FuncGenFoil: Airfoil Generation and Editing Model in Function Space
Authors: Jinouwen Zhang
Junjie Ren
Qianhong Ma
Jianyu Wu
Aobo Yang
Yan Lu
Lu Chen
Hairun Xie
Jing Wang
Miao Zhang
Wanli Ouyang
SHIXIANG TANG
Venue: NeurIPS 2025 poster ()
Keywords: Generative Model
AI for Engineering
Generative Model in Function Space
Flow Model
Primary area: applications
TL;DR: FuncGenFoil is a new generative model that designs and edits high-fidelity airfoils as continuous functions, outperforming prior methods in accuracy and diversity while enabling flexible, resolution-free shape generation.

Abstract: Aircraft manufacturing is the jewel in the crown of industry, in which generating high-fidelity airfoil geometries with controllable and editable representations remains a fundamental challenge. Existing deep learning methods, which typically rely on predefined parametric representations (e.g., Bézier curves) or discrete point sets, face an inherent trade-off between expressive power and resolution adaptability.
To tackle this challenge, we introduce FuncGenFoil, a novel function-space generative model that directly reconstructs airfoil geometries as function curves. Our method inherits the advantages of arbitrary-resolution sampling and smoothness from parametric functions, as well as the strong expressiveness of discrete point-based representations.
Empirical evaluations demonstrate that FuncGenFoil improves upon state-of-the-art methods in airfoil generation, achieving a relative 74.4% reduction in label error and a 23.2% increase in diversity on the AF-200K dataset. Our results highlight the advantages of function-space modeling for aerodynamic shape optimization, offering a powerful and flexible framework for high-fidelity airfoil design.

FunlGenFoil: Airfoil Generation and Editing Model in Function Space
Jinouwen Zhang
Junjie Ren
Qianhong Ma 1.8
Jianyu Wu
Aobo Yang
Yan Lu Lu Chen Hairun Xie
Jing Wang
Miao Zhang Wanli Ouyang Shixiang Tang
"Shnnghai Artificial Intelligence Laboratory
Fudan University
Hong Kong University of Science and Technology The Chinese University of Hong Kong
5 State Key Lab of CAD&CG, Zhejiang University
Shanghai Aircraft Design and Research Institute
7 Innovation Academy for Microsatellites of CAS Shanghai Jiao Tong University
2 hangjj inouwenep jlab. org. on "tangshixiangee jlab. org. en
Abstract
Aircraft manufacturing the jewel in the crown of industry, in which generat- ing high-fidelity airfoil geometries with controllable and editable representations remains a fundamental challenge. Existing deep learning methods, which typi- cally rely predefined parametric representationn (c.g., Bézier curves) or discrete point sets, face an inherent trade-off between expressive power and resolution adaptability. To tackle this challenge, introduce FuncGenFoil, novel function- space generative model that directly reconstructs airfoil geometries as function curves. Our method inherits the advantages of eebitrary-reoouutie sampling and smoothness from parametric functions, as well as the strong expressiveness of discrete point-based representations. Empirical evaluations demonstrate that Func- GenFoil improves upon state-of-the-art methods in airfoil generation, achieving a relative 74.4% reduction in label error and a 23.2% % increase in diversity on the AF-200K dataset. Our results highlight the advantages of function-syy spaccn modeling for aerodynamic shape optimization, offering a powerful and flexible framework for high-fidelity airfoil design.
Introduction
The airfoil inverse design problem serves central aspect of aircraft manufacturing. Traditionally, given geometric requirements, engineers first select the most similar airfoils from well-known airfoil datasets (e.g.. NACA (8) and rely on trial-and-error strategy (5I). Considering the mission of the aircraft, an initial airfoil design that meets these conditions is preliminarily created. Then, through iterative rounds of physical analyses, such as aerodynamic and mechanical evaluations, the airfoil is optimized to achieve improved performance until the specified requirements are met. In practice, such direct design procedures are highly inefficient and time-com consuminn, often taking months. To minimize development and design time, as well as associated costs, automated design methods have been introduced as efficient alternatives in aireraft manufacturing engineering. In particular, machine learning-basse -based design and optimization techniques have gained significant attention. However, before applying these algorithms to airfoil design, is crucial to determine appropriate methods for representing airfoils within these algorithms.
Existing methods for airfoil representation can generally be divided into two categories: parametric- model-based approaches [61] and discrete-poimt-based methods [37 49). First, parametric-moded based methods predefine function families, e.g., Bézier curves 77]. Hicks-Henne curves, and NURBS,
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

---

Control points
Latent function
(a) Airfoil as parametric curves
(b) Airfoil as discrete points
(c) Airfoil as function curves
Figure I: The conceptual difference between FuncGenFoil and previous airfoil representation methods. In many previous approaches, airfoils are represented either as parametric models, as shown (a), or as discrete point-based models, as shown in (b). In contrast, (c) illustrates FuncGenFoills approach, where an airfoil is treated as continuous function mapped from a latent function, enabling generative model in function space.
and leverage mathematical optimization techniques generative models to determine the parameters of these functions for generating a new airfoil. These methods rigorously preserve key geometric properties of the defined function families, e.g., higher-order smoothness. Furthermore, such func- tional representations of airfoils allow arbitrary sampling of control points in real manufacturing, given engineering precision constraints. Despite these benefits, suffer from a significantly reduced design space; i.e., selecting a specific function family excludes other possible shapes, thus limiting the upper bound of airfoil design algorithms. Second, discrete- point-based methods directly generate multiple points to represent airfoil shapes. These methods maximize the airfoil design space but cannot inherently maintain certain important mathematical properties, e.g., continuity. Furthermore, they cannot directly generate control points at arbitrary resolutions, the number of generated points is typically fixed for each model after training.
seaemnttrremodel-bsseeee methods
To address the trade-offs between these two mainstream approaches, we ask: Can we design an algorithm that leverages the advantages of both?
In this paper, we address this question by proposing FuneGenFoil, a novel function dunction-space generative model for airfoil representation (see Figure D. Unlike previous data-driven generative methods in finitedimmensional spaces, e.g., <VAE 25] cGAN 41). difffusion models [19] or flow matching models (35). which directly generate discrete point outputs, we leverage recent advances in diffusion and flow models defined in function space [
23
32
33
42 52] produce infinite-die diment--immnniinall
outputs as general continuous functions. Simultaneously. our approach models airfoil geometry using Neural Operator architectures [1 2 27 31, enabling the generation of diverse airfoils beyond the design spaces of predefined geometric function families, thanks to its general operator approximation capability. Our method combines the advantages of both parametric-mooel--eee based approaches. Due its intrinsic representation in function space, the generated airfoils continuous, and both training and sampling can be performed at arbitrary resolutions, facilitating downstream optimization and manufacturing processes.
and discrete-poing
Specifically, we use the flow matching framework (35), an improved alternative to diffusion models, and FNO (11, a resolution-free free neural operator, as the backbone of our generative model to design FuncGenFoil. During training, we perturb the airfoil into noise function through straight flows and let neural operators learn the airfoil's deconstructed velocity via flow matching. During inference, we reconstruct the airfoil by reversing flow direction starting from Gaussian process. Beyond generation, our method also supports airfoil editing, generating new airfoils from an original airfoil by conforming to certain geometric constraints. This can be accomplished by a few steps of fine-tuning with additional data other than the original airfoil. Constraints can be provided in various forms, such as specifying contour points that the airfoil must pass through or setting geometric constraint parameters, such as thickness.
In summary, our main contributions are threefold: (1) We propose generating airfoil shapes in function space to achieve important properties for aircraft engineering, i.e., arbitrary- control point sampling and maximal design space; (2) We design FuncGenFoil, the first controllable airfoil generative model in function space, which effectively incorporates neural operator architectures into the generative modeling framework; and (3) We further enhance FuncGenFoil with airfoil editing capabilities through minimal adaptations. Experimental results indicate that our proposed method

---

achieves state- of-the-art airfoil generation quality, reducing label error by 74.4% and increasing diversity by 23.2% on the AF-200K dataset, as validated through aerodynamic simulation analysis. In addition, our method is the first to successfully perform airfoil editing by fixing dragging points at arbitrary positions, achieving nearly zero MSE (less than 10 10-7).
2 Related Works
Generative Models. Generative models based on score matching 20 55
and flow matching [4 tee-the-rtt art results in areas such
[6|57] have significantly advanced machine learning, achieving state-of ofate--f-thh-ttte
as image generation [47], text generation [T6], and video generation 22 43 However, most of
these models operate in finit-diimensinnll spaces and rely on fixed discretizations of the data. Such formulations hinder transferability across discretizations and neglect function-level constraints (e.g.. continuity, smoothness), motivating the need for generative modeling in function space.
Neural Architectures for Function Modeling. Designing neural architectures to handle function spaces remains a major research challenge. Standard networks typically assume fixed-size inputs, making them unsuitable for arbitrary resolutions. Implicit neural representations, such SIREN (54) harness random Fourier features 44 to represent continuous and differentiable objects through position embeddings. Similarly, NeRF [40] treats input coordinates as continuous variables, offering flexible resolution for function outputs. Neural operators [ 2 27 B1] and Galerkin transformers S further generalize neural architectures to process sets of points as functional inputs, enabling function- space learning. Recently, neural operator with localized integral and differential kernels has been introduced to improve its capability in capturing local features [38]. To handle complex and variable geometries, the point cloud neural operator has been proposed, which adaptively processes point cloud datasets 633
Generative Models in Function Space. Early Neural Processes 14 15
drew upon Gaussian
processes [46), and later methods such as GASP [1, Functa [IO], and GANO 45 treat data as function evaluations to enable discretization eecretization-indeeeddeee learning. anergy-bbseee and diffusion models [13
along with flow-based approaches like FFM 24 and OpFlow (52). further extend these ideas. Ultimately, developing comprehensive generative models in function space requires defining suitable stochastic processes, score operators, and consistent neural mappings, along with specialized training methods for numerical stability challenges that remain largely unresolved.
32
33 42
Airfoil Design and Optimization. Airfoil design is essential for aircraft and wind turbines. Geometric parameterization supports efficient shape modeling and optimization. Techniques such as Free-Form Deformation (FFD) 12 and NURBS 488 are widely used in CAD/CAE tools (9] [8] but their independent control points can cause instability, high dimensionality, and suboptimal solutions. Modal methods, including Proper Orthogonal Decomposition [3) global modal 4. and compact modal (30) approaches, reduce dimensionality by encoding global features, though they struggle with large or detailed shape changes. Shass-SSapp// Transformation (CST) 28 offers interpretable and differentiable shapes but lacks flexibility for large deformations and is sensitive to parameter choices. Therefore, high-degree-offfrieg ofwedffofffoomo--o representation is needed for robust, nonlinear shape modeling. Unlike general computer vision tasks [26] [6], engineering AI models are domain- domain-specific and often limited by sparse data. Generative models are advantageous here, as they can exploit unlabeled or untested samples. In airfoil design, VAEs, GANs, and diffusion models 7 29 37
58, 61 62), along
with CFD-based mesh representations [59] 60), have been used to map latent spaces to airfoil shapes.
However, most existing methods model discrete airfoil points, limiting flexibility for downstream applications.
FuilGenFoil: Function-Space Generative Model for Airfoils
In contrast to existing airfoil generation methods, FuncGenFoil constructed as function-space generative model capable of producing airfoil geometries as continuous functions rather than discrete points, thereby leveraging the advantages of both aarametric-oodel-eee and discrete-point-bane boset-base methods. In this section, detail the processes for airfoil generation and editing tasks, respectively.
3

---

Integral Operator k40.,,,,9"0000000
Condition & Time Embedding (Optional)
Mc M
Differential Operator
W (*o(Fu,()))
Linear Transform
Whu,6>
Neural Operalor Layer
Condition variables Time
Neural Operator
Airfoil 4
+ t)m
Concatenation
Airfoil & Condition Embedding
Up Linear Projection
(v.
Down Linear Projection
Gaussian Process Ug
U2
Generation Velocity Vg
Gaussian Process Mo
Airfoil U1
Figure 2: Top: Overview of FuncGenFoil's neural network and training scheme. The model Fourier Neural Operator designed for point cloud data, although any other neural operator capable of general-purpose function-space approximation may he used. The model takes as input function 24 (point cloud data at an arbitrary resolution d), optional design condition variables c, and the generation time t.. It then processes this input function consistently and outputs the current velocity operator es (u,, t) for calculating the flow matching loss. Bottom: Inference with FuncGenFoil is conducted by first sampling random latent function from Gaussian process and then reconstructing the airfoil by solving an ODE.
3.1 Airfoil Generation
Airfoil Parametrization. Since airfoil curve (z, y) has circular topology. introduce the
variable a E [0, 1] as the domain of the function, denoting y(ar) - f(x(a) and x(a) +0m(2m))+1
as shown in Figure 7 The entire FuncGenFoil framework is essentially an ordinary differential
equation (ODE))based generative model that produces airfoil curve functions by solving an ODE
along continuous time t [0, 1] as u0 + Simo df, where 41 = starting from some latent function 24.
The velocity operator e gradually transforms a latent function ug sampled from stochastic process P into airfoil function in belonging the target airfoil distribution Q. The velocity operator is a key component, as it must handle function inputs consisting point sets with arbitrary resolution or positions. Specifically, the operator takes input the partially generated airfoil W4 = [on)) u4 (an),, ()(())))))))))))))))())) (o2)....] and outputs a velocity function corresponding to these positions, (on), Hte (01). (022) We realize this by establishing parameterized neural operator vo with model weights 0, constructed as a Neural Operator model capable of processing function-space data arbitrary resolutions in a single unified model, shown in Figure 2 Specifically, FuncGenFoil acts as conditional continuous-time generative model, where the velocity operator ee (we, c, t) takes the noised airfoil Ue, optional conditioning variables c, and generation timestamp & inputs, and consistently outputs the deformation as function. We train ug(u,, c, using Operator Flow Matching (53).
Velocity Operator.
Training. We train uy under the simplest denoising training process. Given an airfoil geometry #1. compute its corresponding noised sample at time as follows:
uelua, = t u1 + (1 t) x tig
and then the ground-truth velocity vi (ux) lng.us given uo and μn can be computed as:
du
vy (e(u)) luo.
dt 21 ug-
(2)
We train up by matching the velocity operator using Flow Matching loss:
LFM = E,-,o,1,, [llve(u:) vo(u,, c, 1)
(3)
For the remainder of this paper, omit the explicit dependence for clarity and convenience.

---

t=
U, +
de (i
Latent function Ag
Uo
Neural Operator 2
Pp EMAD (uf, u²1)) u'1 ag + (o
Neural Operator
Mr
Origin Airfoil M4
U, +
Edited Airfoil ad.
Airfoil editing by FuncGenFoil. Given original airfoil ut. editing requirement ò and target airfoil uf. We first infer its latent function reversely, and make it learnable as. Then sample new airfoil ui and conduct regression in function space via maximum posteriori estimation LMAP. After few iterations of fine-tuning, we can generate edited airfoil uf with high accuracy.
Figure 3:
which according to Conditional Matching Theorem, has same gradients with Conditional Flow Matching loss:
LORM = Fe-10,11,00.u (1e) (we)) -
C,1)1
(4)
VoLru = (0).
(5)
See proofs in Chapter of Flow Matching Guide and Code 36
Inference. Given a trained velocity operator vp. the inference process, i.e., airfoil generation process, equal to deriving airfoil geometry at time í = 1, denoted as u1, based latent coding sampled from the stochastic process P. assumed as a Gaussian Process GP(O, K) in this work, where K is covariance kernel function. Therefore, U1 could be derived by solving the generation ODE numerically (e.g., using the Euler method) as follows:
U1 = ug +
Vo c, dt.
(6)
==
Detailed implementations for training velocity operator and model inference are shown in Appendix Algorithm and Algorithm 2
3.2 Airfoil Editing
The airfoil editing task enables the user to modify parts of the geometry of a given airfoil ui- effectively generating a new airfoil geometry un while preserving the user-edited sections, denoted as wf. We achieve this by finetuning the pretrained model via maximum a posteriori (MAP) estimation, maxp(u; wf) where p(uj u is a probabilistic model that constrains the optimized airfoil ui fulfilling the editing requirements and following the generation prior.
To achieve the constraint probability model, we disentangled with Bayes' Rule:
max maxp uf) = p(uî u') p(u)) maxlog maxlogp(uqluu) + logp(u() - log p(uf), p(u4)
(7)
u/,
where p(uf wi) is a Gaussian measure, so its log term becomes a Mean Square Error (MSE) between uf and Uy constraining the user edited parts in i keeping consistent with uf. We can sample in through solving the velocity operator using Neural ODE method and calculate log p(u)) at the same time using FFJORD method [17] with Hutchinson trace estimator [22]. p(u)) is the prior supported by the trained generative model. p(u{) the marginal likelihood, which does not depend on ui The final optimization target could be written as:
(up o/ logp(u)),
max 20T
(8)
EEA
where a is noise scale for editing, A denotes point indices in edited part of
5

---

For more realistic generation results, the optimization does not directly adjust wi; instead, we optimize U1 indirectly by line-tuning the entire generative model for few iterations. The fine-tuning process is illustrated in Figure 3 while its details are provided Alg. in the Appendix. Specifically, we first initialize nn resample data of W1. by extracting its latent code, denoted as ag., via the inverse of our generative model and re-generating " from this code. Then treat Equation 8 as the loss function to train 0 of the velocity operator and ag simultancously. After the model training, the new U1 is the new edited generation results.
Experiments
4.1 Experimental Settings
Tasks. We evaluate two tasks within the airfoil inverse design problem: conditional airfoil generation and freestyle airfoil editing. In the conditional generation task, the model is provided with set of 11 geometric parameters describing the airfoil geometry. Detailed parameter definitions are provided in Table in the Appendix. The model must generate airfoils that satisfy these geometric constraints. In the freestyle editing task, the model receives an original airfoil and target modification, such as adjusting the position of specific point on the airfoil. The selected point can be located anywhere on the airfoil surface. The model must generate an airfoil reflecting the specified modification.
Metries. We adopt the metrics introduced in AFBench 37 to evaluate the generated airfoils:
Label Error measures the difference between the PARSEC parameters of the generated or edited airfoil and the intended target parameters, calculated as の; = 1px - -psl. where d; is the label error for the i-th parameter, P. the i-th geometric parameter of the generated airfoil, and ps is the corresponding target parameter. Smaller values indicate better alignment with target parameters. To summarize all 11 label errors, we present both the arithmetic mean G, and the geometric mean og for absolute and relative average errors.
Diversity quantifies the variety among generated airfoils, calculated as D= E log det(Ls,), where n is the number of subsets, and Ls, is the similarity matrix of the i-th subset, computed based on Euclidean distances between airfoils within the subset. Higher values indicate greater diversity among generated airfoils.
Smoothness measures the geometric smoothness of generated airfoils, calculated as:
N
M
Distance(P,, Pn-1
where Pa is the n-th point, and - Pa++ is the line segment connecting its adjacent points. The function Distance (Pn, Pa-1 P computes the perpendicular distance from Pn. to this line segment. Smaller values indicate better geometric quality.
Datasets, To benchmark our method, we conduct experiments on three datasets: UIUC [50]. Supercritical Airfoil (Super), and AF-200K. UIUC contains 1,600 designed airfoil geometries. Super focuses on supercritical airfoils and includes approximately 20,000 airfoil samples. AF-200K consists of 200,000 highly diversified airfoil samples.
Baselines. We compare FuncGenFoil with baseline models proposed in AFBench (37, specifically the conditional VAE (CVAE), conditional GAN (CGAN), the modified VAE with PARSEC parameters and control keypoints PP-,, as well as PK-GAN, PKVAE-GAN, the U-Net-based PK-DIFF, and the transformer-based PK-DIT.
Implementation Details. On the AF-200K dataset, we trained for 2 million iterations with a batch size of 2,048 using 8 NVIDIA 4090 GPUs. On the Supercritical Airfoil and the UIUC dataset, we trained for I million iterations with a batch size of 1,024 on single NVIDIA 4090 GPUs. We use Gaussian processes with a Matérn kernel function (v 2.5, L = 0.03) as the prior. Other training hyperparameters are detailed in Appendix
4.2 Main Results
Conditional Airfoil Generation. shown in Table I FuneGenFoil outperforms the strongest baseline method (PK-DIT) across all metrics. For label error, FuncGenFoil achieves reductions in

---

Table 1: Quantitative comparison between FuncGenFoil and baseline methods across different datasets for the conditional generation task. Lahel error, diversity, and smoothness of generated airfoils are reported.
Label Error 4
(10-3)
Method
Dataset
D t
M (10 2)
51
02
da
0g
Cb.
de
0r
0x
636
0:1
6,
CVAE
AF-200K
72.9
52.5
15900
29000
19.1
15.3
46
4131
CGAN
AF-200K
107
85.0
54.4
23200
143
137
59600
25.3
22.3
53
129
7596
217.6
-120.5
7.31
PK-VAE
AF-200K
63.0
47.9
31.3
8620
66
64
17100
13.5
9.3
33
78
2375
106.0
-150.1
5.93
PK-GAN
AF-200K
81.8
63.0
47.0
21030
120
117
32470
22.5
19.6
50
122
4922
179.5
-112.3
3.98
PKVAE-GAN
AF-200K
56.8
31.7
31.0
3650
46
43
12000
9.1
5.1
28
63
1633
77.6
-129.6
2.89
PK-DIT
AF-200K
11.2
32.3
15.4
1050
13
11.5
9790
0.5
0.5
23
24
997
23.5
-93.2
1.04
FunoGenFoil
AF-200K
1.84
17.4
721
47.7
1676
0.45
0.65
160
174
255
14.9
-71.6
PK- VAE
UIUC
80.7
20.9
12.2
12843
36.9
14.0
37263
1.7
1.9
94.8
109.9
4589
69.1
-93.5
7.29
UICC
51.4
11830
25700
11.9
3456
-141.7
FunoGenFoil
UIUC
11.8
698
23.5
12339
0.27
0.44
110.5
115.0
1210
-91.1
VAE
Super
17.5
1735.6
12.1
8131
3.5
1.4
98.3
918
PK-DIT
Super
52.0
35.0
24.0
3010
29
33.2
10500
8.3
2.6
27
1250
58.1
123.44
1.97
FuaoGenFod
Super
0.71
8.23
0.13
201.3
4.72
0.12
174.2
0.09
0.14
34.2
36.7
41,9
3,08
-103,9
1.01
Table 2: Quantitative evaluation of the FuncGenFoil model across different sampling super-resolutions for the conditional generation task. The training resolution is 257.
Label Error 1 (10 $)
Dataset
Resolution
D
M (10-2)
8g
257
0.71
8.23
0.13
201.
4.72
0.12
174.2
000
0.14
34.2
36.7
41.
108
-1039
1.01
Super
513
0.71
8.19
0.12
200.8
4.81
0.12
174.7
0.09
0.15
48.2
47.8
44.1
3.26
-97.5
0.52
1025
0.71
8.29
0.12
202.9
4.3
0.12
1754
0.09
0.15
61.4
57.8
46.5
3.4
.9111
0.37
arithmetic mean label error of 74.4% on AF-200K, 65.0 % on UIUC, and 96.6% on Super. More importantly, it also reduces the geometric mean label error by 36.6% AF-200K, 88.4% on UIUC, and 94.7% % on Super. This substantial decrease in label error underscores the effectiveness of our approach for generating airfoils that more precisely adhere to target geometric parameters. In terms of diversity, FuncGenFoil demonstrates notable improvements, surpassing the best baseline methods by 23.2%. 35.7%. %, and 15.8% on AF-200K. UIUC, and Super, respectively. This highlights the model's superior capability in capturing and generating a broader spectrum of valid airfoil designs. Additionally, the generated airfoils exhibit enhanced surface smoothness, as evidenced by reductions of 4.70 and 0.96 in smoothness values (10-3) on the UIUC and Super datasets, respectively. This improvement is particularly crucial for aerodynamic performance, as smoother airfoil surfaces contribute to reduced drag and improved flow characteristics. The limited coverage of the training dataset constrains the model's effectiveness to a specific operational range, which is detailed in Table 9 A performance comparison between FuncGenFor and classical methods is provided in Appendix D.I.
Freestyle Airfoil Editing. Table 3 shows av- erage performance over 300 editing cases us- ing FuncGenFoil. In each case, the model ad- justs 2 to 4 randomly selected positions on the airfoil surface to target locations over 10 fine- tuning steps, with editing scales ranging from I x 10-4 to 3.2 10-3, Results demonstrate that FuncGenFoil achieves accurate airfoil edit- ing with minimal errors (less than 2.75 X 10-7 MSE) and high surface smoothness (less than 1.16 x 10- smoothness value) after only few fine-tuning steps the edit scale less than 4 x 10 The edit error increases at an accel- erating rate with larger edits but remains relatively low overall. Figure illustrates example editing requirements and corresponding generated airfoils during the fine-tuning stage, demonstrating that our model effectively completes freestyle editing tasks by generating accurate airfoils according to user-specified edits.
Table 3: Quantitative evaluation of the airfoil editing
lask across different editing scales. Mean squared error (MSE) between generated and target airfoils and smooth- ness of generated airfoils are reported.
Dataset
Edit Scale
MSE (10-7)
M (10 (10-2)
0.0001
2.41
1.66
0.0002
2.45
1.15
0.0004
2.75
1.15
Super
0.0008
4,32
1.26
1.35
0.0032
61.7
1.49
Any- Resolution Airfoil Generation. One of the key advantages of FuncGenFoil its ability to generate airfoils at any resolution while maintaining high generation quality. This is achieved
7

---

(A)
(R)
Edited
Figure 4: Examples of eight different FuncGenFoil instances performing airfoil editing 20 training iterations. The orange points and sections represent the editing requirements as constraints; the gray region the original airfoil, while the blue region shows the generated edited airfoil. The results demonstrate that the generated airfoil quickly adapts to the editing requirements within few iterations, achieving natural and smooth function regression. The editing scheme can be completely customized according the user's proference.
Wings Dataset
Generated wings
Lin--o-Dng Ralio
Libl-toDDag Rabo
Figure 5: Left: Comparison of the lift-to-drag ratio histograms for CRM wings in the original dataset and the samples generated by our FuncGenFoil model. Right: Visualization of the pressure coefficient for the generated CRM wings, obtained through acrodynamic simulation.
by representing airfoils as functions and learning risolution-independern function transformations, enabling flexible and consistent airfoil generation across different scales. To evaluate high-resolution generation, we use the model trained on the Supercritical Airfoil dataset at a resolution of 257 and sample new airfoils at resolutions 513 and 1025. We then assess these higher-resolution airfoils using the same metrics as in the conditional generation task. The results are presented in Table 2 We observe consistent generation quality at 2x and even 4x the training resolution, with maximum increase of 4.6x 10 in average label error, decrease of 12.8 in diversity, and an increase of 0.64 10 in smoothness value.
Aerodynamic Simulation. To further assess the physical properties of the generated airfoils and validate the effectiveness of method, analyze their aerodynamic performance using the NASA Common Research Model (CRM) and perform Geynolds-Averager Navier-Stokes (RANS) computational fluid dynamics (CFD) simulations on the generated samples. The CRM dataset contains 135,000 CRM wing geometries along with their corresponding aerodynamic performance, computed using the RANS CFD solver ADflow 33) We pretrain the FuncGenFoil model on all 135,000 CRM wing geometries and generate 500 new CRM wing geometries for RANS CFD evaluation. We analyze the lift-to-drag ratio (L/ L/D) of these newly generated geometries and compare them with the original dataset, as shown in Figure Our results indicate that the L/D distribution of the generated samples closely aligns with that the original dataset, with the highest density occurring around L/D. 19, demonstrating the model's ability to learn and generate physically plausible wing geometries. Additionally, we visualize the coefficient of pressure contours for selected CFD cases in Figure 5] These visualizations confirm that Fuil can generate airfoils with diverse acrodynamic performance characteristics.
4.3 Ablation Study
Kernels of Gaussian Process. The type of Gaussian process kernel used as prior in the model influences the function space into which the data diffused. We perform ablation experiments across several kernel types, including the white noise kernel, the Matérn kernel with smoothness parameter E [1.5, 2.5, 3.5], and the radial basis function (RBF) kernel. We measure the geometric mean o,

---

3.50
White Suiee
3.50
2 0.01
4.50
25
4.22
Maisa 1.5
0.02
4,00
32,9
3,30
Massay 255
3.30
Meitrn 355 RBF
3.30
3.13
3,00
128,
3.08
2.96
3.10
3.10
2.
2.93
2. 1.98
2.96
2.50
2.10
2:20
2.90
2.90
(A) 8g under diverse smoothness
(B) &g under diverse characteristic length
(C) 0. under diverse Fourier mode
Figure 6: (A): Comparison of the geometric mean error og using kernels of different smoothness hut same characteristic length | = 0.03. (B); Comparison of the geometric mean error o, using Matérn kemel v = 3.5 with different characteristic length I. (C): Comparison of the geometric emror og using different Fourier mode FNO using Matérn kemel same 0.03.
Table 5: Quantitative experiments on the impact of ODE solver and time step on sampling quality.
Label Error L (10
+)
ODE Solver
Time Steps
D
M + (10-2)
.7g
4
dg
47
(7g
100
711
0
10
0.71
8.23
0.13
201.3
4.72
0.12
174.2
0.09
0.14
34.2
36.7
41.9
3.08
103.99
1.01
Baler
50
0.62
3.61
0.07
92.5
2,08
0.06
96.8
0.06
0.07
14.5
15.7
20.6
1.60
107.6
0.99
100
0.67
3.13
0.09
83.9
1.84
0.08
90.6
0.07
0.06
12.3
13.1
18.7
1.55
-106.4
0.99
10
0.68
5:29
0.08
161.3
3.12
0.08
178.9
0.06
0.09
21.4
22.4
35.8
2.17
-105.1
1.00
Midpoint
50
0.75
3.82
0.12
137.9
2.34
0.10
153.4
0.08
0.06
14.1
14.9
29.8
1.97
-103,3
0.99
100
0.82
3.94
0.14
125.7
2.22
0.13
137.4
0.08
0.07
14.4
14.2
27.2
2.06
100.6
0.99
10
15.9
46.3
1,21
4406
25.7
1.24
4795
1.17
1.68
333
396
911
36.6
-67.5
3,05
RK4
50
0.92
7.63
0.24
482.7
5.55
0,22
5118
0.16
0.19
39.3
41.3
99.1
4.68
-92.9
1,07
100
0.82
5.65
0.18
160.6
3.21
0.17
170.8
0.10
0.12
22.0
22.8
35.1
2.78
96.9
1,00
under different smoothness, characteristic length, and Fourier mode settings, as shown in Figure We observe that FunGenfoil benefits from employing smoother kernels, which enhance both the smoothness and aceuracy of the generated airfoils. This effect arises because the parameter primarily controls the smoothness of the function space, and the RBF kernel corresponds to the case ~ 00. Additionally, we find that there exists an optimal characteristic length / 0.03 for the kernel. In our experiments, FunGenfoil does not exhibit improved performance with an increased number of Fourier modes. This phenomenon may be due the predominantly low- frefeeuuencc characteristics of supercritical airfoils, such that adding higher modes in the FNO layer introduces unnecessary model complexity. More detailed experimental results are provided in Table IT in the Appendix.
Editing with Prior via ODE Inversion. Table we compare MSE of constraint condition calculations in the airfoil editing task using two initialization schemes: (I) using the latent variables obtained through ODE inversion of the original airfoil as the prior, and (2) using zero prior uo = O without ODE inversion. Both schemes are fine-tuned with the same number of iterations. The ODE inversion-based prior significantly reduces the MSE compared to the zero prior, with maximum reduction of 84.85. This underscores the importance of incorporat- ing original airfoil information during the initialization process.
Table Ablation study on effect dif- ferent latent variable initialization methods for the airfoil editing task.
MSEL(10-T)
Edit scale
ODE in..
w/o ODE
0.0001
2.42 (183.88)
86,3
0.0002
2.46 (184.14)
86.6
0.0004
2.75 (184.85
87.6
ODE Numerical Method. Model inference involves solving an ordinary differential equation (ODE), and the choice of numerical integration scheme affects its performance. Using the same velocity operator Ve trained on the supercritical airfoil dataset and sampling from Gaussian prior with Matérn kernel (v 2.5. I = 0.03), we evaluate FuncGenFoil across various time step sizes and ODE solvers (Euler, midpoint, and fourth-order Runge-Kute Kutta (RK4) methods). Results Table show that FuncGenFoil benefits from larger time steps using Fuler method. However, RK4 may degrade performance at excessively small step sizes, potentially because the model enters regions where the velocity operator is inadequately trained.

---

Conclusion and Limitations
In this work, we tackle the critical challenge of generating high-fidelity airfoil geometries that effec- tively balance expressiveness, smoothness, and resolution flexibility. We introduce FuncGenFoil, function-space generative model leveraging neural operators and flow matching, which represents airfoils as continuous, smooth geometries without resolution constraints while preserving the pressiveness of outperforms state-of offte-of-thhharrr art techniques in terms of label error, diversity, and smoothness, highlighting its potential for validated through aerodynamic simulations. This work paves the way for more efficient, scalable, and versatile airfoil generation, with significant applications in aerodynamic shape optimization for aireraft manufacturing.
methods. Comprehensive experimental results show that FuncGenFoil
airfoil design. Furthermore, the generated wing geometries have been
As for limitations, although we have taken a step toward general object shape modeling function space, the focus of this paper is primarily on airfoils or aircraft wings, which have relatively simple geometries and smooth surfaces yet are highly significant for aerodynamic performance. Thus, our current approach has limitations regarding the scope of geometry. In future studies, if we aim to extend our method to modeling entire aircraft or dealing with objects of general shapes, substantial theoretical work and experimental analysis will be required, especially when a suitable coordinate system is unavailable for describing complex, irregular, or non-smooth geometries. This direction is part of our ongoing efforts.
Acknowledgements
This work was supported by the Shanghai Artificial Intelligence Laboratory, the JC STEM Lab of AI for Science and Engineering, funded by The Hong Kong Jockey Club Charities Trust, the Research Grants Council of Hong Kong (Project No. CUHK14213224) the Natural Science Foundation of China (No. U23A2069), the AI for Science Seed Program of Shanghai Jiao Tong University (Project No. R2SAI44-HYY22 and the AI for Science Program, Shanghai Municipal Commission of Economy and Informatization (No.
Funding:
Support and Collaboration: We thank the team at Shanghai Aircraft Design and Research Institute and Shanghai Jiao Tong University for their valuable discussions and continued support.
References
[1] Anima Anandkumar, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Nikola Kovachki, Zongyi Li, Burigede Liu, and Andrew Stuart. Neural operator: Graph kernel network for partial differential equations. In ICLR 2020 Workshop on Integration of Deep Neural Models and Differemtial Equations, 2019.
[2] Kamyar Azizzadenesheli, Nikola Kovachki, Zongyi Li, Miguel Stu-Schiafiini, Jean Kossaifi, and Anima Anandkumar. Neural operators for accelerating scientific simulations and design. April 2024.
[3] Gal Berkooz, Philip Holmes, and John L. Lumley. The proper orthogonal decomposition the analysis of turbulent flows. Annual Review of Fluid Mechanics, 1993.
[4] Olivier Brüls, Pierre Duysinx, and Jean-Claude Golinval. The global modal parameterization for non-linear model-order reduction in flexible multibody dynamics. International journal for numerical methods in engineering. 2007.
[5] Shuhao Cao. Choose transformer: Fourier or galerkin. NIPS, 34:24924-24940, 2021.
[6] Ricky T. Q. Chen, Yulia Rubanova, Jesse Bettencourt, and David K Duvenaud. Neural ordinary differential equations. In Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, editors, NIPS, volume 31. Curran Associates, Inc., 2018.
[7] Wei Chen and Mark Fuge. Béziergan: Automatic generation of smooth curves from interpretable www-iimensionll parameters, 2021.
10

---

[8] Russell M Cummings, William H Mason, Scott A Morton, and David R McDaniel. Applied com- putational aerodynamics: A modern engineering approach, volume 53. Cambridge University Press, 2015.
[9] III Dannenhoffer, John F. An overview of the engineering sketch pad. In AIAA SciTech 2024 Forum, Orlando, FI., January 2024.
[10] Emilien Dupont, Hyunjik Kim, M. Ali Eslami, Danilo Jimenez Rezende, and Dan Rosenbaum. From data to functa: Your data point is function and you can treat it like one. In Kamalika Chaudhuri, Stefanie Jegelka, Le Song. Csaba Szepesvari, Gang Niu, and Sivan Sabato, editors, ICML, Proceedings of Machine Learning Research. PMLR, 2022.
[11] Emilien Dupont, Yee Whye Teh, and Arnaud Doucet. Generative models as distributions of functions. In Gustau Camps- Calps-Valls, Francisco J. R. Ruiz, and Isabel Valera, editors, AISTA7S. PMLR, 2022.
[12] Gerald E. Farin. Curves and Surfaces for CAGD: A Practical Guide. Morgan Kaufmann, 2002.
[13] Giulio Franzese, Giulio Corallo, Simone Rossi, Markus Heinonen, Maurizio Filippone, and Pietro Michiardi. Continuous-time functional diffusion processes. NeurIPS, 36, 2024.
[14] Marta Garnelo, Dan Rosenbaum, Christopher Maddison, Tiago Ramalho, David Saxton, Murray Shanahan, Yee Whye Teh, Danilo Rezende, and S. M. Ali Eslami. Conditional neural processes, In Jennifer Dy and Andreas Krause, editors, ICML, Proceedings of Machine Learning Research. PMLR, 2018
[15] Marta Garnelo, Jonathan Schwarz, Dan Rosenbaum, Fabio Viola, Danilo J. Rezende, S. M. Ali Eslami, and Yee Whye Teh. Neural processes, 2018.
[16] Itai Gat, Tal Remez, Neta Shaul, Felix Kreuk, Ricky T. Q. Chen, Gabriel Synnaeve, Yossi Adi, and Yaron Lipman. Discrete flow matching. In NeurIPS, 2024.
[17] Will Grathwohl, Ricky T. Q. Chen, Jesse Bettencourt, and David Duvenaud. Scalable reversible generative models with free-form continuous dynamics. In ICML, 2019.
[18] Andrew S. Hahn. Vehicle sketch pad: A parametric geometry modeler for conceptual aircraft design. In 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, 2010.
[19] Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. In H Larochelle, M Ranzato, R Hadsell, M Balcan, and H Lin, editors, NIPS, volume 33, pages 6840 6851. Curran Associates, Inc., 2020.
[20] Jonathan Ho. Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. NIPS, 33:6840-6851, 2020.
[21] Jonathan Ho, Tim Salimans, Alexey Gritsenko, William Chan, Mohammad Norouzi, and David Fleet. Video diffusion models. In S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh, editors, NeurIPS, volume 35, pages 8633-8646. Curran Associates, Inc., 2022.
[22] M.F. Hutchinson. A stochastic estimator of the trace of the influence matrix for laplacian smoothing splines. Communications Statistics Simulation and Computation, 1990.
[23] Gavin Kerrigan, Justin Ley, and Padhraic Smyth. Diffusion generative models in infinite dimensions. Francisco Ruiz. Jennifer Dy, and Jan- willee van de Meent, editors, AISTATS, volume 206 of Proceedings of Machine Learning Research, pages 9538-9563. PMLR, 25-27 Apr 2023.
[24] Gavin Kerrigan, Giosue Migliorini, and Padhraic Smyth. Functional flow matching. In Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, 2024.
[25] Diederik P Kingma. Auto-encoding variational bayes. arXiv preprint 1XXv:1312.6114 2013.

---

[26] Alexander Kirillov, Eric Mintun, Nikhila Ravi, Hanzi Mao, Chloe Rolland, Laura Gustafson, Tete Xiao, Spencer Whitehead, Alexander C. Berg, Wan- Yen Lo, Piotr Dollär, and Ross Girshick. Segment anything, 2023.
[27] Nikola Kovachki, Zongyi Li, Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Neural operator: learning maps between function spaces with applications to pdes. J. Mach. Learn. Res., 24(1), January 2023.
[28] B. M. Kulfan. Universal parametric geometry representation method. Journal of Aircraft, 2008.
[29] Jichao Li, Xiaosong Du, and Joaquim RRA Martins. Machine learning in aerodynamic shape optimization. Progress in Aerospace Sciences, 134:100849, 2022.
[30] Jichao Li and Mengqi Zhang. Adjoint-free aerodynamic shape optimization of the common research model wing. AIAA Journal, 2021.
[31] Zongyi Li, Nikola Borislavov Kovachki, Kamyar Azizzadenesheli, Burigede liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations. In ICLR, 2021.
[32] Jae Hyun Lim, Nikola B. Kovachki, Ricardo Baptista, Christopher Beckham, Kamyar Azizzade- nesheli, Jean Kossaifi, Vikram Voleti, Jiaming Song, Karsten Kreis, Jan Kautz, Christopher Pal, Arash Vahdat, and Anima Anandkumar. Score-based diffusion models in function space, 2025.
[33] Jen Ning Lim, Sebastian Vollmer, Lorenz Wolf, and Andrew Duncan. Energy- based models for functional data using path measure tilting. In AISTATS, 2023.
[34] Yaron Lipman, Ricky T. Q Chen, Heli Ben-Hamu, Maximilian Nickel, and Matthew Le. Flow matching for generative modeling. In ICML, 2023.
[35] Yaron Lipman, Ricky TQ Chen, Heli Ben-Hamu, Maximilian Nickel, and Matt Le. Flow matching for generative modeling. arXiv preprint arXiv:2210.02747, 2022.
[36] Yaron Lipman, Marton Havasi, Peter Holderrieth, Neta Shaul, Matt Le, Brian Karrer, Ricky
T. Q. Chen, David Lape Heli Ben-Hamu, and Itai Gat. Flow matching guide and code. arXiv preprint arXiv:2412.0622 2024.
[37] Jian Liu, Jianyu Wu, Hairun Xie, Guoqing Zhang, Jing Wang, Wei Liu, Wanli Ouyang, Junjun Jiang, Xiamming Liu, Shixiang Tang, et al. Afbench: A large-scale benchmark for airfoil design. In NewrIPS 2024 Dataset and Benchmark Track, 2024.
[38] Miguel Liu-Schiaffin, Julius Berner, Boris Bonev, Thorsten Kurth, Kamyar Azizzadenesheli, and Anima Anandkumar. Neural operators with localized integral and differential kernels. In ICLR 2024 Workshop A14DifferentiaEiiuattnns In Science, 2024.
[39] Charles A. Mader, Gaetan K. W. Kenway, Anil Yildirim, and Joaquim R. R. A. Martins. ADflow-an open-source computational fluid dynamics solver for aerodynamic and multidisci- plinary optimization. Journal of Aerospace Information Systems, 2020.
[40] Ben Mildenhall, Pratul P Srinivasan, Matthew Tancik, Jonathan T Barron, Ravi Ramamoor- thi, and Ren Ng. Nerf: Representing scenes as neural radiance fields for view synthesis. Communications of the ACM, 65(1):99-106. 2021.
[41] Mehdi Mirza. Conditional generative adversarial nets. arXiv preprint arXiv: 2014.
[42] Jakiw Pidstrigach, Youssef Marzouk, Sebastian Reich, and Sven Wang. Infinite-dimensionall diffusion models, 2023.
[43] Adam Polyak, Amit Zohar, and Andrew Brown et al. Movie gen: A cast of media foundation models, 2024.
[44] Ali Rahimi and Benjamin Recht. Random features for large-scale kernel machines. NIPS, 20. 2007.
12

---

[45] Md Ashiqur Rahman, Manuel A Florez, Anima Anandkumar, Zachary E Ross, and Kamyar Azizzadenesheli. Generative adversarial neural operators. TMLR, 2022.
[46] Carl Edward Rasmussen. Gaussian processes in machine learning. In Summer school on machine learning. Springer, 2003.
[47] Robin Rombach, Andreas Blatmannn, Dominik Lorenz, Patrick Esser, and Björ Ommer. High- resolution image synthesis with latent diffusion models. In CVPR, pages 10684-10695, 2022.
[48] I. J. Schoenberg. Spline functions and the problem of graduation. Proceedings of the National Academy of Sciences of the United States of America, 52(4):947-950, October 1964.
[49] Vinothkumar Sekar, Mengqi Zhang, Chang Shu, and Boo Cheong Khoo. Inverse design of airfoil using deep convolutional neural network. AIAA Journal, 2019.
[50] Michael S Selig. Uiuc airfoil database. 1996.
[51] Prashant Sharma, Bhupendra Gupta, Mukesh Pandey, Arun Kumar Sharma, and Raji Nareliya Mishra. Recent advancements in optimization methods for wind turbine airfoil design: A review. Materials Today: Proceedings, 2021.
[52] Yaozhong Shi, Angela F Gao, Zachary Ross, and Kamyar Azizzadenesheli. Universal functional regression with neural operator flows. In NeurlPS 2024 Workshop on Bayesian Decision-making and Uncertainty, 2024.
[53] Yaozhong Shi, Zachary E. Ross, Domniki Asimaki, and Kamyar Arizzadenesheli Stochastic process learning via operator flow matching, 2025.
[54] Vincent Sitzmann, Julien Martel, Alexander Bergman, David Lindell, and Gordon Wetzstein. Im- plicit neural representations with periodic activation functions. Advances in neural information processing systems, 33:7462-7473, 2020.
[55] Yang Song, Jascha Sohl-Dickstein. Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. In ICLR, 2021.
[56] Peng Su, Kun Wang, Xingyu Zeng, Shixiang Tang, Dapeng Chen, Di Qiu, and Xiaogang Wang. Adapting object detectors with conditional domain normalization. In Computer Vision-ECCV 2020: 16th European Conference, Glasgow, UK, August 23-28, 2020, Proceedings, Part 16, pages 403-419. Springer, 2020.
[57] Alexander Tong, Kilian FATRAS, Nikolay Malkin, Guillaume Huguet, Yanlei Zhang, Jar- rid Rector- Brooks, Guy Wolf, and Yoshua Bengio. Improving and generalizing flow-based generative models with minibatch optimal transport. TMLR, 2024. Expert Certification.
[58] Zhen Wei, Edouard R. Dufour, Colin Pelletier, Pascal Fua, and Michaël Bauerheim. Diffairfoil: An efficient novel airfoil sampler based on latent space diffusion model for aerodynamic shape optimization. In AIAA AVIATION FORUM AND ASCEND, 2024.
[59] Zhen Wei, Benoît Guillard, Pascal Fua, Vincent Chapin, and Michaël Bauerheim. Latent representation of computational fluid dynamics meshes and application to airfoil acrodynamics. AIAA Journal, 2023.
[60] Zhen Wei, Aobo Yang, Jichao Li, Michaël Bauerheim, Rhea P. Liem, and Pascal Fua. Deepgeo: Deep geometric mapping for automated and effective parameterization in aerodynamic shape optimization. In AIAA AVIATION FORUM AND ASCEND, 2024.
[61] Hairun Xie, Jing Wang, and Miao Zhang. Parametric generative schemes with geometric constraints for encoding and synthesizing airfoils. Engineering Applications of Artificial Intelligence, 2024.
[62] Aobo Yang, Jinouwen Zhang, Jichao Li, and Rhea Liem. Data-driven aerodynamic shape optimization and multi-fidelity design exploration using conditional diffusion-based geometry sampling method. In ICAS, 2024.
13

---

[63] Chenyu Zeng, Yanshu Zhang, Jiayi Zhou, Yuhan Wang, Zilin Wang, Yuhao Liu, Lei Wu, and Daniel Zhengyu Huang. Point cloud neural operator for parametric pdes on complex and variable geometries. arXiv preprint arXiv:2501.14475 2025.
14

---

NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification:
The main claims made in the abstract and introduction can accurately reflect
the paper's 's contributions and scope.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or
NA answer to this question will not be perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.
is fine to include aspirational goals as motivation as long as is clear that these goals are not attained by the paper.
2. Limitations
Question:
Does the paper discuss the limitations the work performed by the authors?
Answer: [Yes]
Justification:
We address the limitation of our work in the section of limitation of this paper.
Guidelines:
The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors are encouraged to create a separate "Limitations" section their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g.. independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect on the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution is low images are taken in low lighting. Or speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.
While the authors might fear that complete honesty about limitations might used by reviewers as grounds for rejection, worse outcome might be that reviewers discover limitations that aren't 't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
3. Theory assumptions and proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [Yes]
15

---

Justification:
Proof for theoretical result are all given or referenced properly in this paper.
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated or referenced the statement of any theorems.
The proofs can either appear the main paper or the supplemental material, but they appear in the supplemental material, the authors are encouraged to provide short proof sketch to provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental result reproducibility
Does the paper fully disclose all the information needed to reproduce the main ex- perimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
Question:
Answer: [Yes]
Justification: The paper fully disclose all the information needed reproduce the main experimental results.
Guidelines:
The answer NA means that the paper does include experiments. If the paper includes experiments, No answer this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
If the contribution is dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g.. in the case of large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.
While NeurIPS does not require releasing code, the conference does require all submis- sions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) If the contribution is primarily a new algorithm, the paper should make it clear how reproduce that algorithm. (b) If the contribution is primarily
model architecture, the paper should describe the architecture clearly and fully.
(c) If the contribution is a new model (e.g., a large language model), then there should either be way to access this model for reproducing the results or way to reproduce the model (e.g., with an open-source dataset instructions for how to construct the dataset).
(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome describe the particular way they provide for reproducibility, In the case of closed-source models, it may be that access to the model is limited in some way (e.g., registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
5. Open access to data and code
Question: Does the paper provide open access to the data and code, with sufficient instruc- tions to faithfully reproduce the main experimental results, as described in supplemental material?
16

---

Answer: [Yes]
Justification: All codes and data are provided and can be obtained via open access. We provide detailed instructions to faithfully reproduce the main experimental results. See readme file in supplemental material.
Guidelines:
The answer NA means that paper does not include experiments requiring code.
Please the NeurlPS code and data submission guidelines (https://nip)) cc/
wwt.mmissssooooooooooooooooooooi for more details.
While we encourage the release of code and data, we understand that this might not be possible, so "No" an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for new open-source benchmark).
The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https //nips cc/ httppppwllccccc wwtpc///uie/eeeeeeee Code Sww.issionmooooo......o
//nips.cc/public//
for more details.
The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.
The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. only subset of experiments are reproducible, they should state which ones are omitted from the script and why.
At submission time, preserve anonymity, the authors should release anonymized versions (if applicable).
Providing as much information as possible in supplemental material (appended to the paper) recommended, but including URLs to data and code permitted.
6. Experimental setting/details
Question: Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
Answer: [Yes]
Justification:
All the training and test details are given in the paper.
Guidelines:
The answer NA means that the paper does include experiments.
The experimental setting should be presented in the core the paper to a level of detail that is necessary appreciate the results and make sense of them.
The full details can be provided either with the code, appendix, or as supplemental material.
7. Experiment statistical significance
Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Answer: [No]
The metrics of all experiments in this paper is tiny and will not affect main results. Considering the size of the table, we decided not to provide error bars. All results can be reproduced running the evaluation code.
Justification:
Guidelines:
The answer NA means that the paper does include experiments.
The authors should answer "Yes" if the results are accompanied by error bars, confi- dence intervals, statistical significance tests, least for the experiments that support the main claims of the paper.
The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall with given experimental conditions).
17

---

The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)
The assumptions made should be given (e.g.. Normally distributed errors).
It should be clear whether the error bar is the standard deviation or the standard error of the mean.
is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma enror bar than state that they have 96% CI, if the hypothesis of Normality of errors is not verified.
For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out range (e.g. negative enror rates).
If error bars reported in tables plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables the text.
8. Experiments compute resources
For each experiment, does the paper provide sufficient information on the com- puter resources (type of compute workers, memory, time of execution) needed reproduce the experiments?
Question:
Answer: [Yes]
Justification: We provide detailed information on the computer resources we use in Experi- ment Settings section.
Guidelines:
The answer NA means that the paper does not include experiments.
The paper should indicate the type of compute workers CPU GPU, internal cluster, or cloud provider, including relevant memory and storage.
The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
9. Code of ethics
Question: Does the research conducted the paper conform, in every respect, with the NeurlPS Code of Ethics http:sss////ueupsss...tt http://wwwppppp..oooooocccccttta
Answer: [Yes]
Justification:
The research conducted in the paper conform Ethics Guide lines of NeurIPS.
Guidelines:
The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.
If the authors answer No, they should explain the special cireumstances that require deviation from the Code of Ethics.
The authors should make sure to preserve anonymity (e.g.. if there is special consid- eration due to laws or regulations their jurisdiction).
10. Broader impacts
Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
Answer: [Yes]
Justification: The paper discuss both potential positive societal impacts and negative societal impacts of the work performed in Appendix.
Guidelines:
The answer NA means that there is no societal impact of the work performed.
If the authors answer NA or No, they should explain why their work has no societal impact why the paper does not address societal impact.
18

---

Examples of negative societal impacts include potential malicious unintended uses (e.g., disinformation, generating fake profiles, surveillance), faireess considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology is being used intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional unintentional) misuse of the technology.
If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms monitor how system learns from feedback over time, improving the efficiency and accessibility of ML).
11. Safeguards
Does the paper deseribe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
Question:
Answer: [NA]
Justification: The paper poses no such risks.
Guidelines:
The answer NA means that the paper poses no such risks.
Released models that have high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make best faith effort.
12. Licenses for existing assets
Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
Question:
Answer: [Yes]
Justification: Code and data used in the paper are properly credited.
Guidelines:
The answer NA means that the paper does use existing assets.
The authors should cite the original paper that produced the code package dataset.
The authors should state which version of the asset used and, possible, include URL.
The name of the license (e.g., CC-BY 4.0) should be included for each asset.
For scraped data from a particular source (e.g., website). the copyright and terms of service of that source should be provided.
assets are released, the license, copyright information, and terms of use package should be provided. For popular datasets, beaasstts.ppprrsseee paperswws thoode. com datasstt has curated licenses for some datasets. Their licensing guide can help determine the license of dataset.
19

---

For existing datasets that are re-packaged, both the original license and the license of the derived asset (if has changed) should be provided.
If this information is not available online, the authors are encouraged to reach out to the asset's s creators.
13. New assets
Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer: [NA]
Justification:
The paper does not release new assets.
Guidelines:
The answer NA means that the paper does not release new assets,
Researchers should communicate the details of the dataset/code/moddl as part of their submissions via structured templates. This includes details about training, license, limitations, etc.
The paper should discuss whether and how consent was obtained from people whose asset used.
At submission time, remember to anonymize your assets (if applicable). You can either create anonymized URL or include anonymized zip file.
14. Crowdsourcing and research with human subjects
Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
Answer: [NA]
Justification: The paper does not involve crowdsourcing nor research with human subjects
Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
Including this information in the supplemental material is fine, but if the main contribu- tion of the paper involves human subjects, then as much detail as possible should be included in the main paper.
According the NeurIPS Code of Bthies, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
15. Institutional review board (IRB) approvals or equivalent for research with human subjects
Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or equivalent approval/review based the requirements of your country or institution) were obtained?
Answer: [NA]
Justification: The paper does not involve potential risks incurred by study participants.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurlPS Code of Ethics and the guidelines for their institution.
For initial submissions, not include any information that would break anonymity (if applicable), such the institution conducting the review.
20

---

16. Declaration of LLM usage
Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology. scientific rigorousness, originality of the research, declaration is not required.
Answer: [NA]
Justification: LLM is used only for formatting purposes in this paper.
Guidelines:
The answer NA means that the core method development in this research does not involve LLMs as any important, original, non-standard components.
Please refer our LLM policy (https: /ttp///errppppppp/ee cc wwt.c////onnoccss.s.....ooo 2025/ LLM) for what should or should not be described.
21

---

Conns Prejection Bemm Ciede
Pammodortaaein of Nimvll
por Bood!
MUSE
35
Figure Illustration of the airfoil curve paramctcrization employed in FuncGenFoil.
Index
Symbol
Geometric Meaning
1
Rie
leading edge radius
2
Xup
upper crest position x upper crest position
3
Y up
Zoup
upper crest curvature Zexup
5
X
lower crest position x lower crest position y lower crest curvature trailing edge position trailing thickness trailing edge angle up trailing edge angle down
4Y
6
Yoo
P
7
Zxxlo
Y to
9
A Yte te
Zaria
Ote
11
Bie
(a) PARSEC parameters and its geometric meaning.
(b) Demonstration of PARSEC parameters.
Table 6: Eleven PARSEC parameterization in airfoil design.
A Problem Setting for Airfoil Generation
A.1 Airfoil Parametrization
In this paper, we perform a circular parameterization of the airfoil curves, as illustrated in Figure 7 Specifically, we project unit circle onto the x-axis and adopt the angle a as the function domain. The parameterized airfoil function begins the trailing edge on suction side, where a 0, and ends at the trailing edge on the pressure side, where 1. In this manner, we transform the airfoil data from circular topology into a well-defined functional form suitable for reconstruction by generative models.
A.2 Design Parameters of Airfoil Geometry
We adopted the same design condition parameters as provided in AFBench (37], as shown in Table 6 The parameters listed below-including the leading-edge radius, the curvature the maximum thickness of the upper and lower surfaces, and the trailing-edge angle of the airfoil-wee were obtained via curve fitting. Note that our implementation of the trailing-edge angle calculation slightly differs from that of AFBench, as we found the original approximation method not invariant to increased sampling precision. Specifically, AFBench employs B-spline curves for interpolation at the trailing edge and calculates its gradient at the endpoint, which becomes unstable as resolution increases. In contrast, our method assumes the last 2% length of the airfoil curve near the trailing edge nearly straight, and uses linear regression to determine the angle.
22

---

Algorithm Model Training.
Input: data resolution d, data u1 design condition variables (optional).
Parameter:
Gaussian process (O, K) for sampling un-
Output: velocity operator ta.
I: while training... do
sample [O, 1]. uo (O, K) resolution and get {
3:
Compute u2 at resolution and get {vr,s}.
4: Compute at resolution d and get {ue,s].
5:
Compute (Mee f) at resolution d and get {vo,s}-
6:
Minimize vo.. {1.
7: Compute gradient and update
8: end while
9:
return
Algorithm 2 Model Inference.
Input: sampling resolution d, sampling time steps T and steps length dt, latent function (optional), design condition variables e (optional).
Paramefer:
Gaussian process (P, K) for sampling un-
Output: airfoil {yr} } at resolution
I: Let 0, uо GP(O, K) resolution and get {.o.s].
2: while S do
3: Compute ve (ue, t) at resolution d and get {ve,s}.
4: Compute {ue+de,i) {uei)} + {vo,sdt).
5: == dt.
end while
7:
return {y:} = {un,+}
B Algorithm Implementation Details
The algorithm for training the airfoil generative model is presented in Algorithm T and the inference procedure using trained model is described in Algorithm 2 In our implementation, we employed Optimal Flow Matching and utilized Gaussian process with specific kernel functions, such as the RBF kernel or the Matérn kernel, in the latent space of the generative model.
C Training Details
In most of the experiments, we followed the hyperparameters listed in the Table 7 below to train the models, including both the pre-training and fine-tuning stages. On the AF-200K dataset, we increased the maximum number of iterations to 2000000 and the batch size to 2048. For the number of time steps used to solve ODE, we use 10 steps for relative balance between fast sampling speed and good performance, except for the fine-tuning stage, where used 10 steps for training efficiency.
To provide comprehensive analysis of the model's computational requirements. we evaluate its performance relative to the point-based PK-DIT model. Table details this comparison, presenting the wall-clock training time, per-sample inference latency, and peak GPU memory usage. All benchmarks were conducted on single desktop machine equipped with an NVIDIA RTX 4090 and an Intel 13-0KKK
D
More Experiments
D.1 Comparison with a Classical Optimization Method
To establish performance benchmark, we compare method against classical, optimization- based approach for airfoil reconstruction. This benchmark is designed to estimate the near-optimal accuracy achievable with standard parametric representation.
The classical method represents the airfoil using NURBS curve with 24 control points. To create best-case scenario for this method, the optimization for each of the 3825 target airfoils in the
23

---

Algorithm 3 Model Finetuning (Airfoil Editing).
Input:
pretrained neural operator original airfoil function (optional) or latent function (optional), editing requirement 6, editing resolution d, sampling time steps T and steps length dt.
Parameter: Gaussian process (O K) for sampling un. noise level
Output: new airfoil {yi } at resolution d.
I: Inversely sample original airfoil function {wI,s} through ve and get its latent function { {u {
2: Set {ae.s} }
3: while finetuning... do
4: Sample {u, through uo from {-
5: Compute LMAP.
6: Compute gradient and update 0.
end while
8: Sample {ui,}} through th from {aa,s}-
9:
return {y,} = {u1.i}
Table 7: Training hyperparameters and model parameters in conditional airfoil generation tasks
Training hyperparameters
Value
Max learning rate
Batch size
1024
Batch size (AF-200K)
2048
Optimizer (pre-training and fine-tuning)
Adam
Optimizer scheduler
Cosine Annealing
Warmup iterations
2000
Max training iterations
1000000
Max training iterations (AF-200K)
2000000
ODE solver time steps
10
ODE solver type
Euler
Model parameters
Value
Fourier neural operator layers
6
Fourier neural operator modes
64
Fourier neural operator hidden channels
256
Gaussian process kernel
Matérn
Kernel noise scale
1
Kernel order
2.5
Kernel characteristic length
0.03
test dataset initialized with control points configured to match the average airfoil shape. This starting point is already very close to the final target, ensuring stable and rapid convergence. For each target, the optimization was performed using the Adam optimizer for 100 steps with learning rate of 3 x 10 10-4 continuing until the Mean Squared Error (MSE) plateaued. The total computation cost for fitting the entire test
approximately 20 GPU-hours
NVIDIA A800 GPU.
The results of this comparison are summarized in Table TO As expected, the NURBS fitting baseline performs exceptionally well, closely approaching what might be considered a practical upper bound on accuracy for a 24-parameter representation. FuncGenFoil's accuracy competitive with this strong baseline. Analyzing the geometric errors reveals that the NURBS fitting method excels at the trailing edge, likely because the control points can effectively constrain this region. However, FuncGenFoil demonstrates superior performance at the leading edge. The classical method struggles with the high curvature of the leading edge, where the 24 control points lack the flexibility to capture the geometry accurately.
is crucial highlight fundamental difference in the problem setup that heavily favors the classical method. Its optimization begins from well-conditioned starting point, whereas generative models like FuncGenFoii or PK-DIT must learn to construct airfoils from a state maximum entropy (i.e., pure Gaussian noise) a significantly more challenging task. We note that the classical optimization process is delicate; initializing the control points more randomly often caused the optimization to become unstable and diverge.
24

---

Table Time comparison of different generative models.
Wall-clock for
NFEs af
GPU memory
Model
Mean inference time
1 000 epochs
test time
at test
"K-DIT (score matching)
~ 10
50 (DDIM)
220 ms
200 MB
FuncGenFoil (flow matching)
< 6h
10
50 ms
200 MB
Table 9: Effective ranges of the 11 geometric parameters for supercritical dataset.
Index
Parameter
Minimum
Maximum
leading edge radius
0.0073
0.0140
upper crest position
0.3960
0.5200
3
upper crest position
0.0592
0.0784
upper crest curvature
-0.4580
-0.2100
5
lower crest position
0.3180
0.4100
lower crest position y
-0.0589
-0.0414
lower crest curvature
0.3730
0.8050
trailing edge position
-0.0001
0.0001
9
trailing thickness
0,0020
0,0075
10
trailing edge angle up
-0.5514
-0.2254
11
trailing edge angle down
-0.4477
-0.1397
Therefore, this experiment should not be interpreted as a direct comparison. Instead, it provides reference for the best-case reconstruction performance of model-free parametric method under ideal optimization conditions.
D.2 More Ablations Experiments
We conduct a comprehensive ablation study on the choice of Gaussian process kernel types, model modes of Fourier Neural Operators, and characteristic lengths, as shown in Table [I the Matérn kernel, Table 12 for the radial basis function (RBF) kernel, and Table 13 for the white noise kernel, which is commonly used in diffusion or flow models in fnite-dimensional spaces. In general, appropriate selections of these settings can significantly enhance model performance. Specifically, the Gaussian prior should exclude function spaces where the data unlikely to reside, making the prior space as restricted as possible to facilitate effective model training. For example, dataset with high smoothness is unlikely to belong to the function space generated by white noise prior but is more plausibly sampled from a Gaussian prior with an RBF kernel.
In Table 12 we observe that the RBF kernel exhibits similar tendency to the Matérn kernel, with both achieving optimal performance at the characteristic length 0.03. As for the white noise kernel in Table 13 since its characteristic length is effectively O, studied the influence of the noise scale n over the range [1.0, 2.0, 4.0). We found that excessively large diffusion white noise negatively impacts model performance, likely because such noise scales project the function into larger white noise space, making it more difficult for the neural operator to learn the intrinsic dynamics.
D.3 Generated Airfoil Examples
Figure 8 showcases a variety of airfoil geometries generated by FuncGenFoil models. These models were trained on datasets of supercritical airfoils, demonstrating their capability produce diverse and plausible shapes.
E Societal Impacts
This work will accelerate research in AI for Science and Engineering and generally has positive social impacts. On the other hand, AI models for engineering design and optimization could accelerate technological diffusion within society, potentially raising issues related to intellectual property or unauthorized technology transfer entities individuals intending to design specific engineering products or construct items harmful to society.
25

---

Table 10: Performance Comparison: Classical Optimization vs. FuncGenFoil. The classical "NURBS fitting" methad is initialized close the target, representing near-optimal henchmark.
Label Error (x10-
Dataset
Algo.
01
02
04
0%
08
07
09
010
011
0e
0g
NURBS fitting
10.8
17.4
4.5
64.5
9.79
3.56
113.2
1,53
1.72
0.065
0.106
20.6
3.97
Super
FuneGenFoil
0.71
8.23
0.13
201.3
4.72
0.12
174.2
0.09
0.14
34.2
36.7
41.9
3.08
Table 11: FuncGenFoil model performance with different FNO mode and Matérn kernel for conditional airfoil generation.
Label
(10
Modes
V
D D
M 4 (10 -)
01
02
03
0'
On
08
0/10
011
G.
6g
26.7
0.02
-105.7
32
0.01
0.63
5.64
0.09
158.3
156.9
0.07
0.12
27.6
29.2
34.7
2.41
-105.8
1.01
3.5
0.02
5.08
0.10
26.8
22.8
-105,9
0.03
0.53
5.17
0,11
83.5
2.90
0.11
89.5
0.10
0.14
29.4
30.7
22.0
2.28
105.2
0.99
0.01
0.82
8.57
0,10
370.2
0.09
342.5
0.08
0.12
36.3
72.9
3.30
-10400
1.09
1.5
0.02
8.26
0.13
222.9
37.4
48.2
3.12
-103.6
1.04
0.03
0.71
8.18
0.14
183.7
4.82
0.12
207.2
0.09
0.14
34,6
43.3
3.13
104,0
1.02
0.01
0.80
8.77
0.10
291.2
5.23
0.10
277.6
0.08
0.12
35.8
37.7
59.8
3.20
103.7
1.06
2.5
0.02
0.68
8.66
0.13
192.4
4.78
0.11
182.8
0.09
0.12
36.2
37.8
42.2
3.06
103.5
1.02
0.03
0.71
8.23
0.13
201.3
4.72
0.12
174.2
0.09
0.14
34.2
36.7
41.9
3.08
103.9
1.01
0.01
0.88
8.80
0.11
269.3
5.26
0.09
248.9
0.08
0.12
36,5
38.0
55.3
3,20
-103.4
1.05
0.02
0.72
8.52
0.11
177.1
4.84
0.11
174.6
0,08
0.13
33.2
37.4
39.7
2.98
-103.6
1.02
0.03
0.66
7,80
0.12
161.
4.54
0.11
164.2
0.09
0.15
35,1
38.3
37.5
2.96
-103.4
1.01
0.04
0.82
8.19
0.13
170.0
4.45
0,13
159.7
0.09
0.16
36,6
34.8
37.7
3.11
-103.6
1.01
0.06
0.87
8.82
0,16
183.0
5.01
0.13
183.9
0.10
0.17
41.8
41.2
42.3
3.44
-00.8
1.02
0.12
1.39
10.7
0.20
261.0
6.23
0.15
230.3
0.12
0.19
53.4
44.9
55.3
4.27
99.9
1.05
0.01
1.09
10.4
0.11
708.5
7.13
0,10
613.5
0.09
0.11
40,7
41.1
129.3
4.16
-102.8
1.27
2.5
0.02
1.15
10.0
0.11
577.5
6.83
0.11
570.2
0.09
0.14
37.8
42.3
113.3
4.14
-103,4
1.20
0.03
1.05
10.1
0.12
598.0
6.48
0.14
487.7
0.09
0.15
36,9
40.1
107.3
4,22
-103.0
1.23
0.01
1.04
10.3
0.10
628.9
7.05
0.09
576.8
0.08
0.12
39.3
42.0
118.7
4.05
-102.9
1.23
3.5
0.02
0.93
9.94
0.12
560.6
6.81
0.10
484.6
0.09
0.15
41.5
40.8
104.1
4.05
103.2
1.19
0.03
1.11
9.93
0.11
548.2
6.49
0.12
520.3
0.09
0.16
38.6
39.0
105.8
4.16
-103.1
1.23
Table 12: FuncGenFoil model performance with different RBF kernel for conditional airfoil generation.
Modes
Label Error J. (10-5)
.M (10-2)
0.01
0.77
3.67
0.12
223.6
5.05
0.09
174.7
0.08
0.12
35.2
37.1
47.0
3.06
-103.6
1.03
0.02
0.62
8.33
0.12
162.2
4.59
0.11
165.9
0.09
0.17
38.0
42.5
38.4
3.04
-103.2
1.01
61
0.03
0.66
7.80
0.12
147.9
4.13
0.12
154.2
0.09
0.15
33.2
37.9
35.1
2.93
-103.6
1.00
0.04
0.06
1.15
9.97
0.15
220.3
6.21
0.14
215.5
0.10
0.16
38.3
44.2
48.7
3.73
.99.9
LO05
0.12
L12
10.9
0.17
230.5
7.71
0.17
265.6
0.11
0.18
10.0
42.3
51.1
4.14
-95.1
1.08
Table 13: FuncGenFoil model performance with white noise kernel of different noise scale n for conditional airfoil generation.
Label Error J (10-3)
Modes
D
M (10-
01
07
d3
44
0g
de
dy
d7g
d9
011
1.15
7.18
0.09
013.8
5,16
0.07
964.5
0.06
0.08
26.2
26.1
176.8
3.4
-04.7
1.39
7.98
5.94
1268
238.3
-102.7
1.64
4
2.27
9.14
0.12
2015
7.24
0.11
2059
0.09
0.14
41.5
41.4
379.7
5.61
.9.9
2.09
26

---

COOOO..O
O0oo..oo cooooooa
O0oooooa
C0oooooo
Figure 8: A diverse of airfoil geometries generated by FuncGenFoil models trained supercritical airfoil data.
27


Reviews:
[{"review_id": "CuPDdO5SD8", "reviewer": "Reviewer_zbR2", "summary": "The paper presents FuncGenFoil, a novel generative model for airfoil design that operates in function space rather than traditional discrete or parametric representations. By leveraging Fourier Neural Operators (FNO) and Flow Matching, FuncGenFoil enables:\n\n- Resolution-free generation of smooth and continuous airfoil shapes.\n\n- Flexible conditional generation, guided by geometric parameters.\n\n- Freestyle editing capabilities via a MAP-based fine-tuning procedure.\n\nThe method demonstrates significant improvements in label error, diversity, and smoothness over existing state-of-the-art baselines (e.g., CVAE, CGAN, PK-DIT), across multiple datasets (AF-200K, UIUC, Supercritical Airfoil). The model also generalizes well to different sampling resolutions and shows physically plausible aerodynamic performance through CFD simulation.", "questions": "1. **Generality Beyond Airfoils**: Have you considered applying FuncGenFoil to more general or irregular geometries (e.g., full aircraft surfaces or automotive shapes)?\n\n2. **Editing Scope and Flexibility**: How does the method handle conflicting or dense constraints (e.g., multiple overlapping or incompatible edits)?\n\n\n3. **Scalability and Efficiency**: What are the training/inference times compared to point-based models like PK-DIT or diffusion models? It is suggested to include a table comparing computational costs, memory usage, and runtime to help readers assess deployment feasibility.\n\n4. **Design Parameter Interpretability**: Can the latent space or generated airfoils be interpreted in terms of classical aerodynamic design principles?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### Strength\n1. **Novelty and Technical Innovation**: Modeling airfoils as continuous functions in function space, rather than point clouds or fixed parametric families, is a substantial conceptual advancement. Avoids limitations in discrete point set expressiveness and parametric function rigidity. Uses Flow Matching with Neural Operators to support infinite-dimensional generative modeling. Supports conditional generation and geometric editing through a unified framework.\n\n2. **Thorough Experiments**: Strong improvements in label accuracy (e.g., 74.4% reduction on AF-200K), diversity, and smoothness. Extensive ablation studies (kernel choice, ODE solvers, Fourier modes). Evaluation includes classical metrics and physics-based simulation.\n\n### Weaknesses\n1. **Complexity and Accessibility**: The method requires a sophisticated combination of operator learning, Gaussian processes, and differential solvers, which may hinder reproducibility.\n\n2. **Generality of Shape Types**: Although the method is proposed in a general form, the experiments are restricted to airfoils and wings, which are relatively smooth and regular.\n\n3. **Runtime Efficiency**: While performance is strong, training involves millions of iterations and inference requires solving ODEs, which may be costly in time or resources.", "was_revised": true, "final_justification": "During the rebuttal process, the authors' responses have addressed most of my concerns. I remain generally positive about the paper, and my final decision is **Weak Accept**. However, I did not choose **Accept** because I believe the overall quality and potential impact on the community are somewhat limited.", "rebuttal": "Thanks for your feedback and suggestions! I'm glad you appreciated our work.\n\nHere are some responses to your questions and concerns:\n\n---\n\n> Complexity and Accessibility: The method requires a sophisticated combination of operator learning, Gaussian processes, and differential solvers, which may hinder reproducibility. \n\nFirst, we have included the complete code in the supplementary material (zip file) and are committed to open-sourcing our work to ensure reproducibility. Second, our training code is built upon several widely-used open-source packages and maintains a manageable level of complexity. Specifically, our methods are implemented using a standard neural operator library [1]. Notably, we introduce a novel and tractable loss function to model functional distributions over objectives and constraints, which regularizes model outputs in function space. Additionally, we employ a learnable prior during fine-tuning, which adds some complexity compared to other generative model frameworks.\n\n---\n\n> Generality of Shape Types & Generality Beyond Airfoils: Although the method is proposed in a general form, the experiments are restricted to airfoils and wings, which are relatively smooth and regular. Have you considered applying FuncGenFoil to more general or irregular geometries (e.g., full aircraft surfaces or automotive shapes)?\n\nYes, FuncGenFoil can theoretically be applied to a variety of aircraft component surfaces, such as fuselages, engine nacelles, and turbomachinery blades, as long as a suitable coordinate system can be defined for the geometry. In this paper, we focus on airfoil objects as they are important for aircraft systems lift and drag performance.\n\nFor more complex and irregular shapes—such as full aircraft surfaces or automotive bodies—where it is difficult to define a parameterization or coordinate system, further advances in differential geometry and manifold theory would be needed to enhance the FuncGenFoil method. For example, generative models could be used to reconstruct the deformation of general shapes parameterized in geodesic coordinates. This remains an open and challenging problem, and is certainly a promising direction for future research.\n\n---\n\n> Editing Scope and Flexibility: How does the method handle conflicting or dense constraints (e.g., multiple overlapping or incompatible edits)?\n\nFor conflicting constraints, the model does not strictly satisfy all conditions but instead seeks a reasonable airfoil design that balances errors between conflicting objectives. This trade-off is quantitatively determined by the scalar value of the associated penalty term.\n\nFor overlapping or repeated constraints, the model remains effective. Technically, the multiple-constraint loss continues to perform robustly. As shown in Figure 4 and described by Equation 8, even with dense constraints—covering up to 60% of all points—repeating the constraints leads to editing errors of less than 10e−6, which is negligible.\n\n---\n\n> Runtime Efficiency: While performance is strong, training involves millions of iterations and inference requires solving ODEs, which may be costly in time or resources.\n\nOn an NVIDIA 4090 GPU, PK-DIT can be trained for 1000 epochs on the supercritical datasets in approximately 10 hours, while FuncGenFoil completes 1000 epochs in under 6 hours. This efficiency is attributed to our use of the modern flow matching training scheme, which not only accelerates convergence but also improves performance.\n\nFor inference, PK-DIT incurs a higher computational cost due to the use of a large number of function evaluations (NFE = 50) with the DDIM sampling scheme. This can potentially be reduced by employing more advanced diffusion inference techniques, such as DPM-Solver++. In contrast, FuncGenFoil, as a flow matching model, directly generates flows and requires only 10 NFEs to achieve competitive performance.\n\nBoth PK-DIT and FuncGenFoil are lightweight models that can be efficiently executed on consumer GPUs. On a system equipped with an NVIDIA 4090 and Intel 13900K, the average inference time is 220 ms for PK-DIT and 50 ms for FuncGenFoil, with GPU memory usage for both models remaining around 200 MB.\n\n---\n\n> Scalability and Efficiency: What are the training/inference times compared to point-based models like PK-DIT or diffusion models? It is suggested to include a table comparing computational costs, memory usage, and runtime to help readers assess deployment feasibility.\n\n| Model  | Wall-clock for 1 000 epochs (RTX 4090)     | # NFEs at test time    | Mean inference time (RTX 4090 + i9-13900K) | GPU memory at test |\n|-------------|------------|---------------|------------|------------------| \n| PK-DIT (score matching)   | ≈ 10 h                                           | 50 (DDIM)              | 220 ms                                    | ≈ 200 MB           | \n| FuncGenFoil (flow matching)    | < 6 h                | 10       | 50 ms                          | ≈ 200 MB           |\n\nWe will include a table summarizing these computational costs, memory usage, and runtimes in the the paper.\n\n---\n\n> Design Parameter Interpretability: Can the latent space or generated airfoils be interpreted in terms of classical aerodynamic design principles?\n\nThe flow generation path acts as a bridge between the data space and the latent space. The data space consists of numerous airfoils designed according to classical aerodynamic principles by human experts, while the latent space represents a function space with a Gaussian measure. Because the generation path is reversible in both directions, every expert-designed airfoil corresponds to a unique latent function. By leveraging modern interpolation techniques such as spherical interpolation (SLERP)[3] or Latent Optimal Linear combinations (LOL)[4], we can interpolate between latent representations of different airfoils. This allows us to generate new airfoils that transition smoothly between designs, all of which adhere to classical aerodynamic design principles and exhibit similar aerodynamic performance, as illustrated in Figure 5.\n\n[1] Jean Kossaifi and Nikola Kovachki and Zongyi Li and David Pitt and Miguel Liu-Schiaffini and Robert Joseph George and Boris Bonev and Kamyar Azizzadenesheli and Julius Berner and Anima Anandkumar. A Library for Learning Neural Operators. arXiv 2412.10354.\n\n[2] Chen, Ricky T. Q. and Rubanova, Yulia and Bettencourt, Jesse and Duvenaud, David. Neural Ordinary Differential Equations. Neurips 2018.\n\n[3] Ken Shoemake. Animating rotation with quaternion curves. In Proceedings of the 12th annual conference on Computer graphics and interactive techniques, pp. 245–254, 1985.\n\n[4] Erik Bodin and Alexandru Stere and Dragos D. Margineantu and Carl Henrik Ek and Henry Moss. Linear combinations of latents in generative models: subspaces and beyond. ICLR 2025."}, {"review_id": "Ilv07Lcgdh", "reviewer": "Reviewer_Qqnj", "summary": "This paper focus on an important task to generate airfoil that satisify the requirement for several key parameters.\nInstead of generating hyperparameters to define the airfoil shape or its discrete boundary points, this paper proposes representing the airfoil as a closed-form function.\nThen, with the With the usage of operator flow matching and sampling from guassian distribution, it generates the function for the target airfoil step by step.", "questions": "N/A.", "limitations": "N/A.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strength:\n1. The experimental results achieve state-of-the-art performance on AFBench, with a significant boost in accuracy. As shown in Table 1, the error is reduced by orders of magnitude.\n\n2. The writing of this paper is clear and easy to follow. The idea of representing the airfoil as a closed-form function is well-motivated. Due to this, the method can produce the arifoil with any resolutions.\n\n\nWeakness:\n1. A more thorough efficiency study of the proposed method should be provided to better demonstrate its advantages. In particular, a comparison with discrete point-based generation methods would be valuable. \nIntuitively, the proposed approach should offer efficiency gains over these methods, and it would strengthen the paper to provide evidence supporting this.", "was_revised": true, "final_justification": "The generation of airfoil is an important task and this submission. And the idea of operator flow matching provides a great performance on the current benchmarks.", "rebuttal": "Thanks for your feedback and suggestions! I'm glad you appreciated our work.\n\nHere is the response to your concerns:\n\n---\n\n> A more thorough efficiency study of the proposed method should be provided to better demonstrate its advantages. In particular, a comparison with discrete point-based generation methods would be valuable. Intuitively, the proposed approach should offer efficiency gains over these methods, and it would strengthen the paper to provide evidence supporting this.\n\nWe agree that a more thorough efficiency study is beneficial. In short, both PK-DIT and FuncGenFoil have comparable model sizes and utilize MSE-type losses (flow matching for FuncGenFoil, score matching for PK-DIT), resulting in similar computational costs during training. \nOn my current setup, which includes an NVIDIA 4090 GPU, PK-DIT can be trained for 1000 epochs on supercritical datasets in approximately 10 hours, whereas FuncGenFoil completes 1000 epochs in less than 6 hours, benefiting from the modern dataloader provided by the TorchRL library. As for AFBench dataset, it takes around 10 times longer to train both models as the dataset is larger and more complex, but the relative performance remains similar.\n\nFor inference, PK-DIT incurs a higher computational cost due to the use of a large number of function evaluations (NFE = 50) with the DDIM sampling scheme in their original Github implementation. This could be improved by adopting more advanced diffusion inference techniques such as DPM-Solver++. In contrast, FuncGenFoil, as a flow matching model, generates flows directly and requires only 10 NFEs to achieve competitive performance.\n\nBoth PK-DIT and FuncGenFoil are lightweight and can be efficiently run on consumer GPUs. On a platform with an NVIDIA 4090 and Intel 13900K, the average inference time is 220 ms for PK-DIT and 50 ms for FuncGenFoil. GPU memory usage for both models is around 200 MB, which is quite small. We will include a table summarizing the exact computational costs, memory usage, and runtimes in the appendix if the paper is accepted."}, {"review_id": "QMN05Fj6P6", "reviewer": "Reviewer_yJJS", "summary": "This paper presents a novel generative model for generating airfoil geometries (2D contours) based on flow matching. The proposed model includes an optional conditioning variable and supports customized editing on the airfoil geometry. The paper evaluates the proposed model on two tasks: conditional generation and freestyle editing, and compares its performance with multiple baselines on several established datasets.", "questions": "Overall, the paper appears to be in good shape, and I do not have major concerns with it at this time. I am not an expert in airfoil design, though.\n\nThat said, I still have a few questions:\n1. I didn’t find any discussions on the paper’s limitations in the main paper. Please discuss.\n2. Following up on the weakness I mentioned above, is there a way to validate the performance of these explored airfoil designs in the real world?\n3. I’d also like to ask about the potential of this method for 3D airfoil design.", "limitations": "The paper seems to lack discussions on its limitations. I suggest the authors add a paragraph about it.", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. I think the paper tackles a valuable research problem through the lens of modern AI tools. Exploring airfoil geometry designs may have a significant impact, especially if the proposed AI method can discover novel geometries with high-performance aerodynamics.\n2. The technical method proposes a generative model in a function space, rather than the standard Bezier-curve shape space used in airfoil design. This is a novel and timely exploration of diffusion and flow models in this specific problem domain.\n3. I appreciate that the experiments also include aerodynamic simulations of the shapes generated by the proposed generative model. This makes the proposed method more useful in practice.\n\nWeaknesses:\n1. The paper does not present a hardware prototype to confirm the physical fidelity of the results. For example, whether the airfoil geometry proposed by the generative model is more useful than existing designs remains unclear. This is probably beyond the scope of this work, though.", "was_revised": true, "final_justification": "Overall, I think this is a good paper. The problem scope may be niche for a general ML conference, but the solution is interesting and novel, and the downstream application is valuable.\n \nThe rebuttal has adequately answered the weaknesses and questions raised in my original review. I am happy to maintain my original score and recommend acceptance.", "rebuttal": "Thanks for your feedback and suggestions! I'm glad you appreciated our work.\n\nHere is the response to your several concerns:\n\n---\n\n> The paper does not present a hardware prototype to confirm the physical fidelity of the results. For example, whether the airfoil geometry proposed by the generative model is more useful than existing designs remains unclear. This is probably beyond the scope of this work, though. Is there a way to validate the performance of these explored airfoil designs in the real world?\n\nYes, this work is part of a generative design module that has been integrated into modern commercial aircraft design systems in collaboration with aerodynamic experts. Due to commercial confidentiality, we are unable to disclose details regarding the real-world performance of generated or optimized airfoils deployed in industry. Instead, our study utilizes public datasets, on which the proposed techniques demonstrate similar performance trends. As shown in Figure 5, the simulation results for the generated airfoils exhibit trends that closely align with those observed in real-world applications.\n\n---\n\n> I didn’t find any discussions on the paper’s limitations in the main paper. Please discuss.\n\nWe apologize for not including a discussion of the paper’s limitations in the main text due to the 9-page length restriction. However, we have provided this discussion in Appendix E following the main text.\n\nThe primary limitation of FuncGenFoil is that it is applicable only to curves or surfaces that can be parameterized with a suitable coordinate system. For more complex surfaces, where defining such a coordinate system is challenging, FuncGenFoil is not directly applicable. Addressing these challenges will require further theoretical advancements, which we consider an important direction for future research. In addition, we should introduce more kinds of real constraints to the model, such as mechanical constraints and manufacturing constraints, to enhance the model's applicability to real-world engineering problems.\n\n---\n\n> I’d also like to ask about the potential of this method for 3D airfoil design.\n\nFuncGenFoil can be applied to a variety of aircraft component surfaces, such as fuselages, engine nacelles, and turbomachinery blades, as long as a suitable coordinate system can be defined for the geometry. For example, we can generate the 3D main wing surface of an aircraft using a larger dataset of 3D airfoil geometries under cylindrical coordinate system.\n\nFor more complex and irregular shapes—such as full aircraft surfaces or automotive bodies—where it is difficult to define a parameterization or coordinate system, further advances in differential geometry and manifold theory would be needed to enhance the FuncGenFoil method. For example, generative models could be used to reconstruct the deformation of general shapes parameterized in geodesic coordinates. This remains an open and challenging problem, and is certainly a promising direction for future research."}, {"review_id": "Ij67UveXdH", "reviewer": "Reviewer_G9rd", "summary": "# Review of Submission #9911\n\n## Summary\n\nThe paper introduces FuncGenFoil, a generative model to generate airfoil designs/geometries. \nIt is a Flow Matching model with a Fourier Neural Operator (FNO) backbone and is trained on large, open-source datasets (AF-200K, UIUC, Super).\nFuncGenFoil allows both conditional generation and \"freestyle editing\" of geometries. The paper shows that FuncGenFoil outperforms previous deep-learning-based approaches (GANs, Diffusion models) in the conditional generation task. Ablation studies justifying architecture and training/generation choices are provided.", "questions": "- How does the full cost of FuncAirGen compare to a classical method using e.g. NURBS? What is the advantage of FuncAirGen? Are there failure cases of classical methods that it can fix? Are there failure cases of FuncAirGen?\n\n- The paper makes the point that FuncAirGen enables superresolution with smooth, arbitrary resolution shapes. Is this smoothness not a direct consequence of the FNO backbone, which approximates the data with a finite Fourier representation of the signal (geometric curve)? Is the size of the FNO sum a limit/upper bound of FuncGenFoil similar to parametric-model-based methods?\n\n- Why use a FNO backbone instead of typical INRs like SIREN? The paper mentions it is possible in principle, but does not elaborate on this choice further. FNOs are not commonly used in neural geometry processing literature.", "limitations": "See Weaknesses/Questions", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "## Strengths\n\n- Good introduction and related work section, introducing the problem setting (airfoil design for the aviation industry) and previous works on this topic,\nboth \"classical\" computational design (e.g., NURBS, Bézier curves) and deep learning (VAE, GANs, Diffusion) approaches. Good mathematical notation is kept throughout the paper.\n\n- Clear training description: Flow matching based on [1], Gaussian Processes as noise distribution, conditioning on geometric parameters (PARSEC) with an explanatory Figure.\n\n- Clear generation/editing description: Inference with ODE solver and finetuning the model for editing is well explained and shown in the Figure.\n\n- Well-written experimental section, detailing the used metrics, training, and tasks/comparisons.\n\n- Extensive comparison with other DL approaches (VAE, GANs, Diffusion) for conditional generation.\n\n## Weaknesses\n\n- The limitation of classical methods is not really explained; it is just said that picking certain function families limits the design space. But how strong of a limitation is this in practice? Furthermore, neural networks are not random functions [2]. So there is also a clear bias in the shapes/geometries neural networks will learn, depending on the architecture, activation functions, etc, which is well explored in the neural implicit shape and geometry processing literature. We are simply using a different class of function family, but to me it is not clear what underlying problem the authors are trying to solve with this. It looks like just applying a different function class to the problem. I would ask for clarification from the authors on this point.\n\n- This point follows from W1: while the authors provide extensive comparisons to previous DL methods, they provide none to other \"classical\" computational design methods. These should serve as a baseline for the data-driven approaches. \n\n- No comparison methods for the airfoil editing task. Also, the edited airfoils from \"Freestyle Airfoil editing\" are not being tested for their aerodynamic performance. Instead, a new model is trained on a different dataset to test this. It is not clear to me why, please clarify.\n\n- No clear overview of the computational cost of dataset generation, training, finetuning (for editing), and inference is provided. Recent review articles of DL methods for numerical problems (PDEs, simulation, shape/topology optimization) have raised the issue that the full computational costs of DL methods/surrogates are often not provided and compared to SoTA numerical methods (see e.g.[3]). The paper gives the training time, but this should be more thorough. \n\n- I find Figure 4 a bit hard to read/process; the color choices (light green, light blue, light gray) seem suboptimal. Just a minor point.\n\n[1] LIPMAN, Yaron, et al. Flow matching guide and code. arXiv preprint arXiv:2412.06264, 2024.\n[2] TENEY, Damien, et al. Neural redshift: Random networks are not random functions. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2024. S. 4786-4796.\n[3] WOLDSETH, Rebekka V., et al. On the use of artificial neural networks in topology optimisation. Structural and Multidisciplinary Optimization, 2022, 65. Jg., Nr. 10, S. 294.", "was_revised": true, "final_justification": "The main reason for me raising the score are the data range table which shows when the model breaks and the added baseline experiment.", "rebuttal": "Thanks for your valuable feedback and thoughtful suggestions!\n\nWe address your questions and concerns in detail, and hope these responses help clarify our work.\n\n---\n\n> (a) The limitation of classical methods is not really explained; it is just said that picking certain function families limits the design space. But how strong of a limitation is this in practice? (b) Furthermore, neural networks are not random functions [2]. So there is also a clear bias in the shapes/geometries neural networks will learn, depending on the architecture, activation functions, etc, which is well explored in the neural implicit shape and geometry processing literature. We are simply using a different class of function family, but to me it is not clear what underlying problem the authors are trying to solve with this. It looks like just applying a different function class to the problem. I would ask for clarification from the authors on this point.\n\n**For question (a):**\n\nSelecting specific function families, such as NURBS, inherently limits expressiveness, as they cannot exactly represent all smooth functions (e.g., a simple sine curve), and increasing the number of control points adds significant complexity. In practice, such representations restrict the ability to fully explore the design potential of airfoils. Moreover, aerodynamic performance is highly sensitive to curvature and higher-order smoothness, especially in high Reynolds number flows. From a design perspective, it is therefore advantageous to unlock the full functional design space. The function-space generative model we proposed can represent airfoils both as point sets and continuous functions at infinite resolution.\n\n**For question (b):**\n\nRegarding the use of the Fourier neural operator instead of NURBS: as you suggested, this is indeed a choice of a different function class. Our motivation for this choice is that Fourier transforms offer theoretical elegance, as they allow us to operate in the spectral domain and recover functions via inverse transforms with minimal loss, leveraging efficient modern computational algorithms. The Fourier neural operator is also supported by theoretical guarantees as a general function approximator [1], making it a suitable and principled choice for our purposes. We agree that the exploration of alternative neural operators—potentially based on NURBS or Bézier curve —remains an open and interesting research direction. \n\n---\n\n> While the authors provide extensive comparisons to previous DL methods, they provide none to other \"classical\" computational design methods. These should serve as a baseline for the data-driven approaches.\n\nThis paper presents a data-driven and model-based method that enables engineers to more effectively leverage existing high-quality data for airfoil generation. It relies on data.\n\nClassical, model-free methods represent a complementary paradigm rather than a substitute. These methods have the advantage that they do not rely on existing data, but the disadvantage that they cannot extract information from data.\n\nThese two kinds of methods are complementary but not directly comparable. Non-data-driven, model-free methods should only be compared with algorithms that are also model-free. Therefore, they should not be included as baselines in our experiments focused on data-driven methods in Table 1.\n\n---\n\n> (a) No comparison methods for the airfoil editing task. (b) Also, the edited airfoils from \"Freestyle Airfoil editing\" are not being tested for their aerodynamic performance.  Instead, a new model is trained on a different dataset to test this. It is not clear to me why, please clarify.\n\n**For question (a):**\n\nTo the best of our knowledge, as of the submission date, there is no existing research utilizing generative models for the Freestyle Airfoil editing task. We are the first to integrate conditional airfoil generation and freestyle airfoil editing within a unified, consistent framework, achieving highly accurate editing results.\n\n**For question (b):**\n\nFor the airfoil editing task, our primary focus is on whether the model accurately follows the designer’s instructions (i.e., achieves low editing error) while preserving airfoil smoothness. Consequently, we do not evaluate aerodynamic performance for edited airfoils in this paper, as there is no well-established benchmark for comparison. For aerodynamic performance evaluation, we use the NASA Common Research Model (CRM) dataset, a widely recognized benchmark with validated working conditions and extensive use in CFD and aerodynamic studies.\n\n---\n\n> No clear overview of the computational cost of dataset generation, training, finetuning, and inference is provided. Recent review articles of DL methods for numerical problems have raised the issue that the full computational costs of DL methods/surrogates are often not provided and compared to SoTA numerical methods (see e.g.[3]). The paper gives the training time, but this should be more thorough.\n\nWe will include a table summarizing the computational costs, memory usage, and runtimes in the paper. Regarding dataset generation time, we are unable to provide this information as we only utilize publicly available datasets created by other researchers.\n\n|  | Wall-clock for 1 000 epochs (RTX 4090)  | # NFEs at test time  | Mean inference time (RTX 4090 + i9-13900K) | GPU memory at test |\n|--|--|---|--|--| \n| PK-DIT (score matching) | ≈ 10 h  | 50 (DDIM)    | 220 ms  | ≈ 200 MB  | \n| FuncGenFoil (flow matching)| < 6 h  | 10    | 50 ms   | ≈ 200 MB   |\n\n---\n\n> I find Figure 4 a bit hard to read/process; the color choices (light green, light blue, light gray) seem suboptimal. Just a minor point.\n\nThank you for pointing this out. We will improve the color choices in Figure 4 and select a palette with higher contrast to enhance readability.\n\n---\n\n> How does the full cost of FuncGenFoil compare to a classical method using e.g. NURBS? What is the advantage of FuncGenFoil? Are there failure cases of classical methods that it can fix? Are there failure cases of FuncGenFoil?\n\nFor an aircraft engineers, designing an airfoil using traditional NURBS methods requires several minutes of manual work and CAD expertise. Or, using a model-free optimization method takes about 5–10 minutes and thousands iterations to fit a 24-control-point NURBS airfoil, depending on the initial shape condition. In contrast, FuncGenFoil can generate a new airfoil in about one second.\n\nClassical methods may fail when control points are too close together or too few in number, leading to numerical instability and shape errors—particularly near the leading edge where control points are densely clustered. FuncGenFoil avoids these issues by not relying on control points.\n\nFor FuncGenFoil, failures typically occur when generating airfoils far outside the training data range (e.g., specifying an unrealistic leading-edge radius, too small or too large).\n\n---\n\n> The paper makes the point that FuncGenFoil enables superresolution with smooth, arbitrary resolution shapes. Is this smoothness not a direct consequence of the FNO backbone, which approximates the data with a finite Fourier representation of the signal (geometric curve)? Is the size of the FNO sum a limit/upper bound of FuncGenFoil similar to parametric-model-based methods?\n\nThe smoothness of the output is indeed influenced by both the type of operator used and the choice of Gaussian process kernel during operator flow matching. Since FuncGenFoil operates as both a parametric-model-based and point-based method, smoothness and resolution are primarily determined by the number of modes specified for the neural operator. Increasing the number of modes allows the model to capture more information from the data, reducing truncation error and enabling finer control over shape details. While, in theory, a Fourier transform (and its inverse without spectrum truncation) can represent smooth curves with zero error, in practice, capturing high-frequency information may require increasing the model size and mode width.\nIn this paper, performance differences between various model sizes are minor, as demonstrated in Table 8.\n\n---\n\n> Why use a FNO backbone instead of typical INRs like SIREN? The paper mentions it is possible in principle, but does not elaborate on this choice further. FNOs are not commonly used in neural geometry processing literature.\n\nTo model vector operators in infinite-dimensional Hilbert spaces, several options are available, including INRs and neural operators [1]. INRs are a good choice for neural geometry processing due to their ability to perform arbitrary point inference. However, when using INRs in generative models in function space, you must specifically design a functional encoder for function-type inputs to ensure the model is invariant to different resolutions. For example, as a network for the velocity operator $dx=v_{\\theta}(x)$, both input $x$ and output $dx$ can be of any resolution; the original INR can only guarantee the output being of infinite resolution, but not the input. This encoder in the INR should act as an operator in the mathematical sense, since an operator is a transformation between functions and is naturally input-resolution invariant.\n\nNeural operator is widely adopted in AI for science and engineering [1], particularly for solving PDEs and in computational fluid dynamics applications such as weather forecasting [2]. Given their strong theoretical foundation and proven effectiveness in modeling complex physical phenomena, FNO is especially suitable for aerodynamic modeling and research, so we chose the neural operator as the backbone for our approach.\n\n[1]Neural operators for accelerating scientific simulations and design. Nature Reviews Physics, 2024.\n\n[2]FourCastNet: Accelerating Global High-Resolution Weather Forecasting Using Adaptive Fourier Neural Operators. Proceedings of the Platform for Advanced Scientific Computing Conference. 2023"}, {"review_id": "A78isGat4w", "reviewer": "Reviewer_jKo3", "summary": "The paper proposes a generative model for 2D airfoil generation and editing by combining Gaussian Processes and Fourier Neural Operators through a flow-matching training strategy. It combines advantages from both parametric representations, as well as point-based representations of airfoils. Experiments show that the model surpasses baselines in terms of diversity, smoothness and label error in the conditional airfoil generation task. Further, the model allows for constrained editing of the airfoil.", "questions": "- Is there a reason that only PK-DiT is provided as baseline for the UIUC and Super datasets?\n- Could you provide some visualizations of generated airfoils by the model, maybe also for the baselines?\n- As written above, could you elaborate on the model performance on the error of the PARSEC angle parameters, in particular give an explanation why the model struggles?\n- How does the model handle large edits? Is it robust?", "limitations": "Limitations are provided, but only in the appendix. Please move them to the main text.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 2, "strengths_and_weaknesses": "# Strengths\n\nThe paper is clearly written and well structured. Experiments seem to answer the asked research question and show the advantages of the model. Especially the aerodynamic simulation of generated airfoils highlights that the model is able to generate useful geometries. Ablation studies further underline certain design choices and make them transparent.\n\n# Weaknesses\n\nFor the conditional airfoil generation task, only one baseline is provided for the smaller UIUC and Super datasets. I see that the best performing baseline on AF-200K was used for the other datasets as well, but given that the other datasets are way smaller in terms of the number of samples available, it would be interesting to see how at least one more baseline model copes with the limited data regime (that is usually the big constraint in this domain).\n\nThe model seems to have issues learning angles as the label error on the PARSEC parameters $p_{10}$ and $p_{11}$ are quite large relative to baselines. Could you elaborate on that?\n\nThe experiment on airfoil editing shows that the model is able to incorporate user constraints. It seems like the editing scales and perturbations during editing are rather small. It would be interesting to see how robust the model is when asked for larger edits.", "was_revised": true, "final_justification": "The paper can be recommended for acceptance. All points raised in my review were addressed in the rebuttal which lead to me raising the score.", "rebuttal": "Thanks for your feedback and suggestions! I'm glad you appreciated our work.\n\nHere are some responses to your questions and concerns:\n\n---\n\n> For the conditional airfoil generation task, only one baseline is provided for the smaller UIUC and Super datasets. I see that the best performing baseline on AF-200K was used for the other datasets as well, but given that the other datasets are way smaller in terms of the number of samples available, it would be interesting to see how at least one more baseline model copes with the limited data regime (that is usually the big constraint in this domain). Is there a reason that only PK-DiT is provided as baseline for the UIUC and Super datasets?\n\nThank you for your warm suggestion. The original AFBench paper did not report information except for the PK-DiT model, so only PK-DiT was included in Table 1 for the UIUC and Super datasets. However, we trained the PK-VAE model using the original AFBench code and same settings. We have now added new comparison experiments, and the results are as follows:\n\n| Dataset  |  Algo.      |$\\sigma_1$|$\\sigma_2$|$\\sigma_3$|$\\sigma_4$|$\\sigma_5$|$\\sigma_6$|$\\sigma_7$|$\\sigma_8$|$\\sigma_9$|$\\sigma_{10}$|$\\sigma_{11}$|$\\bar{\\sigma}_a$| $\\bar{\\sigma}_g$ |$\\mathcal{D} \\uparrow$|$\\mathcal{M} \\downarrow$|\n|--------|--------|------|------|------|--------|------|------|--------|-----|-----|------|-------|---------------------|--------------------|--------|------|\n| UIUC   | PK-VAE | 80.7 | 20.9 | 12.2 | 12843  | 36.9 | 14.0 | 37263  | 1.7 | 1.9 | 94.8 | 109.9 | 4589                | 69.1               | -93.5  | 7.29 |\n| Super  | PK-VAE | 10.8 | 17.5 | 2.3  | 1735.6 | 12.1 | 3.2  | 8131.5 | 3.5 | 1.4 | 98.3 | 80.6  | 917.9               | 28.4               | -122.8 | 1.38 |\n\nFuncGenFoil outperforms both PK-VAE and PK-DIT on the UIUC and Supercritical datasets, even in the relatively limited data regime.\n\n---\n\n> The model seems to have issues learning angles as the label error on the PARSEC parameters  and  are quite large relative to baselines. Could you elaborate on that? Could you elaborate on the model performance on the error of the PARSEC angle parameters, in particular give an explanation why the model struggles?\n\nThis is because we changed the definition of the trailing-edge angle, as explained in Appendix A.2. In this work, we perform super-resolution inference, increasing the resolution from 257 to 1025 points, as shown in Table 2. We found that the traditional method for calculating the trailing-edge angle is not suitable or numerically stable at higher resolutions, so we updated the definition in our dataset.\n\nThis new definition is not sensitive for the supercritical airfoil dataset (which is the main cluster for modern commercial aircraft design), but it can introduce additional noise and linear regression error for other datasets. As a result, FuncGenFoil shows a higher label error on these particular parameters compared to baselines. However, this deterioration in performance is generally not significant on average, and it has little practical impact on the use of the generated airfoils. The trailing-edge angle can typically be changed very rapidly by adjusting the position of only a small number of design points by a small amount, leaving the main shape of the airfoil largely unchanged, which may suggest that the FuncGenFoil model is focusing more on the overall shape of the airfoil rather than the trailing-edge angle.\n\n---\n\n> The experiment on airfoil editing shows that the model is able to incorporate user constraints. It seems like the editing scales and perturbations during editing are rather small. It would be interesting to see how robust the model is when asked for larger edits. How does the model handle large edits? Is it robust?\n\nHere we provide results for three additional editing scales under the same experimental setting, with the largest being 8 times greater than the original. As shown, the edit error increases at an accelerating rate with larger edits but remains relatively low overall. Given that the average thickness of an airfoil is about 0.1, our editing method remains effective for edit ranges from -5% to 5% under a conservative estimate. In practice, by increasing the finetuning steps and choosing suitable time steps, even larger edit scales or denser constraints are possible, as demonstrated in Figure 4.\n\n| Dataset | Edit Scale | $\\text{MSE} \\downarrow (1e^{-7})$ | $\\mathcal{M} \\downarrow (1e^{-2})$ |\n|---------|------------|--------------|-------------|\n| Super   | 0.0001     | 2.41         | 1.16        |\n|         | 0.0002     | 2.45         | 1.15        |\n|         | 0.0004     | 2.75         | 1.15        |\n|         | 0.0008     | 4.32         | 1.26        |\n|         | 0.0016     | 15.5         | 1.35        |\n|         | 0.0032     | 61.7         | 1.49        |\n\n---\n\n> Could you provide some visualizations of generated airfoils by the model, maybe also for the baselines?\n\nThank you for your suggestion. Due to the new NeurIPS 2025 policy, we are unable to upload PDFs or share external links at this stage. If the paper is accepted, we will include visualizations of the generated airfoils by our model and the baselines both on the public project page and in the appendix of the camera-ready version."}]

Author rebuttal:

Meta-review: The paper introduces a generative model for producing 2D airfoil designs. The novelty lies in directly generating the geometries as function curves, rather than using parametric curves or point sets. This is achieved with an architecture that optimizes a flow-matching criterion using an FNO backbone. The model operates in function space, leveraging recent advances in diffusion and flow-based methods. It also supports airfoil editing, i.e., generating shapes while preserving user-defined sections. The approach is evaluated in two contexts: generating airfoils subject to specified constraints and performing editing tasks. Comparisons are made against several generative baselines on three datasets.

The reviewers agree on the originality of the idea—directly generating 2D curves—and on the contribution of leveraging recent advances in generative models. They acknowledge the evaluation, which demonstrates good performance relative to alternative generative methods. However, they also raised several issues and requested clarifications, including details on computational and memory complexity, efficiency, comparisons with discrete point-based methods and classical design techniques, and analysis of limitations and the effective model range.

In their rebuttal, the authors provided detailed answers to all reviewer questions and included additional experiments, comparing against baselines and analyzing the model’s range. The reviewers consider that their concerns have been resolved, and they all recommend acceptance.
Decision: Accept (poster)