, makes direct summation simple, effective, and reliable choice that avoids costly hyperparameter tuning.
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Table Ablation study on key techniques on ImageNet 256
Configuration
FID@ Epochs 10 (NFE=1,
Collapse
MeanFlow Fixed r MeanFlow (0% FM)
372.3-391.5
Yes
McanFlow (75% FM)
43.03
No
Fixed r McanFlow (75% FM)
15.54
No
Flow Anchoring (Expanded Time Interval)
4.31
No
Interpolation (a(!)-
3.42
No
w/ Residual Clamping
2.86
No
w/ Beta Weighting (3(1)= 1) (FACM)
2.51
No
Table 5: Sensitivity to λPM under two settings. Left: model not pre-finetuned on < < 2. Right:
model pre-finetunl pre-finetuned on <
FM Loss Weight (ApM)
FID (NFE=1, J)
FM Loss Weight (Arx)) FID (NFE=1, D
0.0-1c-8
Collapse
0.0-0.1
Collapse 3,17-3.22
Ic-8-1c-4
2.90-5.88
0.1-10.0
1c-4-10.0
2.90-3.02
10.0-64.0
3.32-4.97
10.0-64.0
3.02-4.58
4.3.1 TRAINING DYNAMICS FACM
We analyze the training dynamics by plotting the total gradient norm under different configurations in Figure(3 Figure Baa) clearly shows that removing the FM objective leads catastrophic gradient spikes, after which the model's output immediately degenerates into pure noise (mode collapse). This confirms hypothesis that pure consistency gradient trap the model in local optimum where sacrifices endpoint fidelity in pursuit of global consistency. Figure 3(bb) further illustrates the effect of our auxiliary techniques. While each individually helps to suppress the gradient norm compared to removing them all, their combined use in our baseline model achieves the lowest and most stable gradient profile, demonstrating their synergistic effect in stabilizing the training process.
00 TM Aschiring' if Scvoral Se Stallity
Di Ffheet of Auntfory Techetiquee
3nd Cipestes
Sunciin tuta beightt with
1.5
40%
Figure 3: (a) The raw gradient norm for pure CM (w/o FM Objective) shows instantaneous spike leading to collapse, while our baseline remains stable. (b) The smoothed gradient norm for ablations of auxiliary techniques. Removing any single technique increases instability.
4.3.2 SCALABILITY ON 14B TEXT-TO-IMAGE MODEL
scales presents significant chal- lenge due to the Jacobian-vector product (JVP) computation. While recent Differential Derivation Equation (DDE) methods (Sun al. 2025 Wang et al. 2025) can yield results comparable to JVP on models up to IB parameters, we observe that they exhibit significant deviation on larger models like our 14B setup. such cases, their computed derivatives become nearly orthogonal or even opposed to the JVP result, indicating an inherent еттоr accumulation that hinders further scalability. To address this, our memory-efficient Chain-JVP provides an accurate and scalable solution. To demonstrate its effectiveness, we applied FACM to distill the 14B parameter Wan 2.2 model. This process successfully accelerated inference from 40 steps to just 2-8 steps. For the experiment, we used pre-trained Wan 2.2 Text-to- Video (T2V) model (Wan et al. 2025) as teacher on an in-house Text-to- to-lma (T2I) dataset (Despite being a T2V model, Wan 2.2 has strong image generation capability from mixed image-vided pre-traine vre-training). Furthermore, self-attention and cross
Scaling continuous consistency models
adapt the model's cross-attentin attention mechanisms to be compatible with JVP computation, following the formulation of (Lu & Song 2024). This adaptation also addresses the correctness of differentiation

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Perfocmance 15. trsinigg
Performance Ptstilattoon
Vnujectary E Fidaeiity Analyass
20:
2--
lxpicced Fime Incornal (3.11 244
25-
2-2
211
3.4
1.-
12
205
601
160
1130
103
230 190
204
1997
Figure 4: (a) Performance of student models (NFE=2) vs. teacher FM model pre-training epochs. (b) Performance vs. distillation epochs. (c) Trajectory fidelity analysis of 14B Wan 2.2 model via flow matching loss. Apart from the conditioning method, all other settings were the same.
for variable-length sequences and with 6 precision. Our visualizations for this experiment are provided in Figure and Appendix AA.1 including comparison against the baseline model, as well models (Labs) 2024b).
as the FLUX.I-Des (Labs 2024a) and the
4.4 FROM CONSISTENCY MODELS TO FLOW MAPPING MODELS
Recent work has increasingly emphasized the advantages of Flow Mapping, where the model learns to predict the average velocity from an arbitrary time f to another time (Sabour et al. 2025 Wang
et al. 2025 Geng et al. 2025 Boffi al., 2025). Flow Mapping requires the model to ensure that the derivative of fo (z,, f, consistency models along the trajectory, demanding that the model's prediction, (av,, r), remains consistent over any time interval [0,r] (detailed in Appendix (A66). We found through experiments on Wan 2.2 that FACM can be easily adapted to be compatible with the Flow Mapping formulation. This is achieved simply by changing CCM from (t) to (f) through zero- vero-initialized time embedder and projection modules, while CFM is maintained in the separate, expanded time domain. As illustrated in Figure ce), the Expanded Time Interval strategy allows the Flow Mapping to be more stably anchored to the teacher's trajectory. If the prediction of the instantaneous velocity field treated merely marginal case of the Auxiliary Time Condition (e.g., (), the FM loss becomes highly unstable, even when increasing the sampling proportion of = r as done MeanFlow. The consistently lower FM loss for the FACM condition demonstrates the superior trajectory fidelity achieved by our decoupled training strategy.
24 + (r t Fa,,,, is zero. This formulation an extension of
5 LIMITATIONS AND FUTURE WORK
Our work highlights two primary areas for future research. First, on large-scale models, a performance gap persists between samples generated in minimal steps (e.g., 1-2) and those requiring more steps (e.g.. 8). Bridging this gap by enhancing the model's expressiveness in the ultra-few-step regime is key challenge. Second, while our Chain-JVP method successfully mitigates the memory bottleneck of the Jacobian- vector product, its computational overhead remains a concern. Optimizing its efficiency is crucial for improving training throughput. Additionally, we found that our acceleration model, even when fine-tuned exclusively on T2I data, can directly accelerate T2V synthesis by targeting only the low-noise diffusion steps (SNR~ < SNR ), all without introducing flickering or detail loss. This could inform future work on efficient, high-fidelity video synthesis.
CONCLUSION
In this work, we introduce the FACM, strategy that addresses the instability of continuous-tin CMs by anchoring the network to the underlying instantaneous velocity field with Flow Matching loss. The core of this Flow-Anchoring approach is an expanded time interval strategy that unifies optimization for single model via continuous target, while functionally decoupling the anchoring and shortcut tasks to ensure high-fidelit and stability. Our method achieves new of the FIDs on both ImageNet 256 (1.70 1.87 at NFE=2). Furthermore, our Chain-JVP overcomes FSDP scalability bottlenecks, enabling us to accelerate 14B model's 's inference from 2x40 to 2-8 steps.
NFE=1 and 1.32 NFE=2) and CIFAR-10 (2.69 at NFE=1 and
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ACKNOWLEDGMENTS
This work was supported in part by the National Natural Science Foundation of China under Grant 62472399.
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A APPENDIX
A.1 THEORETICAL ANALYSIS: STABILITY AND CONVERGENCE OF FACM
This section provides detailed mathematical derivation of the stability and convergence properties of FACM, complementing the intuitive discussion in Section 3
We the same notation as in the main
Student network (online model): Fa
Stop-gradient copy: F-- (xt, t) sg( (o 6)) Instantaneous velocity field (FM anchor): v(v;; t)
CM operator (Consistency target): T[F) u + (1 t) dF de
A.1.1 STABILITY ANALYSIS: PREVENTION OF GRADIENT EXPLOSION
Step 1.1: Structural form the CM gradient.
The CM loss is (0) Ex., Fo(t,,)) t) Utar (It, |)
(16)
where Vur is the CM target derived from the operator T (Eq. 13 and Eq. 2 in the main text).
Differentiating w.r.t. 0 yields the gradient in compact form:
V CLC = VoBm.t
Fo t) - Uma (e |)
2
Ex,,
V
Fo
Utar) (Fg Utar
2
Ex,,
V (Fg -tarr) F
(17)
(Since a grap-gradient target, Va 0)
Em
V Fg (ore,
(Fo(2e,)) () Una (xee 1)
parameter sensitivity
prediction error
where &o denotes the Jacobian F& w.r.t. the parameters 6, and we write e(x,, f) Fo(x,, Unr (ate t).
(18)
Taking norms and using bbll S ||A|] || |b]|
we obtain
LoCel| < E
VoFo 1lop [|e(xx, 0]]
(19)
Thus, the CM gradient norm is governed by two independent factors:
the prediction error which determines the basic scale and direction of the gradient;
the parameter sensitivity V Fo, which acts as multiplicative amplifier.
Step 1.2: Decomposition of the error term. Recall that CM operator for a general field
(ar( () v( () (1 () (a. (2,, Va, Fe (æt, () v(x.,))).
(20)
The error of the online model Fg relative to this target then decomposes as
F
F
(1- F v
(21)
at
finctioo deviation
JVP (spatial Jucohian)
derivative
Hence, the size of is governed by the first-order spatio-tempore derivatives the (roo-grradeet) network Fg-.
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Step 1.3: FACM's stabilization mechanism. FACM combines Flow Matching (FM) with the CM objective, using shared parameters 8, and thereby stabilizes both factors in Eq. (19).
(1) Lipschitz supervision via Flow Matching. The FM loss (Eq. 9 trains Fo CPM) to match the instantancous velocity field v v( t), which is a bounded ground-truth function that does not depend on 0 and is Lipschitz in (ate, t). For standard architectures, the Lipschitz constant of F, with respect to its inputs determined only by the spectral norms of the weight matrices and the activation Lipschitz constants, all of which are shared across time conditions and 2 t. Minimizing the FM loss therefore keeps these spectral norms in a moderate range and induces a global Lipschitz bound
V (x l)pp S Leet
(22)
for all (at,,) c) in the training domain, including both the FM branch (c = - -) and the CM branch (c = t). Here, Luet represents the Lipschitz constant of the network, and || llop denotes the spectral norm (operator norm). In contrast, pure CM supervises Fa with a self-referential target. To satisfy the consistency boundary condition (i.e., mapping æ, to 21 as - 1) under the parameterization zt (1- Fo t). the network output F6 is implicitly forced to approximate the average velocity (x1 (1 - t), which blows up + 1. The CM target is therefore dynamic, potentially unbounded prediction, whereas FM always provides a bounded ground-truth target.
(2) Bounding the error via FM-anchored supervision. Next, show that the prediction error term e in the gradient (Eq. 19 remains bounded under FACM. For clarity, we focus on the deviation between the online model and the FM anchor, Fa vs. v(z,, f), and make explicit use of the expanded time interval strategy (Figure 2). where FM uses the condition CFM 2 but predicts the same physical velocity vxz,,t) t).
We begin with the triangle inequality:
to(,,, v(x,,t)] f|| S Fo(z,,)) t) Fo 2 t)-l
temporal smoothmess of F% Fo 2
(23)
v 2 0
FM emror
The first term measures how much the network output changes when the (time-related) condition goes from to 2 for the same spatial point xt, Using the fundamental theorem of calculus for the time coordinate, we write
à 2 Øग
F - (x 2 -)
Fo ((T) dr,
(24)
where ?(7) interpolates between the CM and FM conditions varies. Taking norms and applying the triangle inequality gives
Fo Fo 2 t)] - < 2- 0T 1 Fol T,
(25)
Since FM constrains the spectral norms of the weights, the partial time derivative || bounded by constant (of order L over the training domain. The integration interval has length at most 2, so
(.)) is
Fo t) Fo 2 t>|l S 2
(26)
The second term in Eq. (23 is precisely the FM error
SPM( t) Fo 2 t) v(z,, t),
(27)
whose squared norm is minimized by LrM and thus has bounded variance. Combining these bounds, we obtain uniform control of the function deviation:
Fa t) vxx,, II| s 2L + [Z,,,
(28)
Finally, because the same spectral- constraints apply to all weight matrices, the spatial Jacobian Va, Fg- and time derivative Ô, Fg in Eq. (2I) are also uniformly bounded by constants of order T Hence, all components of e remain bounded.
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Conclusion (Stability). Putting these results together, we that under FACM:
the parameter sensitivity bounded: Folop I Leeti
the prediction error e is uniformly bounded in norm by constant depending on Lust and the FM error statistics.
Therefore, the CM gradient norm satisfies
VeLcM
S C T
(29)
eliminating the gradient explosion observed in pure CM training.
1.1.2 CONVERGENCE, PROOF
We provide a concise analysis showing that FACM ensures convergence by eliminating target singularities, bounding gradient variance, and enforeing alignment through shared parameters.
Step 2.1: Mechanism of Variance Reduction. average velocity
The pure consistency target implies predicting the
21 2t. 1 1
Vi ( 1)
(30)
As + 1, any variance o² the endpoint estimate æ1 is amplified by (1 (1-t)-?. so
o²
Varlo(<;, t)
(31)
1)2
which makes the pure-CM shortcut objective conditioned near the data endpoint. FACM mitigates this by using relaxed target
Utar (Zt, 1) (1 - a(())) Fo- (at, 1) + a(t) o(2,, t).
(32)
In practice the schedule a(l)) 1 10.5 (Sec. 3) Writing = with 0 < e 1 and
using the Taylor expansion
00. (1 61-8)0.5 5 0
(33)
obtain
(tt) = 1 0.5 f +
= = t) + ((1-)))
(34)
Hence a(t) is asymptotically proportional to (1 (1-t) and
lim a(t)
(35)
2
so the factor ax)) cancels the (1 - 1)- singularity in the average-velocity term up to constant.
Consequently Varl (æ,, () (] remains uniformly bounded over i E [0, 1], providing the core mechanism for variance reduction in FACM.
Step 2.2: Bounded and Reduced Gradient Variance. Locally, Lnorn behaves like rescaled squared & loss, so the CM gradient for one sample (x,, t) satisties
Volcaa(0; x., t)]l & [ f)pp [[ t) (xee 1)1
(36)
Using the stability bound || F opp S Lnei and the uniform boundedness of Berr we obtain a finite second-moment (and hence variance) bound
E [[I S 2 R, (2(((1) E. (o(,,,)) (a 1)1 [4]
(37)
Here the boundedness follows from the variance-reduction mechanism in the previous paragraph, which shows that Var[vtrrax,, t) bound V olop < Lnet from Step 1.3: even if (()) = 1, the right-hand side is finite because both the Jacobian norm and the target variance controlled. The role of 3(1) E [0, 1] is to further reduce the integrated variance: for any non non-negative function q(t).
uniformly bounded in f, together with the global Lipschitz
E, ((())) < E,(g(t)).
(38)
with strict inequality whenever 3(t) < on set of non-zero measure. Since q(t) = Ez, F t)
(a tI], is typically largest near the data endpoint t = 1, choosing decaying schedule (e.g.. cosine) for ((t) suppresses precisely those high-variance contributions, yielding a strictly lower integrated gradient variance than the pure-CM case.
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Step 2.3: Alignment via Shared Parameters. The expanded time interval [0, 2] creates a natural synchronization mechanism between the FM and CM tasks. In the high-SNR region (t Re 1), the schedules (t) and (()) suppress the CM gradients, parameter updates there are dominated by the FM branch, which supervises Fo 2-t) to match the instantaneous velocity field v (0),, t). Because both branches share the same parameters and the stability analysis above bounds the discrepancy between (æ t) and Fa F2 2 t), this FM supervision implicitly guides Fe (Zt, (a t) to align with
the underlying velocity field. Combined with the reduced gradient variance from Step 2.2, this shared- shared-parameter parameter coupling explains the empirically faster and more stable convergence of FACM
compared to pure CM training.
A.2 ON TOTAL DERIVATIVES
In this paper, for network
N (xx,, C(t)) (e.g., Fa), its total derivative along the trajectory 2x(() 21 2o) with respect to l is given by the chain rule:
()ar + lan (with dm
dE
dN C(t)
0N v + VcN. de(l) dx.
(39)
dt
dt
The term dF (ox) is computed for the CM task. Depending on the implementation strategy dt
(Sec. 3.3.2). the conditioning CCM can be ł or tuple (t, 1). In both cases, its derivative with respect to is effectively for the time-dependent component and for any constant component. Therefore, the calculation simplifies to:
dN (2 CCM) ~C
ON
ON δt
v +
(40)
dt
where aN denotes the partial derivative with respect to the explicit time argument(s) encoded in the conditioning.
A.3 NORM L2 LOSs
The CM loss component uses a norm L,2 loss to improve stability against outliers. For model prediction p and a target y, let the per-sample squared error be [pp -yll.. The loss is then calculated as:
Luorn (p. y)
(41)
Ve+c
where e is a small constant. This formulation is equivalent to the adaptive L2 loss proposed in MeanFlow (Geng et al. 2025) with p 0.5, and behaves similarly to Huber loss, being robust to large errors.
A.4 EXPERIMENTAL DETAILS
(a) Pre-training Strategy. Our teacher models are standard Flow Matching models. While FACM distillation works perfectly with a standard, single-condition pre-trained teacher, we find that con- vergence can be accelerated by first familiarizing the teacher with our dual-task conditioning. This optional adaptation can be achieved either by pre-training from scratch with a objective (i.e., replacing the standard time conditioning with our FM-specific formats for 50% of samples) or by briefly fine-tuning a pre-trained FM model with this objective for few epochs. Fur- thermore, to prevent sporadic NaN losses during pre-training, all our LightningDiT implementations incorporate Query-Key Normalization (QKNorm), following updates in the official repository.
(b) Sampling Strategy. Our multi-step sampling (NFE 2 2) follows a standard iterative refinement process. For an N-step generation, we use a simple schedule of N equally spaced timesteps = (i 1)/N for I,..., N. The process starts with pure noise Øg- At each step i, we first compute a one-step prediction e using the model's 's output Fg: क = Zt, + (1 f, Fo (Iz, CCM). If it is not the final step, we generate the input for the next step, Tt.4.. by linearly interpolating between the predicted endpoint and new noise sample, consistent with the OT-FM framework:
Thst 1 + (1 4+ 1) where 2; ~ NN(, I).
(42)
The final output is the prediction from the last timestep, LN-
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(c) Reproduction Details. At the time of main ablations, the official codebases for Mean- Flow (Geng et al. p and sCM (Lu & Song 2024) were not yet available. A JAX implementation of MeanFlow was later released, but without reproducible configuration for its SOTA results. For controlled and fair comparison, we therefore implemented PyTorch reproductions under the exact same environment, teacher, and hyperparameters across methods. Our MeanFlow reproduction follows its two-tiee-vrribble conditioning and log-normal time sampling; following the from-scratch regime, we set this configuration struggled to converge and was therefore not used. For sCM, we incorporated all necessary techniques described in their work, including pixel normalization, tangent warmup. tangent normalization, and adaptive weighting, to ensure stable training. We did not use the TrigFlow proposed sCM, as we believe the specific flow construction is orthogonal to building continuous- time consistency models. We will release our reproductions alongside our code to ensure full reproducibility.
with a 75% probability for optimal performance. In the distillation setting,
(d) Classifier-Free Guidance in Distillation. When distilling a teacher model that supports classifier- free guidance (CFG), we compute both the conditional velocity Voond and unconditional velocity Uuccond from the teacher, and construct the target
uncond + w (Weond Vuncond )
(43)
where w is the guidance weight. During training, the unconditional forward is computed with torch. no-grad (), adding less than 5% overhead, which is consistent with sCM and MeanFlow. To stabilize the high- region, set time threshold fjow and disable guidance for t < hoo (we use =0.05 in our experiments). During pre-training, the null-token probability is 10%, and the under
condition is not dropped during distillation. At inference, FACM uses a single timestamp; CFG, each step requires only one NFE.
(e) Time Sampling Schedule. Following sCM (Lu & Song, 2024), the time t E [0, 1] is sampled according to a schedule that concentrates samples near the data endpoint (t = 1). We first sample a value from log-normal distribution, i.e., In(or) -N mean, P2 ) and then compute as:
arctan(o))
(44)
(f) Weighting Functions. For the CM loss component (Eq. 13 find that the weighting functions (t) = 11 2o5 and 3(t) = cos(t x /2) provide an effective general solution. These functions are crucial for navigating the trade-off between ensuring endpoint quality (in high-SNR regions) and satisfying global consistency (in low-SNR regions).
A.5 DISCUSSION: FROM-SCRATCH TRAINING VS. DISTILLATION
While our method can be trained from scratch and achieves a competitive result (See Table 3). we identify the two-stage distillation paradigm as the more principled and practically superior approach. Attempting to learn both the anchor and the shortcut simultaneously from scratch introduces behicken-and-ege problem, as the model must learn a shorteut based on a trajectory it has not yet accurately modeled. This creates an unstable "moving target" for optimization and incurs higher computational costs. In contrast, distillation from a pre-trained FM teacher provides a fixed, high-quality velocity field, offering much more stable and well-defined learning objective. MeanFlow (Geng et al. 2025) also encounters this problem in the -rom-scratch setting, achieving its optimal performance only by having its objective degenerate Flow Matching task for large portion of samples (c.g., 75%), which further validates our core thesis that a robust foundation in the velocity field is prerequisite for learning stable shorteuts.
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A.6 FLOW MAPPING EQUIVALENCE DERIVATION
Let the Flow Mapping function be Sa (2 1, r) - 2t + (r d
The consistency condition
di So(ws, r, = 0 is equivalent to the learning objective for Fo:
v Fo + (r- dFg dt
dt fo, =
Fo = (r t) dF& dt
Enforcing this for t E [0, r] implies that fo is constant over the interval, thus mapping any point to the endpoint X,:
f,(x,,tr)) t. fx(x,,r,)) X,
A.7 HYPERPARAMETERS
Table Key hyperparameters for our experiments.
Hyperparameter
Value
Hyperparameter
Value
Cifar- 10 Value
Optimizer
AdamW
Batch Size
1024
128
Learning Rate
le-4
Time Sampling (Piemn Paa)
(-0.8, 1.6)
(-1.0, 1.4)
Weight Decay
0
CFG Scale (w)
1.75
1.0 Simple-EDM
EMA Length (Trel)
0.2
Flow Schedule
OT-FM
Norm L2 Loss
le-3
Dropout
0
0.2
CPG fjow
0.05
Adam W Betas (81,82
(0.9, 0.999)
(0.9, 0.99)
A.8 ABLATION ON THE COSINE SIMILARITY TERM
The FM loss in Eq. includes cosine similarity term, which we found to be beneficial for aligning with pre-trained ViE/DiT teachers whose features are trained with representation supervision. Across our ImageNet 256 256 experiments (NFE=1), removing this term consistently degrades FID by 0. 0.2. We therefore keep it as a default component of the FM loss.
A.9 COMPUTATIONAL COST AND RESOURCES
Generation Latency. On single A100 GPU, 2-step FACM sampler takes approximately 70.2 ms per image (including VAE decoding). versus 7062.9 ms for a standard 250-step Euler sampler, translating roughly 100x speed-up in wall-clock time.
JVP Memory and Throughput. Our Chain-JVP introduces no bias to the derivative and is embedded within the FSDP backend, its speed matches standard FSDP forward with differentiation. It reduces peak memory from an OOM error to ~>2GGB for a 14B-parameter model on 80GB A100s. For 5B model, Chain-JVP with FlashAttention2 reduces peak memory from ~6GG 33GB.
We distilled the 14B model using 64 x A100 GPUs. The NFF=8 results reported in the paper were obtained after 5000 steps with a batch size of 512, taking 73 hours. The same setting is reproducible on fewer GPUs via gradient accumulation and FSDP CPU Offload.
14B Distillation Resources.
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A.10
ADDITIONAL VISUALIZATION
FACM (Ours) 14B NEB68
Wan2 A14B NFE-102
FLUXXI1-S 1-Sehell 12B NPE-8
I-Dey NFE-5002
WORLD TURTE
"world turtle," giant carrying fanlasy world swimming through space.
close-up selfie in cracked mirror, the flash highlighting the cracks and the subject's face, moody and introspoctive.
A tiny tree frog clinging to vibrant red leaf, its skin glistening with moisture, rich jungle background bokch.
ONA
OC
1115
Intricate close-up mechanical insect drone, detailed gears sign.
musician playing guitar
City subway station, passing background,
A cyberpunk city street night, painted with thick, switling impasto brushstrokes, the style of Vincent van Gogh.
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FACM (Oun) 14B NPPe
Wan2.2 NPF=410b2
FLUX.I-Schnel B2B NFE-8
FLUX.I-De 12B NFE-50x2
portrait with huir, each strand distinet. thick brushstroke, full of movement.
expressive, thick impasto painting storny seascape, waves crashing heavy, puint, pulette
A portrait of cat, its für suggested with dry brush technique over a soft wash background.
NN-- (poutistt)
NP SEFE-22 looiuiishi
NPR--1
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305 Nar--1
38 NEs--1
NIE-- Iglant
NFE-2 360 (pinni
(oom))
125
NP rchooloo
Aive- Class cahoock
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(anot)
NTC-
NER-
Class MPE-1 escaoonorr
tochooneee
NPR-2
NEE-- Class 933 (cboseeburgerr
983 Gdtioootuugge
0775 (coral
375 Nitecca
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Uncurated 121 generation results of FACM 14B. The generations are based on batch of randomly sumpled prompts. The images from left right generuted with different NPE: 4, and respectively
Love Love Love Love
NONDA
MONDAY
MONOA
MONDA
901
901
901
901
#
4
AER
CHAPTER
AEDO
CHAPTER
CHAPTER
CHAPTER
PET
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A.11 PROMPTS FOR TEASER VISUALIZATIONS
The following are the text prompts used for the text-to-image synthesis examples shown in the top two rows of Figure
A soldier in tactical gear standing next to modified desert-runner muscle car in vast desert under bright sum, digital art.
A surrealist portrait where the person's 's face is composite of various flowers and leaves.
A portrait of girl whose hair is made of flowing, colorful ink, watercolor style.
Close-up of a tarot card, "The World", depicting cyborg wreathed in stars.
A painting of time traveler's footprints through history, each print leading to different era.
A man with a worried expression looking out through the window, overcast lighting.
A florist arranging bouquet fresh flowers, bcautiful combination of colors and scents.
A woman in corner of the library, surrounded by books, studying quictly.
An artist studio, splattered with paint, staring intently large canvas, dramatic lighting. Rembrandt lighting-
A street musician playing the cello in European city square, little girl stops to listen, touching moment.
A.12 THE USE OF LARGE LANGUAGE MODELS (LLMS)
In accordance with ICLR 2026 policy, report the use of Large Language Model (LLM) during the preparation of this manuscript. We used a large language model as writing assistant to help improve the grammar and clarity of our prose. Its role was strictly limited to proofreading; all
scientific ideas, analyses, and conclusions presented are our own. The authors have reviewed the final text and take full responsibility for its content.
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Reviews:
[{"review_id": "aSiO7sGzdS", "reviewer": "Reviewer_AryK", "summary": "This paper proposes Flow-Anchored Consistency Models (FACM), which incorporates a flow matching objective into consistency model training to stabilize training dynamics. The authors demonstrate that the key to effective joint optimization of flow matching and consistency model training is decoupling of flow velocity prediction and mean velocity prediction.\n\nSpecifically, in contrast to MeanFlow and its variants which are trained to predict flow velocity at time $t$ when given time condition $(t,t)$ and mean velocity from time $t$ to $r$ when given time condition $(t,r)$, FACMs are trained to predict flow velocity at $t$ given $2-t$ and mean velocity from time $t$ to $1$ given $t$. In contrast to MeanFlow, whose time conditions are coupled, i.e., $r = t$ for flow velocity, FACM uses decoupled time conditions, i.e., $r \\neq t$ for flow velocity. Additional techniques such as interpolation of the shortcut target with the current EMA network output and scalable chain-JVP implementation are provided to further accelerate training.\n\nThe authors verify the scalability of FACM on CIFAR-10, ImageNet 256x256, and a Text-to-Image dataset. FACM is shown to consistently out-perform several fast baselines such as IMM, MeanFlow, and sCM.", "questions": "**[Q1] Can the authors provide ablations mentioned in [W2] starting from a MeanFlow baseline?** In particular, I am curious how generative performance changes if (a) one fixes $r = 1$. I expect we would obtain a stronger few-step model, as it is solving an easier task compared to MeanFlow, which learns flow maps for all pairs $(t,r)$.\n\n**[Q2] Can the authors provide theoretical or experimental evidence of the hypothesis that instability in CM is caused by the lack of instantaneously velocity field supervision?** The authors could, for instance, plot the variance of the derivative term $dF_{\\theta^{-}}/dt$ without and with flow matching loss.\n\n**[Q3] What is the purpose the cosine similarity term in Eq. (9)?** I believe this loss is redundant, as a cosine similarity is already implicitly contained in the first flow matching objective. If this term plays a non-trivial role, then it should also be included in the ablations.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- **[S1] This paper is original in the aspect that it provides a new perspective into training instability of CMs.** Specifically, the authors hypothesize that CM training instability arises from missing flow velocity supervision, and that one should decouple flow and mean velocity time conditions to mitigate conflict between the two velocities.\n\n- **[S2] This paper is significant in the aspect that it provides a number of techniques for scaling CMs.** The authors provide a number of practical techniques, such as interpolation of shortcut target and current prediction and chain-JVP, which are shown to work on difficult tasks such as ImageNet 256x256 and text-to-image generation.", "weaknesses": "- **[W1] The paper lacks theoretical novelty, in the sense that it is a special case of MeanFlow.** MeanFlow learns a flow map between all time pairs $(t,r)$ for $0 \\leq t < r \\leq 1$, along with a flow matching loss at $t = r$. FACM is a special instance of MeanFlow where a flow map is learned only for time pairs $(t,1)$ for $0 \\leq t \\leq 1$, also with flow matching loss at $t = r$, where $r = 2 - c\\_{FM}$ if one uses expanded time interval proposed in Section 3.3.2. Under this perspective, FACM may be viewed just as a collection of techniques for improved training of MeanFlow.\n\n- **[W2] The paper is missing key ablations of the proposed techniques.** Under the MeanFlow perspective written in [W1], key techniques proposed in this paper can be categorized into (1) fixing $r = 1$, i.e., training only with time pairs $(t,1)$, (2) expanded time interval, and (3) other extraneous techniques such as shortcut target interpolation, residual clamping, and tuning $\\alpha(t)$,$\\beta(t)$ in Eq. (11) and (13). However, authors only provide an ablation of (2), so with all techniques intertwined, it is difficult to judge what is the largest contributor to the final generative performance.\n\n- **[W3] The key hypothesis for instability of CMs, claimed by the authors, is unsupported.** At lines 171-175, the authors write \"However, without a stable anchor in the underlying flow, the model's output $F_\\theta$ quickly begins to drift. ... the derivative term in the identity grows to dominate the ground-truth velocity $v$, effectively diluting its supervisory signal.\" However, there is no supporting theory or experiment to verify this claim.", "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": "mggosqZeOz", "reviewer": "Reviewer_YH7m", "summary": "This paper proposes Flow-Anchored Consistency Models (FACM), a new framework for training consistency models that explicitly anchors the learned consistency mapping to an underlying flow-matching objective. The authors argue that existing continuous-time consistency models are unstable because they discard the flow-field information during training, leading to mode collapse or divergence. FACM mitigates this by jointly optimizing a flow-matching loss (to learn the velocity field) and a consistency loss (to learn efficient one-step mapping), decoupled through an expanded time interval strategy that ensures smooth continuity between the two objectives. The paper further introduces Chain-JVP, a memory-efficient Jacobian–vector product computation compatible with large-scale distributed training (e.g., FSDP), enabling the training of models up to 14B parameters. Experiments on CIFAR-10 and ImageNet-256 demonstrate that FACM achieves state-of-the-art one- and few-step generation performance, outperforming MeanFlow and sCM, and scales effectively to large text-to-image models.", "questions": "1. How sensitive is FACM to the weighting between FM and CM objectives, and does this affect sample quality or convergence speed?\n2. Could the authors clarify the precise role of the extended time interval—would a shared time embedding with gating suffice?\n3. Does the Chain-JVP approach introduce any bias or approximation error compared to full JVP computation?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper identifies a clear and well-motivated problem — instability in continuous-time consistency model training — and provides a principled remedy via flow anchoring.\n2. The formulation is elegant, combining the benefits of flow matching (stability, theoretical grounding) and consistency models (few-step generation) into a unified loss.\n3. The expanded time interval trick is simple yet effective, ensuring a smooth transition between FM and CM regions while avoiding gradient coupling issues.\n4. The Chain-JVP technique is a practical and non-trivial engineering contribution, making second-order consistency training feasible at scale.\n5. Empirical results are strong: FACM outperforms prior methods such as MeanFlow, sCM, and IMM in one- and two-step generation across multiple datasets.\n6. The framework is demonstrated on a 14B-parameter text-to-image model, providing credible evidence of scalability beyond toy datasets.", "weaknesses": "1. The theoretical discussion remains somewhat heuristic. While the “anchoring” intuition is appealing, a more rigorous analysis (e.g., convergence guarantees or stability proofs) would strengthen the contribution.\n2. The relationship between FACM and existing flow–consistency hybrids (e.g., MeanFlow, iCM) could be clarified — in particular, what distinguishes the proposed anchoring mechanism from joint FM/CM training used before.\n3. Some ablations (e.g., varying the relative weight of FM vs. CM losses) are discussed briefly but lack quantitative depth.\n4. The paper does not compare computational costs (training time, FLOPs) against prior CM methods, leaving unclear whether stability comes at significant overhead.", "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": "5G4mwkJsXP", "reviewer": "Reviewer_9H8q", "summary": "This paper identifies the cause of training instability problem for continuous-time consistency models and proposes a solution of Flow-Anchored Consistency Models(FACM) to solve it. The key design is the Flow-Anchoring principle, where a Flow Matching objective is used as a stable anchor and enables the model to learn both the flow's velocity field and the shortcut efficiently. Furthermore a memory-efficient Chain-JVP scheme is proposed which enables scalable FACM training up to a 14B parameter model. Experiment shows that the proposed algorithm achieves state-of-the-art results for few-step image generation task on a set of benchmarks including ImageNet 256x256 and CIFAR-10.", "questions": "1. What is the computational overhead of the proposed Chain-JVP method?\n2. What is the computational cost of distilling a 14B model and what is the GPU resources required?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The analysis on the training instability issue in continuous-time consistency models is clear and convincing\n2. The proposed flow-anchoring solution is elegant and effective, and the process of dual-objective training is clearly presented.\n3. The achieved experiment result is impressive, where in table 1, FACM@NFE=2 outperforms Multi-NFE baseline of LightningDiT.", "weaknesses": "1. Although the paper shows efficiency in terms of reduced NFE, it would be good to also report the total generation time per image.\n2. The individual components of the proposed algorithm including the flow matching loss and consistency model loss are existing work, and the proposed FSDP compatible Chain-JVP scheme still has significant computational overhead", "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": "5sDntyfQFj", "reviewer": "Reviewer_8cpy", "summary": "The paper proposed FACM, a method for stabilizing and improving training of continuous consistency models. The method consists in combining the continuous consistency objective with a flow matching anchor, to encourage the model to learn the underlying flow, which results in improved performance and stability. The method further introduces a novel time embedding strategy and a JVP computation which allows scaling to models with billions of parameters. The method is validated on image generation benchmarks and text-to-image benchmarks and results in competitive performance, beating other one and few-step generative models.", "questions": "- Regarding CFG at sampling, does the model require only one NFE or two NFEs? And if it's the 1 NFE case, is the model only conditioned on the conditioning signal, or does it require extra parameters like in MeanFlow?\n- Still regarding CFG training, how slower is the model, given that it requires one additional forward pass for $v_{cond}$? Also there does not seem to be any information regarding the probability of using the null token for the pretrained/online base model. In addition, $t_{low}$ is only mentioned as hyperparameter but never in the text.\n- What's the reason behind the cosine similarity loss for the FM objective? I am not familiar with this term being used in the flow/score literature, is this part of the contribution or is it commonly used in other FM methods?\n- Are the consistency and FM losses simply summed together? Do you use any weighting term or a percentage of the batch similar to Shortcut Models or MeanFlow?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The method is a simplification of continuous consistency models, by leveraging the features learned with flow-matching training. The novel objective is well motivated and effective, and achieves strong performance. The introduction of the scalable JVP computation is valuable for the research community.", "weaknesses": "The method uses now two forward passes, one of which requires JVP computation, so I am concerned about the training efficiency. Nothing about it is reported in the paper, but I think it is important to mention how slower the model gets when trained with FACM.\nWhile simplified compared to sCM, the method still requires complex formulations and weighting functions, for which no ablation is provided.\n\nAs a minor note, the authors claim that the MeanFlow code is not available, while it was already by the time of the submission deadline.", "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)