Generative adversarial network for inverse design of airfoils with flow control devices

1.   Introduction

In recent years, an exponential growth in the use of deep learning–based methods has occurred. This is attributable to various technological and scientific advances, including increased computational capacity, the maturation of algorithms, and the availability of massive datasets for model training. As a result, deep learning has emerged as a powerful tool across multiple fields and sectors.

In the domain of fluid dynamics, numerous authors have employed these methods to address various tasks related to the modeling, control, and design of fluid dynamic systems. Most research of deep learning applied to fluid dynamics focuses on system modeling, demonstrating the validity of these methods for a wide variety of systems, such as airfoils ^[1,2,3], flow control devices ^[4,5], vehicles ^[6,7], thermal systems ^[8], chemical reactions ^[9], and many others. Beyond modeling, multiple authors ^{[10,11,12,13,14]} have leveraged these methods for aerodynamic performance enhancement through control strategies; many others, such as Lee et al. ^[15], Wang et al. ^[16], and Mehrjardi et al. ^[17,18], have focused their efforts on the design of fluid dynamic systems, aiming to optimize the performance of these systems. However, the application of deep learning to system design remains a relatively unexplored and evolving area of research.

Within the field of fluid dynamic system design, inverse aerodynamic design is particularly noteworthy. Inverse aerodynamic design consists of determining the shape of the geometry based on information about the desired aerodynamic behavior. As explained in Yilmaz and German ^[19], traditional approaches rely on potential flow and surface pressure distribution to infer geometry. In contrast, modern techniques aim to generate designs based on broader performance metrics and more complex flow characteristics. These methods typically adopt an optimization-based approach, wherein a cost function is defined to optimize one or more performance measures, subject to constraints that account for the physical and practical limitations of the system under study.

Inverse aerodynamic design presents numerous challenges, including the following ^[19]:

• The representation and parameterization of the surface shape through discretization or basis functions that allow sufficient freedom to develop good designs, while avoiding poorly thought-out designs.

• The choice of suitable cost functions that strike the right balance between the objectives of the designer, which are often contradictory and require trade-offs.

• The identification of all constraints necessary to achieve realistic solutions, including those tacit to human experience and difficult to codify, to avoid the tendency of optimizers to exploit weaknesses in the problem formulation rather than achieve realistic results.

• The detection and treatment of situations where no solution exists, or more than one solution satisfies the specified performance criteria and constraints.

Deep learning has opened new possibilities for addressing key limitations of traditional inverse design methods, including the need for time-consuming iterative simulations, the difficulty of optimizing under complex physical constraints, and the lack of flexibility in capturing highly nonlinear relationships between performance and geometry. Although in some studies, such as that of Sekar et al. ^[20], alternative architectures for inverse design have been explored, generative models such as generative adversarial networks (GANs) are particularly well-suited for this type of problem, since they offer a promising alternative by learning these relationships directly from data, enabling fast and accurate design generation once trained. As explained by Mehrjardi et al. ^[17,18], in a GAN framework, the generator produces geometries that satisfy the desired performance criteria, while the discriminator ensures that the generated geometries resemble real-world designs. This combination encourages the generator to produce geometries that not only meet performance targets but also conform to learned geometric constraints, enhancing the quality of the samples created by the generator and avoiding the generation of unrealistic outputs.

For instance, Oh et al. ^[21] proposed a GAN-based design automation framework that generates a diverse set of structurally optimized designs, with a particular emphasis on aesthetic quality and design novelty. This work serves as a benchmark for the topic under discussion, but it lacks aerodynamic considerations. In contrast, Shu et al. ^[22] presented a method based on a GAN for 3D geometry generation, with a self-updating training scheme where performance-evaluated designs are reintroduced into the training set. This approach addresses functional feasibility through physics-based validation, yet it does not explicitly tackle the inverse design problem.

Regarding problems closely related to the present study, some authors have used GANs for inverse airfoil design. For example, Achour et al. ^[23] used a Conditional GAN (CGAN) for airfoil optimization based on aerodynamic coefficients and airfoil area; Chen et al. ^[24] and Tan et al. ^[25] designed CGAN architectures with interpolation layers for better geometry representation; Wang et al. ^[26] proposed a generative model for supercritical airfoil generation, based on the wall Mach number distribution; and Yilmaz and German ^[19] performed inverse airfoil design based on polar data using another CGAN.

This study advances the field by proposing the use of a CGAN for the design of 4-digit NACA airfoils with Trailing Edge (TE) flaps, considering both flow conditions (Reynolds number and angle of attack) and desired aerodynamic performance (drag and lift coefficients). In comparison with existing works that mainly focus on airfoil shape design based on desired aerodynamic performance, this study integrates flow conditions and geometric modifications by means of flow control devices into the design process, offering a novel approach to aerodynamic optimization. By incorporating TE flaps into the generated geometries, this work extends the applicability of GAN-based inverse design to more complex and functionally adaptive aerodynamic configurations.

Traditional inverse design methods rely on iterative optimization loops that require multiple evaluations of CFD models, which can be computationally expensive and time-consuming. In contrast, a data-driven approach, such as the one proposed in this study, can generate candidate geometries in a single forward pass through a trained model, enabling near-instantaneous design generation. This is especially advantageous in scenarios requiring rapid design exploration, real-time system adaptation, or integration into control loops, where speed and flexibility are critical.

The remainder of the manuscript is organized as follows: Section 2 explains the methodology followed to perform the investigation, describing the conducted numerical simulations, which are used to train the CGAN, and the architecture of this network; Section 3 shows the results obtained in this study, comparing the geometries generated by the CGAN with the real ones; and Section 4 summarizes the main findings of this research.

2.   Methodology

2.1.   Numerical simulations

In this study, 4-digit NACA airfoils with a Trailing Edge (TE) flap were considered. The front part of the airfoil is defined as the fixed part and remains static for all the cases; the rear part of the airfoil is defined as the flap, whose rotation angle ($\theta$) depends on the case. Figure 1 shows a schematic view of the tested case.

Figure 1.  Dimensions and modifiable parameters of the tested case.

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Different simulations of the case under discussion were carried out, with the objective of obtaining a diverse dataset for training the CGAN. In these simulations, a chord length of $c = 1m$, flap rotation angles between $\theta = 0°$ and $\theta = 30°$, and flap lengths between $L = 0.2c$ and $L = 0.3c$ were considered. Regarding the flow, Reynolds numbers between $Re = 0.1·{10}^{6}$ and $Re = 2·{10}^{6}$ and angles of attack of the flow between $\alpha = -5°$ and $\alpha = 5°$ were considered. XFOIL ^[27] software was used to perform the simulations, from which drag (${C}_{D}$) and lift (${C}_{L}$) coefficients were calculated. For training purposes, only the cases whose solution achieved convergence were selected, for a total of 47,645 cases.

Following the study of Achour et al. ^[23], a cosine distribution was used for the representation of the airfoil, so that the most significant areas—the Leading Edge (LE, which refers to the front-most portion of the airfoil) and the Trailing Edge (TE, which refers to the rear portion of the airfoil)—contained more sampling points for a more accurate representation. This is important because these regions exhibit the most complex geometric curvature and have a critical influence on aerodynamic performance. Mathematically, the x-coordinates were defined as shown in Eq (1).

where $N$ represents the total number of discretization points.

With the selected representation, the x-coordinates were kept constant for all cases, while the y-coordinates were specific to each geometry. This simplifies the design and training of the CGAN, since it only has to generate the y-coordinates, and also avoids the generation of physically implausible outputs, such as self-intersecting surfaces. The selected distribution contains 80 x-coordinates, for a total of 160 points. Figure 2 shows an example of the points extracted for the airfoil representation.

Figure 2.  Example of the points selected (red crosses) for the representation of an airfoil.

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2.2.   CGAN architecture

For the generation of airfoils with flow control devices, a CGAN architecture was considered. The generator of this CGAN generates the coordinates of the airfoil from a noise signal, the flow characteristics ($\alpha$ and $Re$), and the desired aerodynamic characteristics (${C}_{D}$ and ${C}_{L}$). As previously explained, the x-coordinates were kept constant for all cases, so it only generates the y-coordinates. The discriminator labels the coordinates passed to it as real or generated, receiving the data of the real airfoils, the generated airfoils, and the characteristics of each case. Figure 3 shows the CGAN architecture used for this study. For designing and training, the proposed CGAN MATLAB software ^[28] was used, with its Deep Learning Toolbox ^[29].

Figure 3.  Architecture of the used CGAN.

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The generator is a convolutional neuronal network (CNN) with two input layers, one for the noise signal and another one for the problem characteristics. Both inputs are projected so that the dimensions match the features of the network, and then the obtained layers are concatenated. The size of the layers is then increased by one-dimensional transposed convolutions, doubling the size of the data while halving the number of filters until the dimensions of the airfoil are reached. These transposed convolutions are performed with a kernel of 5 in the first layers and a kernel of 3 in the last layers, and all of them are followed by ReLU activation functions. Figure 4 shows the architecture of the generator.

Figure 4.  Architecture of the generator.

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The discriminator is also a CNN that receives two inputs, the airfoil coordinates and the problem features. These inputs are concatenated after projecting the problem features, in order to match the dimensions of the layers. After the concatenation, a dropout layer sets input elements to zero with a probability of 0.25, in order to avoid overfitting. Subsequently, convolutions are performed, with a kernel of 5, followed by leaky ReLU activation functions with a slope of 0.2. In contrast to the generator, in this case the size of the data is halved, while the quantity of filters is doubled, aiming to reduce the data until a single scalar value is obtained. In the last layer, a convolution with a kernel of 10 is performed, and the sigmoid activation function is used, so that the output determines the probability that the airfoil is real (value close to 1) or generated (value close to 0). Figure 5 shows the architecture of the discriminator.

Figure 5.  Architecture of the discriminator.

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To train the CGAN, the data was split into 90% training and 10% testing, a batch size of 128 was selected, and the Adam optimizer ^[5] was used with a learning rate of 0.0002. Training was performed until the generator and discriminator losses reached convergence, with the generator loss being lower than the discriminator loss. Aiming to avoid overfitting, L2 regularization with a weight decay of 0.001 was applied, and losses were monitored on the test-set.

2.3.   Generalization

An additional dataset was created to test the capacity for generalization of the proposed neural network. The objective of this dataset was to test the ability of the CGAN to transfer the acquired knowledge to conditions not seen during training. To this end, the cases considered in this dataset include extreme conditions not covered by the original dataset, these being $-7°\le \alpha < -5°$, $5° < \alpha \le 7°$, $Re = \mathrm{50,000}$, and $Re = \mathrm{5,000,000}$. The combination of these conditions results in the dataset for generalization, for a total of 42,092 cases.

3.   Results

A qualitative comparison of the results, an aerodynamic performance analysis, and a computational performance analysis were carried out in order to assess the ability of CGAN to generate airfoils with flow control devices. For these analyses, only the cases of the test-set and generalization-set are considered, which are the cases that the neural network has not seen during training.

3.1.   Qualitative comparison

A qualitative comparison of the airfoils generated by the CGAN and the real ones was carried out in order to evaluate the capacity of this network to generate this kind of geometry. In this comparison, different random cases of the test-set, with different flap angles, were considered. Figure 6 shows the comparison of real and generated geometries.

Figure 6.  Comparison of the real airfoils and the ones generated by the CGAN. (a) $\mathrm{\theta } = 0°$; (b) $\mathrm{\theta } = 10°$; (c) $\mathrm{\theta } = 20°$; (d) $\mathrm{\theta } = 30°$.

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The results show that the designed CGAN is able to generate geometries of high similarity with the real ones, adequately representing the main characteristics of the airfoils. The generated airfoils present certain discontinuities, which can be easily solved by different techniques, such as the interpolation used by Chen et al. ^[24]. The thickness of the generated airfoils is slightly higher than the real airfoils for the thinner airfoils, and slightly lower for the thicker airfoils. In general terms, the camber is well defined, both in value and position.

Concerning the flap, a correct representation of the flap is observed for all samples. However, in some cases, the start of the flap is not properly defined, being a continuation of the airfoil. This is more noticeable for low flap angles.

3.2.   Aerodynamic performance of generated airfoils

The aerodynamic coefficients provided by the CGAN-generated airfoils were analyzed, since the main objective of the neural network is to generate geometries that meet the established aerodynamic requirements. This analysis focuses on the relative error (${\epsilon }_{r}$), calculated following Eq (2), of the coefficients provided by the generated airfoils compared to the reference coefficients defined as input to the network, evaluating the error for different flow conditions with the objective of identifying the scenarios in which the CGAN performs better and worse. Tables 1 and 2 show the relative error of the ${C}_{D}$ and ${C}_{L}$, respectively, of the generated airfoils.

where $C$ represents the aerodynamic coefficients (${C}_{D}$ or ${C}_{L}$, depending on the error that is calculated) and $k$ is an adjusting scalar (with a value of 0.001) that is added for cases in which $C$ is equal to 0, in order to avoid infinite relative errors.

Table 1.  Mean relative error ± standard deviation of relative error (both in %) of the ${\mathrm{C}}_{\mathrm{D}}$ of generated airfoils.

$\alpha$	$Re$
	$5 \times {10^4}$	${10^5}$	$2.5 \times {10^5}$	$5 \times {10^5}$	${10^6}$	$2 \times {10^6}$	$5 \times {10^6}$
[$-7°$, $-5°$)	3.65 ± 0.51	3.38 ± 0.95	4.80 ± 1.38	7.87 ± 0.91	14.07 ± 3.7	18.36 ± 3.11	23.42 ± 5.8
[$-5°, -3°$)	3.69 ± 0.8	2.99 ± 0.42	3.11 ± 1.11	9.19 ± 1.5	13.79 ± 2.95	16.81 ± 3.41	22.17 ± 4.24
[$-3°, -1°$)	4.08 ± 1.19	2.94 ± 0.74	4.19 ± 0.65	10.40 ± 2.78	16.98 ± 2.57	18.62 ± 3.87	24.03 ± 4.67
[$-1°$, $1°$)	5.76 ± 2.12	3.87 ± 1.24	3.02 ± 0.54	7.69 ± 2.12	16.46 ± 3.82	17.16 ± 3.43	23.91 ± 5.13
[$1°$, $3°$)	4.42 ± 0.76	3.44 ± 0.98	2.41 ± 0.59	5.89 ± 0.95	15.45 ± 3.64	14.94 ± 3.51	21.52 ± 3.42
[$3°$, $5°$]	4.99 ± 0.83	3.62 ± 0.91	3.65 ± 1.12	5.72 ± 1.14	11.95 ± 1.96	14.30 ± 3.61	22.27 ± 3.89
($5°$, $7°$]	5.96 ± 1.78	4.04 ± 0.56	3.96 ± 1.03	5.28 ± 0.76	10.34 ± 3.26	14.70 ± 3.12	24.51 ± 5.02

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Table 2.  Mean relative error ± standard deviation of relative error (both in %) of the ${\mathrm{C}}_{\mathrm{L}}$ of generated airfoils.

$\alpha$	$Re$
	$5 \times {10^4}$	${10^5}$	$2.5 \times {10^5}$	$5 \times {10^5}$	${10^6}$	$2 \times {10^6}$	$5 \times {10^6}$
[$-7°, -5°$)	3.62 ± 0.68	6.33 ± 1.14	5.78 ± 1.29	7.54 ± 1.39	8.66 ± 1.6	9.97 ± 2.02	10.46 ± 2.11
[$-5°, -3°$)	8.21 ± 1.85	9.82 ± 1.74	11.41 ± 2.46	12.13 ± 2.3	12.41 ± 2.55	14.93 ± 2.79	13.73 ± 2.39
[$-3°, -1°$)	17.21 ± 3.17	15.74 ± 3.48	15.55 ± 3.08	14.46 ± 2.57	17.60 ± 3.41	18.08 ± 3.81	19.15 ± 3.88
[$-1°, 1°$)	20.81 ± 4.02	13.53 ± 2.71	14.47 ± 2.96	14.54 ± 2.43	15.74 ± 2.93	16.44 ± 3.19	18.13 ± 3.44
[$1°$, $3°$)	9.42 ± 1.91	3.70 ± 0.81	5.04 ± 1.08	5.93 ± 1.18	6.13 ± 1.25	7.34 ± 1.52	8.42 ± 1.7
[$3°$, $5°$]	4.86 ± 0.93	1.98 ± 0.44	1.99 ± 0.41	2.69 ± 0.58	2.99 ± 0.63	3.50 ± 0.81	5.68 ± 1.09
($5°$, $7°$]	4.40 ± 0.97	4.27 ± 0.84	3.58 ± 0.77	3.49 ± 0.71	4.58 ± 0.96	4.89 ± 1.03	5.33 ± 1.02

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The results indicate that the aerodynamic characteristics of the airfoils generated by the CGAN closely approximate the desired properties in all cases, exhibiting low relative errors in both aerodynamic coefficients.

For ${C}_{D}$ ​, the relative error increases with $Re$. This behavior is attributed to the fact that, as previously discussed, the geometric feature most affected during airfoil generation is thickness. Consequently, since ${C}_{D}$ ​ tends to decrease with increasing $Re$, the relative error becomes more sensitive to slight variations in thickness at high $Re$ values. Under the same $Re$ conditions, variations in $\alpha$ do not significantly affect the relative error.

In contrast, ${C}_{L}$ ​ is primarily influenced by $\alpha$. The largest relative errors for ${C}_{L}$ ​ occur at low $\alpha$ (particularly between -3° and 1°), where ${C}_{L}$ ​ approaches zero. In this range, minor geometric deviations can lead to disproportionately large relative errors. Although increasing $Re$ within the same $\alpha$ range also leads to an increase in relative error, its effect on ${C}_{L}$ ​ is considerably less pronounced than on ${C}_{D}$.

Regarding the generalization-set, the results demonstrate that the proposed CGAN successfully generalizes to extreme conditions not encountered during training, maintaining low relative errors comparable to those observed in the test set. For varying values of $\alpha$, the network demonstrates good generalization performance across both extremes. In the case of different $Re$ values, the network generalizes significantly better at low $Re$ than at high $Re$, which is consistent with the results obtained with the original test-set.

3.3.   Performance analysis

A comparison of the computational time required by each method was carried out. In this case, a fair comparison between the computational time required by CFD-based tools and the proposed CGAN cannot be made, since CFD simulations are responsible for performing forward design, i.e., providing results from a given geometry, while CGAN is responsible for performing the inverse design, i.e., providing a geometry for given characteristics. Therefore, as the number of CFD simulations required to obtain the design that meets the desired characteristics is variable, the time and computational resources required by CFD are case-dependent.

However, CGAN is able to provide extremely fast designs, generating the 4765 geometries corresponding to the test set in 0.8435 s, which is equivalent to $1.77 \times {10^{ - 4}}$ s per geometry, whereas obtaining CFD results of a single case using XFOIL ^[27] requires an average of 0.54 s. In both cases, a single core of the Intel Xeon Gold 5120 CPU was used.

4.   Conclusions

In this study, the use of a CGAN was proposed for the inverse design of airfoils with flow control devices, particularly 4-digit NACA airfoils with a TE flap. The designed CGAN receives the flow characteristics ($Re$ and $\alpha$) and the desired aerodynamic characteristics (${C}_{D}$ and ${C}_{L}$) and designs a geometry that meets the provided input characteristics.

The results show that the CGAN is able to generate geometries with a strong resemblance to the real ones, being able to design the airfoil and flap properly, with very similar aerodynamic properties, and being extremely fast. Additionally, the proposed CGAN is able to generalize adequately to extreme situations not seen during training, making it feasible to use this approach for a wider variety of situations, apart from the ones in which training conditions are considered. Regarding the considered flow parameters, the CGAN struggles to obtain the desired ${C}_{D}$ for high $Re$ cases and the desired ${C}_{L}$ for $\alpha$ values close to 0; these are the cases in which higher relative errors are obtained. Nevertheless, these errors are considered acceptable given the conditions encountered in these scenarios.

Despite the advances made in this study, several promising directions remain open for future research. One avenue involves extending the current CGAN framework to support more complex geometries, such as multi-element airfoils or other fluid dynamic systems. This would require adapting the network to work with more sophisticated data representations, such as point clouds or mesh-based surfaces. Additionally, incorporating physics-informed loss functions (considering geometric or aerodynamic constraints) could improve the physical plausibility of the generated designs. Another important direction for future research is the integration of geometric regularization layers within the generator architecture, instead of just smoothing the output values. These approaches would enforce smoothness and continuity in the output geometries by design, thus addressing issues related to performance sensitivity caused by shape discontinuities. Such representations also allow for more compact and structured encoding of airfoil geometries, potentially improving both model interpretability and training efficiency. All these potential research directions, alongside many others, highlight the promising and impactful nature of inverse design and optimization of fluid dynamic systems using deep learning tools.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors are grateful for the support provided by SGIker of UPV/EHU. This research was developed under the frame of the Joint Research Laboratory on Offshore Renewable Energy (JRL-ORE). This work has been partially supported by the Government of the Basque Country, program: Elkartek MOREDIGITAL; Grant No.: KK-2024/00117 and DeepBask Grant No.: KK-2024/00069. U.F. G. was supported by the Mobility Lab Foundation, a governmental organization of the Provincial Council of Araba and the local council of Vitoria-Gasteiz and by Government of the Basque Country, ITSAS-REM Grant No.: IT1514-22.

Conflict of interest

Unai Fernandez-Gamiz is a guest editor for the special issue of the ERA and was not involved in the editorial review or the decision to publish this article. All authors declare that there are no competing interests.