FireFlow: Fast Inversion of Rectified Flow for Image Semantic Editing
Authors: Yingying Deng
Xiangyu He
Changwang Mei
Peisong Wang
Fan Tang
Venue: ICML 2025 poster (2025)
Keywords: Rectified Flow
Image Editing
Primary area: deep_learning->generative_models_and_autoencoders
TL;DR: An 8-step inversion and 8-step editing process works effectively with the FLUX-dev model. (3x speedup with results that are comparable or even superior to baseline methods)

Abstract: Though Rectified Flows (ReFlows) with distillation offer a promising way for fast sampling, its fast inversion transforms images back to structured noise for recovery and following editing remains unsolved. This paper introduces FireFlow, an embarrassingly simple yet effective zero-shot approach that inherits the startling capacity of ReFlow-based models (such as FLUX) in generation while extending its capabilities to accurate inversion and editing in **8** steps. 
We first demonstrate that a carefully designed numerical solver is pivotal for ReFlow inversion, enabling accurate inversion and reconstruction with the precision of a second-order solver while maintaining the practical efficiency of a first-order Euler method. This solver achieves a $3\times$ runtime speedup compared to state-of-the-art ReFlow inversion and editing techniques while delivering smaller reconstruction errors and superior editing results in a training-free mode. The code is available at [this-URL](https://github.com/HolmesShuan/FireFlow-Fast-Inversion-of-Rectified-Flow-for-Image-Semantic-Editing).

FireFlow:
Fast Inversion of Rectified Flow for Image Semantic Editing
Yingying Deng'
Xiangyu He? Changwang Mei
Peisong Wang?
Fan Tang
Source
Origami
[culpuue
[R]Suilding
[R]ning
[+]oobbball
[+]Hot
Source
[C]Cortonn
[R]
[R]Grochet dog
[RJwer rings
[R]Smile
[+]Mak
Source
Flower
[2] Rainbow
Source
[R]Chocolate bear
Source
Coco cola"
Figure 1. FircFlow for Image Inversion and Editing in 8 Steps. Our approach achieves outstanding results in semantic image editing and stylization guided by prompts, while maintaining the integrity of the reference content image and avoiding undesired alterations. [+1/[=] means adding removing contents, [C] indicates changes in visual attributes (style, material, or texture), and [R] denotes content or gesture replacements
bstrract
ation while extending its capabilities to accurate inversion and editing in steps. We first demon- strate that carefully designed numerical solver is pivotal for ReFlow inversion, enabling accurate inversion and reconstruction with the precision of second-order solver while maintaining the practical efficiency of a first-order Euler method. This solver achieves a 3x runtime speedup com- pared to state-of-the-art ReFlow inversion and editing techniques while delivering smaller recon- struction errors and superior editing results in training- free mode. The code is available at this- URL.
Though Rectified Flows (ReFlows) with distilla- tion offer promising way for fast sampling, its fast inversion transforms images back to struc- tured noise for recovery and following editing re- mains unsolved. This paper introduces FireFlow, an embarrassingly simple yet effective zero-shot approach that inherits the startling capacity of ReFlow-based models (such as FLUX) in gener-
Uniiersity of Science and Technology Beijing, Beijing, China Institute of Automation, Chinese Academy of Sciences, Beijing, China Nanjing University of Science and Technology, Nanjing, China 'Institute of Computing Technology. Chinese Academy of Sciences, Beijing, China. Correspondence to: Xiangyu He <hexiangyu17 @ mails.uca ucwll.uuca.eee. ........ca.cdu.cc......
1. Introduction
Proceedings of the ndnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
The ability to accurately and efficiently invert generative models is critical for enabling applications such as semantic

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FireFlow: Fast Inversion of Rectified Flow for Image Semantic Editing
image editing, data reconstruction, style transfer and latent space manipulation (Deng al., 2024a; Hertz et al., 2022; Tumanyan et al., 2023; Deng et al., 2024b). Inversion, which involves mapping observed data back to its latent represen- tation, serves as the foundation for fine-grained control over generative processes, Achieving a balance between compu- tational efficiency and numerical accuracy in inversion is particularly challenging for diffusion models, which rely on iterative processes to bridge data and latent spaces.
Table I. Comparison of recent training-frce inversion and editing methods based on FLUX, including inversion/denoisingg steps, NFEs (Number of Function Evaluations) for both inversion and editing, local truncation error orders for solving ODE, and the need for a pre-trained auxiliary model for editing. Our approach offers
simple yet effective solution address the challenges.
Methods
Add-it
RF-Solver
RF-Inv.
Ours
Steps NFE
30
15
28
8
60
60
18
Diffusion models have long been used for high-quality im- age generation (Ramesh et al., 2022; Rombach et al., 2022:
Aux. Model
V
w/o (
w/o
w/o
Local Error
O
O
(AA))
Podell et al., 2024; Zhang et al.. 2023; Deng et al., 2025) and inversion (Hertz et al., 2022; Parmar et al., 2023) due to their ability to capture complex distributions through stochastic differential equations (SDEs). Nevertheless, the nonlinear solver introduces undesirable drift in the reverse trajectory, thereby compromising the accuracy of reconstruc- tion and the fidelity of bhiting-by-inversonnn In contrast, rectified Flow (ReFlow) models (Liu et al., 2023), which replace stochastic sampling with ordinary differential equa- tions (ODEs) for faster and more efficient transformations. This motivate closer investigation of ReFlow-based mod- els, partieularly in the context of inversion and editing, to develop simple and effective methods.
(Tewel et al., 2025) for Add-it, (Wang et al., 2024) for RF-
Solver, (Rout et al., 2025) for RF-Inv.
2. Preliminaries and Related Works
2.1. Rectified Flow
Rectified Flow (Liu et al., 2023) offers a principled approach for modeling transformations between two distributions, To and π based on empirical observations Xo ~ To and X1 ~ ア The transformation is represented as an ordinary differential equation (ODE) over a continuous time interval E [0, 1]
d2, - v(z, dl,
(I)
ReFlow models possess an underutilized advantage: well- trained ReFlow model learns nearly constant velocity dy- namics across the data distribution, ensuring stability and bounded velocity approximation errors. However, existing inversion methods for ReFlow models fail to fully exploit this property (Rout al., 2025; Wang al., 2024). Current approaches rely on generic Euler solvers that prioritize each step's 's computational efficiency at the expense of accuracy or incur additional costs to achieve higher precision. As a result, the potential of ReFlow models to deliver fast and accurate inversion remains untapped.
where Zo To is initialized from the source distribution, and 7, Ti is generated at the end of the trajectory. The drift v R2 x [0, 1] - R is designed to align the trajectory of the flow with the direction of the linear interpolation path between X, and X1 This alignment is achieved by solving the following least squares regression problem:
min E [V. |1(X: -Xo) - uo (X,, t)12 dt dt:
(2)
where X, = tX,,+ + (1 (11-t))x X) denotes the linear interpolation path between Xu and X1.
In this work, we introduce a novel numerical solver for the ODEs underlying ReFlow models, addressing the challenges of inversion and editing. Our method achieves second-order precision while retaining the computational cost of a first- order solver. By reusing intermediate velocity approxima- tions, our approach reduces redundant evaluations, stabilizes the inversion process, and fully leverages the constant ve- locity property of well-trained ReFlow models. As shown in Table 1, our approach is the first to provide a solver that strikes an optimal trade-off between accuracy and efficiency, enabling ReFlow models to excel in inversion and editing tasks. By combining computational efficiency, numerical robustness, and simplicity, our method offers a scalable solution for real-world tasks requiring high fidelity and real- time performance, advancing the utility of ReFlow-based generative models like FLUX.
Forward process seeks to transform samples X, ~ mo to match the target distribution क A direct parameterization of X, is given by the linear interpolation X, = tX + (1 t) No which satisfies the non-causal ODE:
dX, = (Xi - -Xo) dt.
However, this formulation assumes prior knowledge of X1, rendering it non-causal and unsuitable for practical sim- ulation. By introducing the drift vi ((,,t) rectified flow causalizes the interpolation process. The drift v is fit to approximate the linear direction X1 - Xo resulting in the forward ODE with Xo To,
dX, = v( ((.,, 1) di, LE [0,
(4)
This causalized forward process enables simulation of Z, without requiring access to X2 during intermediate time steps.
2

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FireFlow: Fast Inversion of Rectified Flow for Image Semantic Editing
Reverse process generates samples from Ti by reversing the learned flow. Starting from X1 1 the reverse ODE is given by negating the drift term:
ments in image inversion and attention manipulation tech- nologies (Hertz al., 2022; Cao et al., 2023; Meng et al.,
2022; Couairon et al., 2023; Deutch et al., 2024; Xu et al.,
2024; Brooks et al., 2023). Training- ffreen editing methods typically employ a dual-network architecture: one network is dedicated to reconstructing the original image, while the other focused on editing. Prompt-to-Pront
dxx -xX,, dt, E [1.
(5)
This process effectively "undoes" the transformations ap- plied during the forward flow, enabling the generation of Xg that follows the original distribution xo- The reverse process guarantees consistency with the forward dynamics by leveraging the symmetry of the learned drift v.
(Hertz et al., 2022) manipulates the cross-attention maps within the editing pipeline by leveraging features from the recon- struction pipeline. Plug-and-Play (Tumanyan et al., 2023) method substitutes the attention matrices of self-attention blocks in the editing pipeline with those from the recon- struction pipeline. Similarly, MasaCtrl (Cao et al., 2023) modifies the Value components of self-attention blocks in the editing pipeline using values derived from the reconstruc- tion pipeline. Additionally, the Add-it (Tewel et al., 2025) method utilizes both the Key and Value components of self- attention blocks from the source image to guide the editing process effectively. With respect to FLUX models, RF- Editing (Wang et al., 2024) reuses Value components from MMDiT (generated during inversion) for reconstruction- based editing, while RF-Inversion (Rout et al., 2025) uses the original image as prior for guiding the forward and reverse process in semantic editing. We introduce FireFlow, novel solver that can effectively integrate these techniques (and can also be used with the vanilla ReFlow model) to achieve promising results with fewer steps.
2.2. Inversion
The inversion of real images into noise feature space, as well as the reconstruction of noise features back to the orig- inal real images, is prominent area of research within diffusion models applied to image editing tasks (Lin et al., 2024; Brack et al.. 2024; Miyake et al., 2023; Ju et al., 2024; Zhang et al., 2022; Luuberman-Spiegllae
et al.,
2024; Cho et al., 2024). The foundational theory of De- noising Diffusion Implicit Models (DDIM) (Song et al., 2021) involves the addition of predicted noise fixed noise during the forward process, which can subsequently be mapped to generate an image. However, this approach encounters challenges related to reconstruction bias. To address this issue, Null-Text Inversion (Mokady et al., 2023) optimizes an input null-text embedding, thereby correct- ing reconstruction errors at each iterative step. Similarly, Prompt-Tuninge Inversion (Dong al., 2023) refines con- ditional embeddings to accurately reconstruct the original (Miyake et al., 2023) replaces the null-text embedding with prompt embeddings to expedite the inversion process. Direct-Inversion (Ju et al., 2024) incorporates the inverted noise corresponding to each timestep within the denoising process to mitigate content leakage.
3. Motivation
image. Pegative-Prompt-Ineessi
The ReFlow model operates under the simple assumption that X, evolves linearly between Xo and X1, corresponding to uniform linear motion. Drawing an analogy to physics, it is natural to extend this linear motion to accelcrated motion by incorporating an acceleration term:
du: dt
dX. dt =(,,,t)
In contrast to the aforementioned Stochastic Differential Equation (SDE)-based formulations, rectified flow models that utilize ordinary differential equations (ODEs) offer a more direct solution pathway. RF-Inversion (Rout et al., 2025) employs dynamic optimal control techniques derived from linear quadratic regulators, while RF-Solver (Wang et al., 2024) utilizes Taylor expansion to minimize inversion errors in ODEs. Nevertheless, achieving superior inversion results typically necessitates an increased number of genera- tion steps, which can lead to significant computational time and resource expenditure. In this paper, we propose a few- step ODE solver designed to balance effective outcomes with high efficiency.
=a(,,,t)
(6)
where X(41 X, + a + Ja,tt and is is equiva- lent to v(X,, () for simplicity. Recent works have empiri- cally shown that training- strategies (Park et al., 2024; Chen et al., 2024) for solving Equation (6) improve cou- pling preservation and inversion over rectified flow, even with few steps. Moreover, a training-free method (Wang et al., 2024) leveraging pre-trained ReFlow models has also demonstrated the utility of the second-order derivative of v in achieving effective inversion, essentially aligning with the principles of accelerated motion.
However, this observation appears counterintuitive. A well- trained ReFlow model, such as FLUX, generally assumes that v approximates the constant value X, - Xo. Thus, the acceleration term a, = dv,/ dt theoretically approaches zero, as the learning target for i is constant.
2.3. Editing
Image editing utilizing a pre-trained diffusion model has demonstrated promising results, benefiting from advance-
3

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FireFlow: Fast Inversion of Rectified Flow for Image Semantic Editing
Connection to High-Order ODE Solvers: Instead of treating a continuous term, we reinterpret it through the lens of high-order ODE solvers. Using the finite-difference approximation at = (Vi+Ae ut)/ we can rewrite the equation as:
evaluating more points within the interval [4, + 1], leading to higher NFE per step, which negates any reduction in the number of steps and fails to improve overall computational efficiency.
For instance, the midpoint method achieves a local error of O ((tt)) and a global error of ((Ii). Formally, it proceeds as follows:
1 X1+1 = X1 a + 20, = =X, + 2 (Ur + Vtt
2
which corresponds to the standard formulation of the second- order Kunge-Kutta method. This high-order approach al- lows fewer steps (or equivalently, larger step sizes At) to achieve the same accuracy as Euler's method, since the global error of p-th order method scales as (Att)). This enables larger At while maintaining the same er- ror tolerance E. Similarly, if we approximate at using a = (1++ jAt - ( the resulting position update becomes:
Al vo (X,,,)),
X 1+ + = X +
(8)
2
At
X++1 = Xe+ At uo (X, 1+ 2e, t +
(9)
2
This scheme requires two NFEs per step: one to compute X as and another for uo (X, (4+40 -r, 1+ + effectively dou- bling the cost compared to the Euler method. The midpoint method leverages 4 + to provide more accurate estimate A
of X--1-x than Ut, which inspires us to seek an alternative with lower computational cost.
X1+1 X₂ + Ue+ Aar
which corresponds to the standard midpoint method, another second-order ODE solver.
A Low-Cost Alternative:
The training objective of Re- Flow implies that a well-trained model satisfies ug (X,, t) R -X)-Xo) for all Leveraging this property, the most effi- cient approach would replace " with to. enabling one-step generation proposed in the original ReFlow method (Liu et al., 2023). However, this simplification makes it difficult to incorporate conditional priors, as multi-step iteration is no longer required.
Impact on ReFlow Inversion: It is well-established that the global error in the forward process of ODE solvers benefits from higher-order methods. Likewise, inversion and reconstruction tasks also exhibit improved performance with high-order solvers, as they better preserve original image details during the inversion of ReFlow models. We formalize this property in the following statement.
To maintain a multi-step paradigm, we propose modified scheme that replaces i, with previous - 1-step midpoint velocity ( rather than d This approach is formalized as:
Proposition 3.1. Given a p-th order ODE solver and the ODE d.X. (x(X,,t) t). the dynamics of the reverse pass
dx. = - 1 -)) t) which is Lipschitz continuous satisfy dt with constant L. The perturbation AT at t = T propagates backward to 0. The propagated error satisfies:
(x(X,,,) = vo (o(((-1)) a (t 1) + A
(10)
2
A S - 1 |A
load from memory
Al
Implication. The inversion error || |rll introduced during the p-th order numerical solution propagates into the re- verse pass, experiencing a slight reduction scaled by the Lipschitz constant L of the learned drift (-(,, t). Despite this reduction, the overall reconstruction error || for the original image remains asymptotically of the same order, O(At)). ). where At represents the integration step size. Con- sequently, high-order solvers are preferred in ReFlow to achieve accurate inversion and editing with fewer steps.
8- A1 = Xx +
(X
(11)
2
At
X Xx + Al un Ar
(12)
run & save to memory
In this simple scheme, only one NFE is required per step', matching the computational cost of the Euler method. The key question, then, is whether this scheme retains the second- order accuracy of the original midpoint method.
For the local and global truncation error, we derive that if og (X,, t) is well- and varies smoothly with respect to both X and f, the proposed scheme achieves the same truncation error the standard midpoint method. This ensures that the modified approach retains the benefits of
4. Method
Challenges with High-Order Solvers: While the use of high-order solvers is theoretically promising, it fails to yield practical runtime speedups. For a parameterized drift u,,,,, t), the runtime is determined by the Number of Func- tion Evaluations (NFEs), i.e., the number of forward passes through the model vo (X,, High-order solvers require
'Specifically, we perform two NFEs t to initialize the conditions Uo and 1/+ at for subsequent iterations. Python-style pseudo-code is provided in Sec.D.

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FireFlow: Fast Inversion of Rectified Flow for Image Semantic Editing
30
30*
33
30
-0
(a) Euler Method (NFE 20)
(b) Midpoint Method (NFE 20)
(c) Ours (NFE = 20)
Figure 2. Results 2D synthetic dataset. We evaluate the performance of 2 2-Rectified Flow using the Euler solver, midpoint solver, and our proposed approach on a 2D synthetic dataset. The source distribution тo (orange) and the target distribution I (green) are parameterized as Gaussian mixture models. For the Euler method, the number of sampling steps set N 20, corresponding to an NFE of 20, Our approach generates samples that align more closely with the target distribution, achieving a better match in density and structure. Additionally, the trajectories of the samples exhibit greater straightness, adhering closely to the ideal of lincar motion.
Agp merreindeer Ben
stances. Consequently, is expected to exhibit smaller overall error while maintaining the same runtime cost as the first-order Euler method.
Pe-
I
Theorem 4.2. Consider a ReFlow model governed by ODE:
dx = 0g (X,t), t). where vo vx(X,t) t) is smooth and bounded, di and the solution Xt evolves over a time interval [0, T
Fiee
(a) Inversion and Reconstruction with Step=10
The modified midpoint method, defined in Equation (12) achieves the same global truncation error O(Ai²) as the standard midpoint method, provided the reused velocity satisfies: |]6o( (X,, t)
10
100
vo (X,, t)ll & O(At).
To highlight the advantages of our approach, we conduct experiments on synthetic data following the setup in (Liu et al., 2023). As shown in Figure 2, the transport trajec- tories generated by our method are straighter, leading to improved accuracy while maintaining the same NFE the Fuler method, and even surpassing the performance of the standard midpoint method.
(b) Inversion and Reconstruction with Step=20
Figure Illustrations of the approximation error in velocity (l0s well II) as it evolves with inversion steps (left subfigures) and denois- ing steps (right subfigures), with At included as reference.
second-order accuracy while operating the computational cost of first-order solver.
Numerical Results and Discussion To empirically validate the theoretical assumption that the reused velocity approxi- mation error [] (X,, t) ug (Xe, t)ll is bounded by O(At), we conducted numerical experiments and analyzed the rela- tionship between the approximation error and the time step size At FLUX-dev model during inversion and recon- struction. The results are summarized in Figure 3, which depicts the average approximation error across different step sizes, with the shaded area representing +1 standard deviation.
Proposition 4.1. Let (x(Xe. denote the reused velocity approximation in Equation 10, and og (X(e))) denotes the exact velocity time t. Then, the approximation satisfies the error bound: [(ōg(X,,t) vo (xxx))))) S O(At), under the following conditions:
The temporal error is directly propor- tional to the time step At, stemming from smoothness of vo X, 1) in the time domain.
1) Temporal Error:
The data exhibit the following key trend that the approx- imation error grows approximately linearly with the step size C At, consistent with the theoretical bound ((At). Despite the occasional spikes and inherent variability of the error (illustrated by the shaded standard deviation), the trend and magnitude of the error remain well-controlled and stable across most steps, further validating the robustness of the reused velocity approximation in practice.
2) Spatial Error: The spatial error is dominated by O(At),
due to the boundedness of Oug ax
We formally prove in the appendix that when two required conditions are satisfied, leading to the following result: our modified midpoint method achieves the same truncation error as the standard midpoint method under these circum-
5

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FireFlow: Fast Inversion of Rectified Flow for Image Semantic Editing
Root Meon Squarg Emro (RMNSE)
Vanilla ReFlow
Table 2. Quantitative results for Unconditional Image Generation on 10. Our approach achieves performance comparable to that of a second-ordes solver, but with reduced computational cost.
Ours RF-Salver
2.73x Speedup
ODE Solver
Steps(()
NFE(L)
FID(L)
IS(1)
5
Smallest Reconstruction
0%. Speaduup
Euler
15
15
5.67
9.20
Euler
10
10
5.83
9.03
Evaluations (NFT)
Reduction
Midpoint
5
10
5,45
9.27
Ours
5
6
5.35
9.26
Figure 4. Image reconstruction errors versus denoising NFE: Our approach, compared to the first-order vanilla ReFlow inversion and sccond-order RF-solver, achieves lower reconstruction errors and demonstrates faster convergence with respect to NFE.
Table 3. Quantitive results on Text-to-Image Generation. The FLUX- dev uses Ist-order ODE Solver, the RF-Solver and our approach use Ist-order ODE Solver. Compared with FLUX-dev and RF-Solver, our approach has higher FID and CLIP Score.
Image Semantic Editing: To ensure simplicity and a fair comparison with the second-order solver method, we follow the approach in (Wang al., 2024), where the value features in self-attention layers during the denoising process are placed with pre-stored value features generated during the inversion process, serving as a prior. Subsequently, a refer- ence prompt is used as guidance to achieve semantic editing. Leveraging superior image preservation, our method does not require a careful selection of timesteps for applying the replacements, as suggested in the original paper. The in- version and denoising sampling processes are detailed in Algorithms 1 and 2.
Methods
Steps
NFE(J)
FID(J)
CLIP Score(t)
FLUX-dev
20
20
26.77
31.44
RF-Solver
20
40
25.54
31.39
RF-Solver
10
20
25.93
31.35
Ours
11
25.16
31.42
to-image generation (T2I) and 30 steps for reconstruction. For editing, RF-Solver varies the number of steps by task, using up to 25 steps. In contrast, achieves comparable or better results using 8 steps. The ablation study is shown in Section E.
find that our approach
5. Experiment
5.2. Text-to-image Generation
5.1. Implementation Details
Conditional Image Generation: We compare the perfor- mance of our method against the vanilla rectified flow and the second-order RF-solver on the fundamental T2I task. Following the setup in RF-solver, we evaluate a randomly selected subset of 10K images from the MSCOCO Caption 2014 validation set (Chen et al., 2015), using the ground- truth captions as reference prompts. The FID and CLIP scores for results generated with fixed random seed of 1024 are presented in Table 3. In summary, our method de- livers superior image quality while maintaining comparable text alignment performance.
Baselines: This section compares FireFlow with DM inversion-based editing methods such as Prompt-to-Promp (Hertz et al., 2022), MasaCtrl (Cao et al., 2023), Pix2Pix- zero (Parmar et al., 2023), Plug-and-Play (Tumanyan al., 2023), DiftEdit (Couairon et al., 2023) and Directlnversion (Ju et al., 2024). We also consider the recent RF inver- sion methods, such as RF-Inversson (Rout et al., 2025) and RF-Solver (Wang et al., 2024).
Metrics: We evaluate different methods across three as- pects: generation quality, text-guided quality, and preserva- tion quality. The Fréchet Inception Distance (FID) (Heusel et al., 2017) is used to measure image generation quality by comparing the generated images to real ones. A CLIP model (Radford et al., 2021) is used to calculate the simi- larity between the generated image and the guiding text. To assess the preservation quality of non-edited areas, we use metrics including Learned Perceptual Image Patch Similar- ity (LPIPS) (Zhang et al., 2018). Structural Similarity Index Measure (SSIM) (Wang et al., 2004), Peak Signal-to-Noise Ratio (PSNR), and structural distance (Ju et al., 2024).
Unconditional Image Generation: We adhere to the pro- tocol established in the original ReFlow paper (Liu et al., 2023) for unconditional image generation on CIFAR-1O. uti- lizing the open-source 1-Rectified Seeetified-Floww-dee -loowddistlll weights. The results, as shown in Table 2, compare various ODE solvers, and our method demonstrates consistent performance with fewer steps and higher scores.
5.3. Inversion and Reconstruction
Quantitative Comparison: We report the inversion and re- construction results on the first 1K images from the Densely Captioned Images (DCI) dataset (Urbanek et al., 2024), us- ing the official descriptions as source prompts. The results,
Steps: Since the number of inference steps can significantly impact performance, we follow the best settings reported for the RolSel to ensure a fair comparison: 10 steps for text-
6

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FireFlow:
We empirically compare the conver- gence rates of different numerical solvers during reconstruc- tion, as shown in Figure 4. For fair comparison, we use the demo "boy" image and prompt provided the RF-solver source code. Our approach achieves the lowest reconstruc- tion error with the fastest convergence rate, offering up to 2.7x speedup and over 70% error reduction. Figure fur- Fast Inversion of Rectified Flow for Image Semantic Editing
Table Compare approach with other editing methods PlE-Bench. Our method demonstrates superior performance in terms of background preservation and CLIP similarity.
Structure
Background Preservation
CLIP Similarity!
Method
Model
Steps
NFE,
Distance
PSNR
SSIM
Whole
Edited
Prompi2Prompt (Hertz ct al., 2022)
Diffusion
0,0694
17.87
0.7114
25.01
22.44
50
100
Pix2Pix-Zero (Parmar et al., 2023)
Diffusion
0.0617
20.44
0.7467
22.80
20.54
50
100
MasaCurl (Cao al., 2023)
Diffusion
0.0284
22.17
0.7967
23.96
21.16
50
100
PnP (Tumanyan et al., 2023)
Diffusion
0.0282
22.28
0.7905
25.41
22.55
50
100
PnP-Inv. (Ju et al., 2024)
Diffusion
0.0243
22.46
0.7968
25.41
22.62
50
100
RF-Inversion (Rout et al., 2025)
ReFlow
0.0406
20.82
0.7192
25.20
22.11
28
56
RF-Solver (Wang et al., 2024)
ReFlow
0.0311
22.90
0.8190
26.00
22.88
1.5
60
Ours
ReFlow
0.0284
23.29
0.8287
26.01
22.96
1.5
32
Ours
ReFlow
0.0271
23.03
0.8249
26.02
22.81
8
18
2114
NPE66
Source Image yaueg bry playing with tay
Flan
airpline
greassy front hawn suburban
N
fluffy.
Prompt
Vanilla ReFlow
RF-Inversion
RF-Solver
Ours
Figure 5. Qualitative results of image reconstruction. Our approach achieves faster convergence and superior reconstruction quality compared to baseline ReFlow methods utilizing the FLUX model. Difference images showing the pixel-wise variations hetween the source image and the reconstructed images also provided.
significantly smaller compared to baseline methods, align- ing with the quantitative results.
Table Quantitative results for inversion and reconstruction us- ing the FLUX-dev model (excluding the DDIM baseline). NFE includes both inversion and reconstruction function evaluations. Steps computational costs are kept comparable across compar- isons. Reconstruction is performed without leveraging latent
Convergence Rate:
features from the inversion process.
Steps
NFE
LPIPS
SSIM
PSNR
DDIM-Inv.
50
100
0.2342
0.5872
19.72
ReFlow-lne Inv.
30
60
0.5044
0.5632
16.57
RF-Solver
30
120
0.2926
0.7078
20.05
ther illustrates the results when other approaches are fully converged at denoising NEE 60.
RF-Inversion
30
0.4480
0.4599
16.22
Ours
30
62
0.1579
0,8160
23.87
ReFlow-Inv.
0.8145
0.3828
15.29
5.4. Inversion-based Semantic Image Editing
RF-Solver
5
20
0.5010
0.5232
14.72
Rn-errsioonn
9
18
0.7574
0.3742
13.37
Quantitative Comparison: We evaluate prompt-guided
Ours
8
18
0.4111
0.5945
16.01
editing using the recent PIE-Bench dataset (Ju et al., 2024), which comprises 700 images across 10 types of edits. As shown in Table 4, we compare the editing performance in terms of preservation ability and CLIP similarity. Our method not only competes with but often outperforms other approaches, particularly in CLIP similarity. Notably, approach achieves high-quality results with relatively few editing steps, demonstrating its efficiency and effectiveness in maintaining the integrity of the original content while producing edits that closely align with the intended modifi-
shown in Table 5, demonstrate that our approach achieves significant reduction in reconstruction error, whether com- pared at the same number of steps (yielding approximately 2x speedup) or at the same computational cost.
Qualitative Comparison: As shown Figure 5, our ap- proach provides an efficient and effective reconstruction method based on FLUX. The drift from the source image is

---

FireFlow: Fast Inversion of Rectified Flow for Image Semantic Editing
[R]
[c]Plush
[c]Real
[+]biWild gooserd
[R]Child
[+]Sun
CR]
[+]5now
WR]e
Source
Ours
RF-Solver RF-Inversion DiffEdit
MasaCtrl
PnP-Inv.
PnP
Pix2Pix-7erro P2P
Figure 6. Comparison with State-of oftth----tee-rttt editing methods. As shown in the comparison (original images in Row edited results in subsequent rows), our approach delivers the most accurate and high-quality edits.
cations.
Qualitative Comparison: We present the visual editing re- sults in Figure 6, which are consistent with our quantitative findings. Our method highlights fundamental trade-off between minimizing changes to non-editing areas and hancing the fidelity of the edits. In contrast, methods like P2P, Pix2Pix-Zero, MasaCtrl, and PnP often struggle with inconsistencies relative to the souree image, as evident in the 3rd and 6th rows of the figure. Additionally, these methods frequently produce invalid edits, as shown in the 7th and rows. While PnP-Inv excels preserving the structure of the source image, often fails to effectively apply the desired edits. Rectified flow model-based methods, such as RI--Inversion and So-Sol, deliver better editing re-
Table 6. Per-image inference time for ReFlow inversion- based edit- ing measured an RTX 3090.
Resolution
Steps
Time Cost Speedup
Vanilla ReFlow
512 x 512
28
23.76s
1.0x
RF Inversoso
512 x 512
28
23.36s
1.02
RF-Solver
512 x 512
15
25.31s
0.94
Ours
512 512
7.70s
3.09
Vanilla ReFlow
1024 1024
28
72.10s
1.0x
RI--Inversion
1024 1024
28
71.35s
1.01
RF-Solver
1024 x 1024
15
78.80s
0.92x
Ours
1024 x 1024
8
24.52s
2.94x
8

---

FireFlow: Fast Inversion of Rectified Flow for Image Semantic Editing
sults compared the aforementioned methods. However, they still face challenges with inconsistencies in non-editing areas. Overall, our method provides a more effective solu- tion to these challenges, achieving superior results in both preservation and editing fidelity.
pix2pix: Learning to follow image editing instructions. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, CVPR 2023, Vancouver, BC, Canada, June 17-24, 2023, Pp. 18392-18402. IEEE, 2023.
Cao. M., Wang, X., Qi, Z., Shan, Y., Qie, X., and Zheng, Y. Masactrl: Tuning- frre mutual self-attention control for consistent image synthesis and editing. In Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), pp. 22560-22570, October 2023.
Inference Speed: In Table 6, we compare the inference time of FireFlow with several recent models. The number of steps is based on those reported in the original papers provided in open-source code. FireFlow is significantly faster than competing reflow models and does not require an auxiliary model for editing.
Chen, T., Gu, J., Dinh, L., Theodorou, E., Susskind, J. M., and Zhai, S. Generative modeling with phase stochastic bridge. In The Twelfth International Confer- ence on Learning Representations, 2024. URL https: / //openreviee ew net/ fww.mmidddtttoo dit.ummiiddtttteeeddd...
6. Conclusion
We proposed a novel numerical solver for ReFlow models, achieving second-order precision at the computational cost of a first-order method. By reusing intermediate velocity approximations, our training-free approach fully exploits the nearly constant velocity dynamics of well-trained Re- Flow models, minimizing computational overhead while maintaining accuracy and stability. This method addresses key limitations of existing inversion techniques, providing scalable and efficient solution for tasks such as image re- construction and semantic editing. Our work highlights the untapped potential of ReFlow-based generative frameworks and establishes a foundation for further advancements in efficient numerical methods for generative ODEs. We also provide a discussion of the limitations Section F.
Chen, X., Fang, H., Lin, T., Vedantam, R., Gupta, S., Dollár, P., and Zitnick, C. L. Microsoft COCO cap- tions: Data collection and evaluation server. CoRR, abs/1504.00325, 2015. URL http://arxiv.or abs/1504.0033 00325.
Cho, H., Lee, J., Kim, S. B., Oh, T., and Jeong, Y. Noise map guidance: Inversion with spatial context for real image editing. In The Twelfth International Conference on Learning Representations, ICLR 2024, Vienna, Austria, May 7-11, 2024. OpenReview.net, 2024.
Couairon, G., Verbeek, J., Schwenk, H., and Cord, M. Diffedit: Diffusion-based semantic image editing with mask guidance. In The Eleventh International Confer- ence on Learning Representations, ICLR 2023, Kigali, Rwanda, May 1-5, 2023. OpenReview.net, 2023.
Acknowledgement
This work was partially supported by the National Key R&D Program of China (No. 2022ZD0160304), the Beijing Science and Technology Plan Project (No. Z231 100005923033) and CCF- Tencent Rhino-Bird Open Research Fund.
Deng. Y., He. X., Tang, F., and Dong, W. Z*: Zero-shot style transfer via attention reweighting. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 6934 6944, June 2024a.
Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
Deng. Y., He, X., Tang, F,, and Dong, W. Z-star+: A zero-shot style transfer method via adjusting style dis- tribution, 2024b. URL https: http://axxxvvv/// abs/ 2411.19231
Deng. Y., He, X., Tang, F., and Dong, W. Z-magic: Zero- shot multiple attributes guided image creator, 2025. URL https: / http///axvvvrrr/// abs 250.. 12124.
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Heusel, M., Ramsauer, H., Unterthiner, T., Nessler, B., and Hochreiter, Gans trained by a two time-scale update rule converge local nash equilibrium. In Guyon, I., von Luxburg, U.. Bengio, S., Wallach, H. M., Fergus, R., Vishwanathan, S. V. N., and Garnett, R. (eds.), Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, December 4-9, 2017, Long Beach, CA. USA. pp- 6637, 2017.
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2022.

---

FireFlow: Fast Inversion of Rectified Flow for Image Semantic Editing
A. The Pseudo-code for Inversion and Editing
Algorithm Solving ReFlow Inversion ODE
Require: Discretization sleps N, reference image Xo. prompt embedding network Φ, Flux model ((()))))) time steps
[o,,..,
Ensure: Structured noise X1
1: Initialize vio (Xio) = v(Xto: to, p()):pp)
[Run}
2: Ato = t
Aoot Xta + 5e00 Uta (Xto)
4: Initialize U 1 3 (X )ots) = v v(x to toci to + 2lo, (())))
(Run & Save to GPU Memory)
5: Xe, = Xto + Ato tto seo (X (
6: for = ::N-1 1 do
7: 0,, (X,,)) s- 1111tt - (X, (x )
(Load from GPU Memory)
8:
Al; = -4
9:
3e, = =Xe; + A/44 01; (Xo:)
Jut (X,,)) = xX,,, 1,- Jar, t + S1,, (():pp)
(Run Save to GPU Memory)
11:
X1,11 = =x, + A4 o4 بدي + }
12:
if==N - then
12:
Save ViN", to storage.
13:
end if
14:
end for
return X1, Viy in Self-attention Layers
Algorithm 2 Solving ReFlow Denoising ODE (Editing)
Require: Discretization steps N, reference text "prompt", structured noise X1, prompt embedding network 4, Flux model (1))) time steps/ L = [IN-1. ..... fo], pre-computed Vix", in Self-attention Layers during inversion
Ensure: Edited image Xu
I: Initialize -tx- (1(w-1)= v v(X 1I IN 1, (promppt):
{Replace Viddit with Vine. in Self-attention Run}
2: AtN-1= fN- fN-1
3: A = XtN-1 ANN VeN I
4: Initialize +xx-tt/ee-- (X, Sy--+t-Ae-
r(X, ex forit//en-a -ANN-I AN-1 + 1Atx-1 #(prompt); sp) (Replace Vedit with Vine.
Self-attention & Run Save to GPU Memory]
5: X = XIy-1 + AtN-1 Vlx 1+ ia (X itb 1)
6: for i= 22
7: (X,,) + 1144t]1 (844thAcu
[Load from GPU Memory)
8: At;=ti-: ti-1 t4
9:
Xi+tA1, = -X,, + A44 0, (Xi,)
10: Uut (Xut)an) = v(x,,- Jan + 1 Ats, (promppt): 9)
Run Save GPU Memory)
X1,1 = Xi, Al u te, (X,, (x,,)
12: end for
13: return Xa
B. Technical Proofs
This section provides detailed technical proofs for the thcoretical results discussed in this paper.
B.1. Proof of Proposition 3.1
Proof. The reverse ODE is given by:
dr -- ().
(13)
dt
Let n. Troe be the true solution of the reverse ODE, starting from =.Tre and a Penurbed be the solution starting from Nuumerical _uuuee + Ar. The initial condition difference is:
Penurbed (T) x. (TTe((T) AT.
(14)
12

---

FireFlow: Fast Inversion of Rectified Flow for Image Semantic Editing
Define the error (1) as the difference between the perturbed and true solutions:
(t) Perturbed (t) x True (t).
(15)
The dynamics of (t) follow from the reverse ODE:
da(()
Perurbod (1) dx (1)
(16)
dt dt dt
Substituting the reverse ODE for each term:
dA(t)
v0 Perturhed (t).t) + =xxx True (t),
(17)
dt
Using the Lipschitz continuity of v(x, t), we have:
()x)))((x"iin)) || (t), v1x (t), (t). = LIIA(t)II
(18)
where L is the Lipschitz constant of t.. Thus,
da(t)
(19)
dt
Based the definition of the derivative of norm:
d|
A(t)
dl(t)
A(t)
da(t)
dd(()
L|)(())
(20)
dt
|A(I]
di
(A())
dt
di
This can be rewritten as: Integrate both sides from t = T (initial condition) to t = 0 (final condition):
(
dd.
(21)
|
d (A)I T
& ["
Ltt.
(22)
(
TT
Thus, the inequality becomes:
In ||())) In | < - LT.
(23)
Simplify:
In | S In [[()I - LT.
(24)
Exponentiate both sides to remove the logarithm:
|A()]] & (I(T)) LT
(25)
B.2. Proof of Proposition 4.1
Proof. Define the time interval between and t - 1 as At for simplicity, the reused velocity at is given by:
2/4 2
(o(X,, = = 1)
(26)
where X X(I-1)+ a is computed recursively using:
Al
X (0-11444 -X-II 2
(Xe-1,t- 1). 0o
(27)
The exact velocity at is:
(28)
13

---

FireFlow: Fast Inversion of Rectified Flow for Image Semantic Editing
To quantify the difference [| (X,,) ",(X,,))], expand the reused velocity 6o(X,; () around the exact velocity vo ()) Using Taylor series expansion, expand vo (X (1-))) 1) ) around X, and
+ (t
00 (I-() (1- + A R (X
Oua (X (X(11))+ Xx) + Oug ( Al + 2
At
+ +OAA))
(29)
0x
dL
The temporal difference is:
At At 2
At
(30)
2
Thus, the temporal contribution to the velocity difference is:
At Oug
(31)
2 01
The leading term introduces an error of ((1)).
The spatial difference is:
((1-1)+ - Xx.
(32)
Using the recursive relationship:
At
(1-1)) X1-1 + 0g (x 1).
(33)
2
and the local truncation error of Euler method
X X A vo (x
+) O(A())
(34)
we subtract:
At
X (1-11)+ -x, = -t -(
(35)
2
Simplify:
At
X (1-1)+ -
Xx
2 10) (-
(36)
Substitute this into the spatial term:
dee (X 1)+ - X,)
At Eug
(0o (Xi-1,tt-1)) 2
(37)
ax
2 ax
Combine both temporal and spatial difference:
At dug (fa (XL-1,, 11) 1))
At
Oog
Oog
() 00 (X,,,))) t) uo ()) 2 ox
+
0g (X.-1, 1)) +(AF).
(38)
2
at
ax
Let &, =1000(x,11) - -vo(X,,()1 f)1. From the above analysis and triangle inequality:
Al
S 8 O(A1).
(39)
Unfold the recursion:
At O(At) .....
& O(At)) + o(At)+ +
(40)
2
2
This is geometric series with common ratio At 2
summing to:
At &
8, s (At)
= =O(At)-
(41)
1 At'
kmm
For small At, T- ~ 1 + At
2, So:
6, = =16 [X,,, 1) (x. (11 S S((A).
(42)
14

---

FireFlow: Fast Inversion of Rectified Flow for Image Semantic Editing
B.3. Proof of Theorem 4.2
Proof. Define the time interval At for simplicity, our modified midpoint method updates X as:
At )
X2+1 Xx + AL- uo (X+ 2r, ,
(43)
where:
At
X x = Xx
20 t),
(44)
2
Substituting x+ the velocity term becomes:
00 (x , 1 2 ss (X(,), + At Oug
At doa
00 (X 1) + O(AI))
(45)
ot
2 ax
Under the assumption ((,, - vo (Xe, t) s ((t)), we write:
60 8)
(X,,t) + 50,
(46)
where ||ōo|| S O())). Substituting this into the velocity term expansion:
Dug 0x
Oug
Oug sv.
00 (X,, t) = 0x
uy (X,, t) +
(47)
0x
Since ||ōد|| & (()t), the additional term At 5xōr de contributes an error of O which is the same order as 2
higher-order terms in the expansion. Thus, the velocity term becomes:
At 2
(X,, At Oup
At dug 00 (X,, () O(AI))
89 X 4 2 +
(48)
ōt
2 ox
Substituting the expanded velocity back into the updated equation:
X = Xx At
(10(X(,t)+ +
At Oug
+
At Oug
2o (x +xxx))+((()))))
(49)
2 Ot
2 ox
Simplify:
At Oup
2 dog
X1+1 = Xx, + At (Xe,tt)
+
2o (X,,) +
(50)
2 ôt
2 0x
The exact solution to the ODE is:
At Ovg
At Oug
X(t + At) = X(t)= + At ug (x t)
+
+
Uy (X,,t + (OAP))
(51)
2 Ōt
2 ax
The modified midpoint method's update matches the exact solution up O (AAP), confirming that the local truncation remains O(At). Since the local truncation error is O(At)), the global error accumulates over ((1Att) steps, resulting in O(AP).
C. Empirical Convergence Rate
We now present comparative analysis of reconstruction errors for different solvers, considering up to 60 Number of Function Evaluations (NFE). Our method exhibits rapid convergence a consistently low reconstruction error, demonstrating its efficiency in minimizing reconstruction loss. The vanilla ReFlow model, implemented with first-order Euler method, suffers from a slow convergence rate, implying higher computational cost to achieve comparable accuracy. Notably, the RF-solver, which incorporates a second-order truncation error, displays a non-monotomi convergence behavior. Although it initially converges more quickly than vanilla ReFlow, the reconstruction error begins to increase after approximately 25 steps. This suggests that the second-order approximation in the RF-solver may introduce instability or over- ver--itting beyond this point, emphasizing the robustness and efficiency of our proposed method.
15

---

FireFlow: Fast Inversion of Rectified Flow for Image Semantic Editing
70
Vanilla ReFlow
Root Mean Square Error (RMSE)
80
Ours
RF-Solver
80
40
30
20
10
10
20
30
40
50
80
Number of Function Evaluations (NFE)
Figure 7. Visualization of the convergence of different order inversion and reconstruction method. With 60 NEE, our approach still enjoys the lowest reconstruction error and the fastest convergence speed.
D. Python-style Pseudo-Code
In this section, we provide Python-style pseudo-code to elucidate the core algorithmic steps of our proposed approach. This approach, while conceptually simple, delivers substantial performance improvements. Specifically, initialization of the variable is performed as None in the first iteration. Subsequently, each iteration requires only single invocation model evaluation function, thereby maintaining computational cost per iteration equivalent that the baseline ReFlow model employing the first-order Euler method. However, evidenced by the convergence analysis presented in Figure approach achieves significantly accelerated rate of convergence.
hat_velocity = None
for t_curr, t_prev in zip (timestepss[--1) timesteps [1:1):
if hat_velocity is None:
velccity model (X, t_curr)
else:
velocity = hat_velocity
Xmii = X (t_prev t_curr) / 2 velocity
velocity_mid model mi_iid, t_curr + (t_prev t_curr) / 2)
hat hat_velocity velocity_mid
10
- (t_p t_ourr) velocity_mid
11 return
E. Ablation Study
Editing Steps: An ablation study was conducted to evaluate the influence of the number of editing steps on the editing performance. The number of steps was varied from 2 to 12, and the corresponding results are presented Figure 8. The analysis reveals that with only two editing steps, the editing prompts are not effectively incorporated, resulting in suboptimal performance. However, significant improvement in performance observed as the number steps increases. Notably, the performance achieved with eight steps is comparable that obtained with 10 steps, indicating that eight steps provide sufficient capacity for effective editing. Consequently, a value of eight editing steps selected for use in all subsequent experiments.
Inversion-free Editing: Here, we apply our modified midpoint method to solve the controlled ODEs employed by other ReFlow- based editing methods. This compatibility underscores FireFlow's robustness handling the ODE challenges inherent ReFlow models, regardless whether they occur during the forward or reverse diffusion process. Visual comparisons highlighting the superiority of our approach are shown in Figures
16

---

FireFlow: Fast Inversion of Rectified Flow for Image Semantic Editing
Source
2 steps
4 steps
6 steps
8 steps
10 steps
12 steps
Figure 8. Ablation Study on the Number of Editing Steps.
(a) cat
(b) tulip
tiger
(d)
(c) pink flower
(f) in the woods
(g) Source Images
(h) silver sculpture
(i) lion
(i) cat
(k) fox
teddy hear
(m) the city
(n) FlowEdit Results
(o) silver sculpture
(p) lion
(g) cat
(r) fox
(S) teddy bear
(1) in the city
(u) FlowEdit FireFlow Results
Figure 9. Illustration on the effect of FireFlow solver when coupled with other controlled ODE.
17

---

FireFlow: Fast Inversion of Rectified Flow for Image Semantic Editing
PILLA
(a) Source Image
(b) Black Cat
(c) Source Image
(d) Raising Hands
(e) Source Image
(f) Storm-trooper
Figure 10. Illustrations FireFlow Failure Cases.
PHLLI
(a) Source Image
(b) Black Cat
(c) Source Image
(d) Raising Hands
(e) Source Image
(f) Storm-trooper
Figure 11. Illustrations on FireFlow with feature addition in Self-attention.
F. Limitations
We empirically observe that approach struggles with editing tasks involving changes to object colors uncommon scenarios in natural images. As illustrated in Figure 10, the cat's 's color remains unsatisfactory after editing. Similarly, in less common scenes, such when the person's 's head is not visible in the image, the editing results are poor.
Another example involves an uncommon description, such as "a [storntrooper] with blue hair wearing a shirt," which also yields unexpected results. We attribute these issues to the simplicity the editing strategy, which involves only replacing the feature the self-attention module:
Qudit Kedit
Self Attmedit Softmax(
Vinv.
(52)
Vd
This approach appears insufficient for handling such scenarios.
We empirically find that incorporating K feature addition in the self-attention module can resolve these problems. Formally,
Self_Attneati = Softmax Qedit (Kedst + Kinv.)
Vedit
(53)
Vd
However, this comes at the cost of diminished preservation of the original structure and background details. Details can be found in Figure 11. We also include an quantitative analysis different editing strategies Table 7. The results consistent with the illustrations. It is evident that Equation 53 belongs the category of "cross-attention." focusing merging features from the self-attention module during inversion with the corresponding features generated during the denoising process. This concept has been extensively discussed in diffusion model (DM) editing methods. Table presents only the foundational attempts in this direction. We hope this will inspire further research and advancements in future work.
18

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FircFlow: Fast Inversion of Rectified Flow for Image Semantic Editing
Table 7. Comparison on different editing methods. Results on PIE Bench are reported. Guidance terms indicate the guidance ratio settings used the FLUX model during the denoising process.
Structure
Background Preservation
CLIP Similarity!
Method
Guidance
Steps
NFE
Distance
PSNR1
SSIM
Whole
Edited
Add
[1.2,2....]
0.0590
17.72
0.7340
27.01
23.84
8
18
Add Q + Add K
[1.2,2....
0.0537
18.35
0.7520
26.82
23.60
8
18
Add Q + Add K + Add V
[1,2,2,...
0.0416
19.63
0.7805
25.95
22.92
8
18
Add Q
[1.1,2...]]
0.0530
18.78
0,7580
26.99
23.85
15
32
Add Q Add K
[1.1,2....]
0.0486
19.30
0.7721
26.68
23.59
15
32
Replace V
[2.2....]
0.0271
23.03
0,8249
26.02
22.81
8
18
Add Q
[2.2...]]
0.0710
16.49
0.7077
27.33
24.09
8
18
Add K
[2.2....]
0.0738
16.41
0.7066
27.25
24.01
8
18
19


Reviews:
[{"review_id": "xevqzY1WNe", "reviewer": "Reviewer_MGiL", "summary": "This paper introduces ​FireFlow, a fast inversion and editing method for Rectified Flow (ReFlow) models, designed to enable accurate image reconstruction and semantic editing with minimal computational overhead, enabling fast, high-fidelity image editing while fully leveraging the model's inherent linear motion properties. The approach advances practical applications of ReFlow-based generative models without architectural modifications or training.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Q1: Results on PIE-Bench are reported with fixed seed? Variance across multiple runs is not reported.\n- Thank you for your valuable feedback. To clarify, T2I generation starts from random noise determined by seed, while inversion-based editing starts from a fixed image as the initial point in ODE and solving ODE does not involve randomness. Nonetheless, we have re-run the experiments 5 times using different random seeds:\n\nPIE-Bench|Structure Distance↓|PSNR↑|SSIM↑|CLIP Whole↑|CLIP Edited↑\n-|-|-|-|-|-\nReported in the paper|0.0271|23.03|0.8249|26.02|22.81\n5-runs Mean|0.0271|23.03|0.8249|26.02|22.81\n5-runs Std|0.|0.|0.|0.|0.\n\nQ2: Minor gaps in dataset diversity and human evaluation.\n- Thank you for your comment. We evaluated FireFlow using three public datasets—MSCOCO Caption, Densely Captioned Images (DCI), and PIE-Bench—across T2I, inversion, and editing tasks. While additional experiments could further support our findings, the current results already show significant improvements over recent methods. Notably, PIE-Bench includes 10 diverse editing tasks, including style transfer, as requested.\n- Regarding human evaluation, we are conducting a crowdsourced blind experiment. Due to time constraints, we couldn’t gather enough voting results for the rebuttal. However, we will include the user study results in the appendix for a more comprehensive evaluation.\n\nQ3: Without controlling for NFEs or steps for baseline methods, will the claimed speedup (e.g., \"3× runtime speedup\") be exaggerated?\n- Thank you for your insightful question. To clarify, we did not intentionally increase the NFEs or steps in the baseline methods to exaggerate the reported speedup. We followed the standard settings for a fair comparison. Besides, we conducted an ablation study on RF-Inversion and RF-Solver, which shows that reducing the number of NFEs or steps leads to higher image editing failure rates. This demonstrates that the selected steps are more optimal for editing tasks.\n\nMethod|steps|NFEs|Structure Distance↓|PSNR↑|SSIM↑|CLIP Whole↑|CLIP Edited↑|Qualitative Analysis\n-|-|-|-|-|-|-|-|-\nRF-Inv.|28|56|0.0406|20.82|0.7192|25.20|22.11|Better trade-off\nRF-Inv.|14|28|*0.0677*|*18.88*|*0.7160*|26.03|23.20|**Failed to preserve original content**|\nRF-Solver|15|60|0.0311|22.90|0.8190|26.00|22.88|Better trade-off|\nRF-Solver|8|32|0.0211|24.93|0.8597|*24.47*|*21.25*|**Failed to conduct editing**\n\nQ4: More discussions on the cause of limitation should be given. Any solutions for these cases?\n- Thank you for your valuable comment. As a fast, training-free inversion-based editing method, we believe the limitations of our approach can be attributed to the following factors, along with potential solutions:\n  - Practical conditions vs. Proposition 4.1: As shown in Table 4, increasing the number of steps contributes to smoother velocity, which in turn improves performance. Additionally, more powerful ReFlow models offer more accurate velocity estimations, helping to bridge the gap between the theoretical assumptions and real-world scenarios. This is reflected in Table 7, where SD3.5 performs better than SD3.\n  - Editing technique compatibility: Our method works as a fast solver for ReFlow models, which makes it compatible with other ReFlow-based editing techniques. However, it inherits both the strengths and weaknesses of those methods. As such, improving the design of the editing technique itself will further enhance the performance of our approach.\n  - Prompt suitability: It is well-known that prompt engineering plays a crucial role in image generation. The official prompt for PIE-Bench is relatively brief, which could limit further improvements in editing quality. A more sophisticated vLLM-based prompt generator could better align the prompt with the editing task and yield better results.\n\nQ5: Experiments focus on ​semantic edits (e.g., object replacement). Complex edits like stylization, multi-object manipulation is unexplored.\n- Thank you for pointing that out. We apologize for the bias in the illustrations, which mainly focus on object replacement and may have been misleading. In fact, our approach performs well on more complex tasks, including stylized generation and multi-object manipulation. For additional results, please refer to the following [anonymous link](https://ibb.co/F4XzBrbM).\n\nQ6: Can this method be extrapolated to further accelerate by approximating higher-order methods through more caching intermediate steps?\n- Thank you for your insightful suggestion. We are actively exploring this avenue and have already found higher-order solvers compatible with FireFlow, enabling faster inversion with as few as 4 steps. We are also working towards a theoretical proof that a class of numerical solvers can be accelerated in this way.\n- However, we believe the current submission is self-contained, and we view this as an exciting direction for future work. We appreciate your suggestion and plan to explore it further in subsequent research.", "claims_and_evidence": "The claims are ​largely supported by theoretical analysis and extensive experiments. The core contributions (efficient high-order solver, empirical superiority) are valid and impactful. Some suggestions for revisions:\nThe evaluation on PIE-Bench uses a fixed seed; variance across multiple runs is not reported.\nInvolving humen assessment would strength the contribution.", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria are well-aligned with the problem of efficient inversion and semantic editing for ReFlow-based generative models.  Evaluation criteria are ​appropriate and comprehensive, covering quality, accuracy, and speed. Minor gaps in dataset diversity and human evaluation could be addressed in future work but do not undermine the core contributions. ​", "theoretical_claims": "The numerical experimental results and theoretical results are not perfect matches, there are slight inconsistencies at some points, but I agree with the author that the overall trend is consistent. The findings presented in this paper contribute to this field.", "experimental_designs_or_analyses": "Most results are compelling, several aspects of the experimental design and analysis warrant closer scrutiny:\n- The baselines (e.g., RF-Solver, RF-Inversion) are compared at different step counts and NFEs. Without controlling for NFEs or steps, the claimed speedup (e.g., \"3× runtime speedup\") may exaggerate the method's efficiency.", "relation_to_broader_scientific_literature": "The key contributions of FireFlow build upon and extend several lines of research in generative modeling, numerical methods for ODEs, and image inversion/editing, would benefit researchers in related fields.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Strengths \n- The authors propose a modified midpoint method that achieves ​second-order accuracy with first-order computational cost, a creative adaptation of ODE solvers tailored for ReFlow's constant velocity dynamics. This is a significant departure from prior ReFlow inversion methods (e.g., RF-Solver, RF-Inversion) that either sacrifice accuracy for speed or require higher NFEs.\n- Establishes error bounds for velocity reuse (Proposition 4.1) and proves global truncation error equivalence to standard midpoint methods (Theorem 4.2), advancing the theoretical understanding of ReFlow inversion.\n- Logical flow from motivation to theory, method, and experiments are well-structured. Algorithms 1–2 and synthetic 2D experiments clearly illustrate the solver's mechanics. Code availability and detailed ablation studies (e.g., step-size vs. error in Figure 3) enhance reproducibility.\n\nWeaknesses\n- The statistics are reported with fixed seed. \n- More discussions on limitations should be given. Experiments focus on ​semantic edits (e.g., object replacement). Complex edits like style transfer, multi-object manipulation is unexplored, limiting the perceived versatility of the method.\n\nFireFlow’s ​innovative solver design and ​strong empirical results make it a compelling contribution to fast inversion/editing in ReFlow models. While limited comparisons and narrow task scope slightly weaken the narrative, its theoretical rigor, efficiency gains, and practical applicability position it as a valuable advancement for generative ODEs.", "other_comments_or_suggestions": "Line 72: This motivates a closer investigation. The format of Table 3 could be revised.", "questions_for_authors": "How to enhance the proposed method for more challenging editing tasks?\n\nCan this method be extrapolated to further accelerate by approximating higher-order methods through more caching intermediate steps?", "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": "iiKNSRPinS", "reviewer": "Reviewer_3efD", "summary": "The authors proposed a second-order ODE solver to speed up inversion and reconstruction of flow-based model. It reused saved midpoint velocity so the computation efficiency remains the same as first-order ODE. It is proven to be faster in speed and higher in quality, and also benefit image editing by a large margin.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate the authors for carefully verifying our theoretical claims, especially given the heavy review process. We will do our best to address your concerns.\n\nQ1: It is a fully training-free method which does not involve 'learning' methods which are better aligned with ICML.\n- We respect the reviewer’s opinion on whether a fully training-free method fits within the scope of machine learning. However, we believe that generative models are a prominent topic in the field of machine learning, and the ReFlow model, as a pioneering work in this area, warrants attention. FireFlow, with its efficiency gains, demonstrates practical applicability, which aligns with the interests of the machine learning community, hence aligns with ICML.\n\nQ2: Whether it will be better to estimate the next midpoint velocity by measuring the current velocity and previous velocity to propagate, achieved by Numerical Extrapolation (using the previous two velocity values from the network to extrapolate next midpoint)?\n- Thank you for your insightful suggestion. If we understand correctly, the proposed numerical extrapolation involves using the previous two velocity values, $v_t$ and $v_{t-1}$, generated by the network to approximate the next midpoint velocity as $\\hat{v}\\_{t+\\frac{1}{2}\\Delta t}=v_t + 0.5(v_t-v_{t-1})$ and updating the state as $X_{t+1}=X_t+\\Delta t\\cdot\\hat{v}_{t+\\frac{1}{2}\\Delta t}$. To evaluate this approach, we have conducted an ablation study on step selection, and our approach continues to outperform the others, with results on PIE-Bench presented below:\n\nMethod|steps↓|NFEs↓|Structure Distance↓|PSNR↑|SSIM↑|CLIP Whole↑|CLIP Edited↑\n-|-|-|-|-|-|-|-\nNumerical Extrapolation|15|30|0.0398|22.48|0.7913|**26.14**|22.64\nNumerical Extrapolation|8|16|0.0658|21.45|0.7504|26.04|22.32\nOurs|**8**|**18**|**0.0271**|**23.03**|**0.8249**|26.02|**22.81**\n\nQ3: Whether the proposed method will downgrade to 1st-order ODE if the velocity estimation of the midpoint is not accurate enough, whether the method can be well generalized to other ReFlow models, especially when $v$ is not accurate enough.\n- Thank you for your insightful question. We recognize that the robustness of our approach, particularly when velocity estimation is less accurate, is an important aspect to explore further. Since the accuracy of $v$ directly impacts image quality, we assess our method on earlier-generation, medium-sized Stable Diffusion models, such as SD3-medium, which has a significantly lower ELO score than the FLUX model and serves as a meaningful test case. We apply different ODE solvers to ReFlow-based editing, and the results on PIE-Bench are as follows:\n\nMethod|Order|steps↓|NFEs↓|Structure Distance↓|PSNR↑|SSIM↑|CLIP Whole↑|CLIP Edited↑\n-|-|-|-|-|-|-|-|-\nRF-Inversion|1st-order|28|56|0.0464|19.75|0.6951|25.20|**23.16**\nOurs|2nd-order|**8**|**18**|0.0476|19.41|0.6871|25.16|23.11\nOurs|2nd-order|15|32|**0.0439**|**20.07**|**0.7019**|**25.30**|23.07\n\n- While the speedup rate is lower than FLUX, our method consistently outperforms 1st-order ODE across most metrics, achieving approximately a 2× speedup while maintaining high-quality results. Given that the accuracy of $v$ is directly linked to image quality, we believe it is more meaningful to focus on state-of-the-art models with higher accuracy and better generation capabilities—such as the FLUX model, as discussed in our paper.\n\nQ4: The paper does not introduce enough novelty in inversion and editing, with problems like all other inversion-based method like poor identity preservation.\n- We respectfully disagree with the claim that our work lacks novelty in inversion and editing. Our fast ODE solver for ReFlow introduces a fundamentally new approach to accelerating generative models, with no direct overlap with existing acceleration methods. This is substantiated by both rigorous theoretical analysis and extensive experiments, demonstrating clear improvements.\n- As with any training-free editing method, certain failure cases are expected, and claiming otherwise would be unrealistic. This is precisely why we have transparently acknowledged limitations in the appendix. However, occasional challenges in identity preservation should not overshadow the consistently strong performance demonstrated on widely recognized benchmarks. Our results indicate that FireFlow surpasses prior methods in efficiency and effectiveness.\n- More importantly, our core contribution extends beyond individual editing performance—it establishes a new direction for accelerating ReFlow models. This is not merely an incremental improvement but a principled advancement, supported by both theoretical foundations and empirical validation.", "claims_and_evidence": "Yes. The claims in the paper are proven with evidences.\n- For example, the authors claimed that the inversion efficiency can be improved with the proposed second-order method. Compared with midpoint method, it reduces half of the NFEs.\n- The authors claims the accuracy remains similar to standard midpoint methods. It is proven in the appendix given some constraints.\n- The authors also showed results on PIE-bench and visual results to prove the editing quality.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "The main theoretical claims are proposition 4.1 and theorem 4.2. They are intuitively correct, however, they may have some potential issues to the best of my knowledge. \n\n- I am not sure whether the proposed method will downgrade to 1st-order ODE if the velocity estimation of the midpoint is not accurate enough, saying v is not well trained. \n- Whether it will be better to estimate the next midpoint velocity by measuring the current velocity and previous velocity to propagate, instead of fully relaying on the velocity estimation?\n\nBut given the constraints and assumption, the proof is sound and correct.", "experimental_designs_or_analyses": "The experiments have no obvious flaw. One concern is all the methods are evaluated using FLUX model which is somehow well trained. So it is not clear whether the method can be well generalized to other ReFlow models, especially when v is not accurate enough.", "relation_to_broader_scientific_literature": "I am not confident that this method is novel enough. First, it is a fully training-free method which does not involve 'learning' methods which are better aligned with ICML;  Second, the 2nd-order or 3rd-order ODE can be also achieved by Numerical Extrapolation (using the previous two velocity values from the network to extrapolate next midpoint) which also only needs 1 NFE. Some proper comparison might be needed to mitigate the concerns of large delta_t and inaccurate v estimation. Third, the paper does not introduce enough novelty in inversion and editing. The proposed method remains having problems like all other inversion-based method like poor identity preservation.", "essential_references_not_discussed": "Not found.", "other_strengths_and_weaknesses": "See above.", "other_comments_or_suggestions": "See above.", "questions_for_authors": "See above.", "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": "xY0p5drCHJ", "reviewer": "Reviewer_ekVQ", "summary": "This paper proposes a low-cost alternative method for a second-order ODE solver aimed at Rectified Flows. Compared to the common second-order ODE solver, which requires $2T$ NFEs, this method only needs $T+1$ NFEs while maintaining the sampling quality of the second-order ODE solver. The core idea is to replace $v_t$ in the standard midpoint method with the midpoint velocity $v_{(t-1)+\\frac{\\Delta t}{2}}$ from the previous $t-1$ step (Eq.8 vs. Eq.10), so that intermediate values from $t-1$ step can be cached during the denoising process and loaded in the t-th step, avoiding the need for 2 NFEs calculation at each step.\n\n## Update After Rebuttal\nThe author's reply solved most of my problems, but I still feel that this work has some shortcomings in innovation, and I will keep my score.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate the reviewer’s detailed feedback and constructive suggestions for improving our paper. Below are our responses to your concerns:\n\nQ1: As shown in Table 2, the effect of the 20-step method should be compared again to observe how much better its solver performance is compared to the synchronous number.\n- Thank you for your valuable suggestion. We have now reported all three methods with the same 20 steps, and our approach continues to outperform the others, as shown below:\n\nMethods|Steps↓|NFE↓|FID↓|CLIP Score↑\n-|-|-|-|-\nFLUX-dev|20|20|26.77|31.44\nRF-Solver|20|40|25.54|31.39\nOurs|20|21|25.13|31.44\n\nQ2: Some chart captions are too simple, such as Table 2, Table 4, and Figure 6.\n- Thank you for pointing that out. We will revise the descriptions for Table 2, Table 4, and Figure 6 to make the captions more complete and self-explanatory.\n\nQ3: In Figure 2, why is the result of a low-cost approximation of the Midpoint method even better than the Midpoint method under the same NFE?\n- To clarify, the standard Midpoint method requires 2 NFEs per step, whereas our approach requires only 1 NFE per step. This means that, under the same computational cost, our method effectively has more steps. Using more steps (i.e., smaller step sizes) generally improves accuracy by reducing truncation error, which is bounded by the step size, in the discretization of the continuous ODE. This explains why our low-cost approximation achieves better results despite using the same total NFE.\n\nQ4: Except for the approximation, what are the most valuable contributions of this paper, according to the author?\n- We believe our work challenges the conventional notion that high-order ODE solvers must be computationally expensive. Beyond mere approximation, we introduce a new direction for accelerating ReFlow ODEs with a solid theoretical foundation and a fully training-free approach. Our method is distinctly different from training-based distillation and traditional model compression techniques. The modified Midpoint method simply serves as a starting point, we are actively exploring a broader group of high-order methods to further enhance efficiency.\n\nQ5: Under different seeds, what is the comparison of the results produced by inversion and editing, and do they have diversity?\n- We would like to clarify that, unlike T2I which begins from random noise, inversion-based editing and reconstruction start from a fixed original image as the initial point in the ReFlow ODE. As shown in Algorithm 1 and 2, solving the inversion and denoising ODE do not involve any randomness, resulting in no diversity across different seeds. For a quantitative analysis, please kindly refer to our response to Q1 for Reviewer MGiL. \n\nQ6: When NFE is extremely small (1-4), how is the performance? Some distillation-based methods seem to be able to achieve inversion and generation with fewer steps.\n- Thank you for your insightful question. Our current modified midpoint method is not designed for extremely small NFEs (1–4) and the results are unsatisfactory. However, we are actively exploring this avenue and have found higher-order solvers compatible with FireFlow, enabling faster inversion with as few as 4 steps, yet this exploration seems to extend beyond the scope of this submission.\n- Regarding distillation-based methods, FireFlow is a fully training-free approach, making direct comparisons with training-based methods less fair. Besides, efficient solver introduces a novel perspective for accelerating ReFlow models, offering a distinct and complementary direction to existing techniques.\n\nQ7: Why does this article indeed compare with methods such as dpmsolver++, unipc, etc., and why can't these methods be adapted to the sampling method of rectified flow?\n- DPM-Solver++ and UniPC rely on diffusion model ODEs, which contain an analytically tractable term. ReFlow lacks this structure, making these solvers ineffective. Specifically:\n  - Fast solvers for diffusion models exist because traditional diffusion trajectories are **curved** and inefficient, requiring precise numerical integration. In contrast, **ReFlow straightens these paths**. Since DPM-Solver++ and UniPC refine curved trajectories through multi-step corrections, they are unnecessary for ReFlow. Instead, as demonstrated by FireFlow, an efficient solver for ReFlow should focus on minimizing discretization error without requiring multiple steps.\n  - Formally, both DPM-Solver++ and UniPC leverage the diffusion model ODE structure $\\frac{dx}{dt}=f(t)x+g(t)e\\_\\theta(x,t)$ where the linear term $f(t)x$ allows for **partial analytical integration**, enabling efficient numerical solvers. However, ReFlow follows a different ODE $\\frac{dx}{dt} =v_\\theta(x,t)$ which **lacks an explicit drift term** $f(t)x$. As a result, the analytical simplifications used by DPM-Solver++ and UniPC do not apply, making these fast solvers inapplicable without a fundamental redesign.", "claims_and_evidence": "This paper claims that the proposed method can maintain the sampling efficiency of a 1st-order ODE solver while achieving the sampling quality of a 2nd-order ODE solver. This claim is mainly verified through a comparison of its sampling quality-NFEs with those of the 1st-order Vanilla ReFlow and the 2nd-order RF-solver. The experimental comparison includes qualitative and quantitative results such as t2i, inversion then reconstruction, and inversion then prompt-guided editing.\n\nAccording to the experimental results of the report and my understanding of the field, this claim has been verified.", "methods_and_evaluation_criteria": "This paper proposes a method that is meaningful for accelerating the sampling of rectified flows. The method is meaningful for accelerating rectified flow sampling, which helps to improve efficiency while ensuring the quality of sampling, thereby achieving fast inversion and prompt-guided editing.", "theoretical_claims": "The core contribution of this paper lies in improving the algorithm in Eq.8~Eq.9 to Eq.10~Eq.12, where Eq.8~Eq.9 are the standard midpoint method.\nBy the way, the proof of proposition 3.1, 4.1, and Theorem 4.2 is also provided in the supplementary materials, providing a theoretical basis for the methods presented in this paper.", "experimental_designs_or_analyses": "This paper first verifies the effectiveness of the method on synthetic mixed Gaussian data through a toy example, and then validates the effectiveness of the method for T2I, inversion-reconstruction and inversion-editing experiments. \nFrom the perspective of accelerated generation without training through solving the ODEs solver, the experimental design is reasonable.", "relation_to_broader_scientific_literature": "The most relevant to this paper is the previous method RF-Solver [1]. RF-Solver has proven that expanding the first-order ODE solver of vanilla Rectified Flow into a second-order form will help improve the accuracy of inversion. The contribution of this paper lies in reducing the NFEs of the second-order ODE solver while maintaining the quality of sampling.\n\n\n[1] Wang, J., Pu, J., Qi, Z., Guo, J., Ma, Y., Huang, N., Chen, Y., Li, X., and Shan, Y. Taming rectified flow for inversion and editing. arXiv preprint arXiv:2411.04746, 2024.", "essential_references_not_discussed": "Based on my understanding of the field, this paper discusses most methods of accelerating rectified flow by solving solvers.", "other_strengths_and_weaknesses": "Strengths: This paper relies on FLUX, a 12b t2i model, and it is meaningful to study its inversion acceleration.\n\nWeaknesses: Some experimental validations are not sufficiently thorough. For example, as shown in Table 2, the effect of the 20-step method should be compared again to observe how much better its solver performance is compared to the synchronous number.", "other_comments_or_suggestions": "Some chart captions are too simple, such as Table 2, Table 4, and Figure 6, which should have more complete and self-evident captions.", "questions_for_authors": "1. Through the toy example in Figure 2, it can be found that the method proposed in this paper is better than the Midpoint method under the same NFE, but according to my understanding, the method proposed in this paper is a low-cost approximation of the Midpoint method. Why is the result even better than the Midpoint method?\n2. In addition to the approximation of Eq.10~12 v_{t} and the introduction of cache mechanisms, what are the most valuable contributions of this paper, according to the author?\n3. Under different seeds, what is the comparison of the results produced by inversion and editing, and do they have diversity?\n4. When NFE is extremely small (1-4), how is the performance? Some distillation-based methods seem to be able to achieve inversion and generation with fewer steps.\n5. Why does this article indeed compare with methods such as dpmsolver++, unipc, etc., and why can't these methods be adapted to the sampling method of rectified flow?\nI will re-evaluate my score based on the author's reply.", "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": "t9Y5uwVv4E", "reviewer": "Reviewer_DAG3", "summary": "This work proposes a modified midpoint method, which is a well-known second order numerical solver. One drawback of midpoint method is that it needs an additional model evaluation for each sampling step, which makes the overall solver slow. This work instead proposes to piggyback on the previous model prediction in the first step of midpoint method. This reduces the number of model evaluations per step to 1. Surprisingly, the authors show that this modified midpoint method retains the  local error and global error of the second order methods while being computationally efficient. The authors demonstrate advantages of this approach in image editing tasks as well as in image reconstruction and inversion.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate the authors for carefully verifying our theoretical claims and constructive comments. Responses to your concerns as follow:\n\nQ1: Include the original midpoint method's results in Table 4 and its inference time in Table 5.\n- Although both ours and Midpoint are 2nd-order solvers, they exhibit different numerical stability when applied to ReFlow ODE. Since $v_\\theta(x,t)$ predicted by neural network is not perfectly accurate, numerical solvers handle the resulting noise differently. Ours demonstrates robustness to these inaccuracies, leading to slightly better results.\n\nSolver|steps|time(s)|Structure Distance↓|PSNR↑|SSIM↑|CLIP Whole↑|CLIP Edited↑\n-|-|-|-|-|-|-|-\nMidpoint|15|24.6|0.0307|22.94|0.8208|25.93|22.88\nMidpoint|8|13.3|0.0318|22.45|0.8122|26.02|22.89\nOurs|8|7.7|0.0271|23.03|0.8249|26.02|22.81\n\nQ2: Fig. 3 should include error analysis of the original midpoint. Fig. 4 and 7 can include RMSE of the original midpoint.\n- Thanks for valuable suggestions. We hope to clarify that Fig. 3 illustrates the approximation error between the estimated midpoint velocity and true midpoint velocity. The original midpoint method does not involve such an approximation, so an error analysis is not applicable. Regarding Fig.7 with RMSE values, we revised the illustration of RMSE versus ODE steps \"for better understanding of the effects of modification of the first substep of the midpoint method\" at [link](https://ibb.co/mrBDMkTT).\n\nQ3: Can authors include the prompts used in Fig. 6?\n- Due to this year’s strict character limit for rebuttal, we hope to include the prompts in appendix to make room for addressing your other insightful comments. We appreciate your understanding.\n\nQ4: How many sampling steps were used to compute time cost in Table 5?\n- We used 8 steps for FireFlow, 15 steps for RF-Solver, and 28 steps for RF-Inversion, consistent with Table 4. We appreciate your suggestion and will include an additional column to indicate the number of steps.\n\nQ5: Table 4 mentions steps for RF-Solver as 15 but line 328 mentions it as 25.\n- We hope to clarify that line 328 states, “*up to 25* steps.” According to RF-Solver's official code, step number is not fixed and up to 25. Since 15 steps results in a computational cost comparable to RF-Inversion, we report 15 in Table 4 for a fair comparison.\n\nQ6: In Fig. 2, how were the pairs of points for plotting the transport maps/trajectories chosen? The number of points in transport maps are fewer than scatter plots. \n- We strictly follow the [official instructions](https://colab.research.google.com/drive/1CyUP5xbA3pjH55HDWOA8vRgk2EEyEl_P?usp=sharing) provided by the original ReFlow paper for visualizing Fig. 2. According to their code, the first 30 pairs of points are selected to draw the transport trajectory, while all pairs are used for the scatter plot.\n\nQ7: Could authors elaborate why trajectories of midpoint method are less straight than the proposed method?\n- Please kindly refer to Reviewer ekVQ’s Q3, where we explain why FireFlow performs better.\n\nQ8: Alg. 1 and 2 mention $V_{t_{N-1}}^{inv}$ in self-attention layers. Could you elaborate the context of self-attention layer? Isn’t $V_{t_{N-1}}$ simply the prediction of ReFlow model? How does this relate to the discussion in Lines 1008-1012?\n- We hope to clarify that $V_{t_{N-1}}^{inv}$ refers to the $V$ feature in $\\text{softamx}(\\frac{QK}{\\sqrt{d}})V$ within the self-attention layer of ReFlow model, whereas $v_{t_{N-1}}(X_{t_{N-1}})$ denotes the prediction of network. Algorithms 1 and 2 mentioned $V_{t_{N-1}}^{inv}$ as it involved in the editing technique introduced by RF-Solver, which we follow to ensure a fair comparison  (detailed in “Image Semantic Editing” section). We observe that using $V_{t_{N-1}}^{inv}$ can lead to certain failure cases, and Lines 1008-1012 discuss its limitations along with some simple alternatives.\n\nQ9: Have authors tried using proposed method for unconditional image generation from ReFlow models?\n- Thanks for your suggestions. We follow the original ReFlow paper’s protocol for unconditional image generation on CIFAR10, using open-source 1-Rectified-Flow-distill weights. Our method performs well:\n\nSolver|steps↓|NFEs↓|FID↓|IS↑\n-|-|-|-|-\nEuler|15|15|5.67|9.20\nEuler|10|10|5.83|9.03\nMidpoint|5|10|5.45|9.27\nOurs|5|6|5.35|9.26\n\nQ10: Can FireFlow be used for solving SDEs similar to original Midpoint method? Do any assumptions in the proof not hold?\n- In fact, neither FireFlow nor the original Midpoint can be directly applied to SDEs because they do not properly handle the stochastic term. The noise term $dW_t$ in SDE representing a Wiener process is nowhere differentiable, rendering approximations that rely on smoothness invalid. Besides, if someone tries to use a midpoint method, would implicitly assume a Stratonovich interpretation, while SDEs are defined in the Itô sense. Standard ODE solvers don’t account for the correction term required when switching between these interpretations.", "claims_and_evidence": "Some important quantitive experiments are missing an ablation/baseline — the original midpoint method. The proposed method modifies the original midpoint method with an approximation, but the tradeoffs between the two choices are difficult to infer from the provided quantitative results. Note that I understand that the proposed method needs only one NFE per sampling step as opposed to two NFEs of the original midpoint method, but there should be additional results as described below for a complete overview of the tradeoffs between the two:\n    1. Table 4 should include quantitative results with the original midpoint method as well.\n    2. For better understanding of the effects of modification of the first substep of the midpoint method, Figure 3 should ideally include error analysis of the original midpoint method. Similarly, Figure 4 and Figure 7 can also include RMSE values of the original midpoint method.", "methods_and_evaluation_criteria": "Yes, the proposed methods and evaluation methods make sense. \n\nThe paper considers PIE-Bench for editing tasks, and DCI (Densely captioned image) dataset for inversion and reconstruction tasks. The paper reports quantitative performance on standard metrics for T2I models such as PSNR, SSIM, FID, CLIP similarity etc.", "theoretical_claims": "Yes. I checked the proof of Proposition 3.1, 4.1 and 4.2. The proofs seem correct.", "experimental_designs_or_analyses": "The experiments considered in this work are sound.", "relation_to_broader_scientific_literature": "The proposed method seems to be general enough to also have implications beyond the specific task of semantic editing considered in this work which is in designing efficient numerical solvers for flow ODEs for unconditional image generation. However, this paper does not include this additional analysis as it is beyond the scope of the research problem considered in this paper, but it is an interesting direction nonetheless.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Strengths:\n1. The proposed method is attractive as it provides the advantages of a second order method while being computationally more efficient. \n2. The results included in the paper seem promising as this method seems to have both improved qualitative and quantitative performance compared to prior SOTA methods. \n\nWeaknesses:\nAs mentioned above, the authors should also provide the ablation study against the original midpoint method for various applications to better understand the tradeoff between the two.", "other_comments_or_suggestions": "1. In Figure 6, can the authors include the prompts that were provided for editing?\n2. How many sampling steps were used to compute time cost in Table 5 for other methods besides Vanilla Reflow? Is this 8 steps for the proposed method and upto 28 steps for RF Solver? The exact number of steps will help understand the speedup better. Perhaps this can be indicated in an additional column or in the text. In addition, can we also include results for the original midpoint method to better demonstrate the inference time advantages (I presume it must be 2x of the proposed method)?\n3. Minor (Clarification): Table 4 mentions NFEs for RF-Solver as 15 but the text (line 328) mentions it as 25.", "questions_for_authors": "1. In Figure 2, How were the pairs of points for plotting the transport maps/trajectories chosen? The number of points in the transport maps are fewer than the number of points in the scatter plot that shows the generated points from the target distribution. Also, could the authors elaborate why the trajectories of midpoint method are less straight then the proposed method (as the proposed method is expected to have a larger error constant, even though the error is of the same order)\n2. Algorithms 1 and 2 mention $V_{t_{N-1}}^{inv}$ in self-attention layers. Could you elaborate the context of self-attention layer? Isn’t $V_{t_{N-1}}$ simply the prediction of the ReFlow network? How does this relate to the discussion in Lines 1008 - 1012 in the appendix? \n3. Have the authors tried using the proposed method for unconditional image generation from ReFlow models? This will be an interesting and useful addition to the appendix.  \n4. Can the proposed method be used for solving SDEs similar to the original midpoint method? Do any of the assumptions in the proof not hold for SDEs (e.g. smoothness assumptions)? These limitations if any, should be discussed.", "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:

Meta-review: All reviewers support the acceptance of the paper, with one Accept and three Weak Accepts. The main contribution is a numerical solver for the ODEs in RectifiedFlow models, which accelerates model inversion and reconstruction. The contribution is thus closely tied to this specific class of models. As the reviewers acknowledge the benefits of the proposed method, I recommend acceptance. The authors are encouraged to incorporate the points raised during the rebuttal into the final version.
Decision: Accept (poster)