Airfoil Shape Generation and Feature Extraction Using the Conditional VAE-WGAN-gp

Abstract

A machine learning method was applied to solve an inverse airfoil design problem. A conditional VAE-WGAN-gp model, which couples the conditional variational autoencoder (VAE) and Wasserstein generative adversarial network with gradient penalty (WGAN-gp), is proposed for an airfoil generation method, and then, it is compared with the WGAN-gp and VAE models. The VAEGAN model couples the VAE and GAN models, which enables feature extraction in the GAN models. In airfoil generation tasks, to generate airfoil shapes that satisfy lift coefficient requirements, it is known that VAE outperforms WGAN-gp with respect to the accuracy of the reproduction of the lift coefficient, whereas GAN outperforms VAE with respect to the smoothness and variations of generated shapes. In this study, VAE-WGAN-gp demonstrated a good performance in all three aspects. Latent distribution was also studied to compare the feature extraction ability of the proposed method.

1. Introduction

In mechanical design, it is desirable to design shapes that indicate the required performance. This task is called an inverse design problem. Recently, deep generative models such as generative adversarial networks (GANs) and variational autoencoders (VAEs) have been used for inverse designs [1,2]. Both GANs and VAEs can generate desired shapes that satisfy these requirements [3]. In VAEs, designers can choose the shapes to generate by analyzing latent vectors [2], which is difficult in GAN. In this study, to enable latent engineering in GAN models, a variational autoencoder generative adversarial network (VAEGAN) model, a combination of VAE and GAN is used for inverse designs.

One approach to an inverse problem and optimization is to use a surrogate model [4,5]. The surrogate model is constructed using data, and it outputs the performance parameters from the input shapes in a short time. The authors in [6] used surrogate models for turbine cooling hole optimization problem. However, the number of design parameters was large, and it was difficult to use it for inverse problems. Hence, they conducted a principal component analysis to reduce dimensionality, but since PCA was necessary, the shape diversity was low and accuracy was low. Hence, when a certain shape indicates a certain performance requirement, designers can search from the surrogate models. The surrogate model is used for inverse problems and optimization in industrial applications such as turbine cooling holes [7]. Deep neural networks are also used as surrogate models to predict performance [8,9]. However, the shapes obtained from surrogate models are like those obtained from a training dataset, and completely different shapes cannot be obtained.

Deep generative models have recently been used to solve inverse problems. The GAN [10] is a widely used generative model. GAN-based inverse design is utilized in various industrial fields [11,12,13,14,15]. Apart from these industrial applications, the airfoil design task is an important benchmark problem for inverse design method [16,17,18]. The performance of the output shape sometimes differ from the required labels. The desired accuracy depends on each task. In [19], a flow machinery is designed using DNN as a regression model. In the literature, the error was less than 5 to 10%. In the GAN-based industry problem [15], the target accuracy is 5 to 10%. Those target accuracy is indeed not high. Hence, in real applications, the method is used to obtain good initial solutions for further numerical optimization. For example, in turbine blade optimization [20], it needs a good initial solution to start, but how to obtain such a solution is not trivial.

In order to improve accuracy, one approach is to couple physical models to NN models. The authors in [21] proposed physics-informed neural networks (PINNs) that use the residual of the physical equations in the loss functions. When PINNs are used, the physical model has to be implemented within the NN model to conduct backpropagation through the physical calculations. Therefore, general purpose software cannot be used with PINNs. However, in the inverse design, general purpose software is used in most cases. In the airfoil design benchmark problem, XFoil [22], the general purpose software for subsonic airfoils, is utilized, and hence, PINNs cannot be used. To overcome this limitation, refs. [23,24] proposed and utilized PG-GAN to the inverse design.

In airfoil designs, because the aim is to obtain shapes that satisfy specific aerodynamic requirements, it is important to obtain a smooth shape such that a numerical analysis, such as fluid analysis, can be conducted. For this purpose, a conditional GAN is used for the airfoil inverse design [25,26]. However, one of the issues with GAN models in airfoil generation is that the output shapes are not smooth, and fluid analysis is not applicable. Savitzky–Golay smoothing filters [27] are used in [26], but the filter changes the trailing edge. To overcome this issue, a conditional Wasserstein GAN with a gradient penalty (cWGAN-gp) [28,29] is employed, and it outputs a smoother airfoil than the ordinary GAN models as reported in [3]. The uncertainty of generated shapes is also evaluated [30]. However, GAN models generate data from random latent vectors, and the output shape is chosen randomly.

The VAE [31] is another generative model. It is used to generate three-dimensional models [32,33] and the airfoil [1]. The VAE is also used for airfoil generation tasks; ref. [2] reported that two different types of shapes are mixed when using VAE models, and such mixing is not possible in the GAN models. One of the differences between the GAN and VAE lies in the latent space. In the GAN model, the generator generates data from a random latent vector or noise vector, whereas in the VAE model, feature vectors are extracted from the training dataset, and data are embedded in the latent space. Hence, in the GAN model, the output shape is chosen randomly using the random noise vector.

VAEGAN [34] is a GAN-based neural network that uses a VAE model as its generator. The VAE part of the VAEGAN enables feature extraction from data, whereas the entire model is based on the GAN architecture. The VAEGAN was applied to the airfoil generation task in [25], but although the latent space of the VAEGAN is essentially different from that of the GAN, the latent space has not been discussed in airfoil generation tasks. In this study, the VAEGAN was coupled with cWGAN-gp to increase the smoothness of shapes, and it was then applied for the airfoil generation task. The latent space was compared with the cWGAN-gp model; from the comparison, the latent space is well-organized.

The remainder of this paper is organized as follows: Section 2 introduces the GAN, VAE, and VAEGAN models. The conditional VAEGAN model with the Wasserstein distance and gradient penalty for airfoil generation is proposed in Section 3. Numerical experiments are presented in Section 4, and conclusions are presented in Section 5. In order to follow the notation of existing literature, G and $f_w$ both represent the generator network, and D and $g_{\theta }$ represent the discriminator network.

2. Conditional GAN, VAE, and VAEGAN Models

2.1. Conditional GAN and WGAN-gp

GAN [10] consists of a generator network G and discriminator network D as illustrated in Figure 1. The generator generates data that mimic the training data, which are also called real data, and the discriminator distinguishes real data from fake data. The generator and discriminator minimize and maximize the loss function, respectively.

$$\begin{array}{cc}\hfill & \underset{G}{min}\underset{D}{max}{\mathcal{L}}_{GAN}(G,D).\hfill \end{array}$$

(1)

The loss function is defined as follows:

$$\begin{array}{cc}\hfill & {\mathcal{L}}_{GAN}(G,D)=E_{\mathit{x}\sim p_r\left(\mathit{x}\right)}\left[log\left(D\right(\mathit{x}\left)\right)\right]+E_{\mathit{z}\sim p_z\left(\mathit{z}\right)}\left[log\left(1-D\left(G\right(\mathit{z}\left)\right)\right)\right],\hfill \end{array}$$

(2)

where $\mathit{x}$ denotes real data, and $p_r\left(\mathit{x}\right)$ is the probability distribution of the real data. $\mathit{z}$ denotes a random noise vector sampled from probability distribution $p_z\left(\mathit{z}\right)$. G and D denote the generator and discriminator networks, respectively. $G\left(\mathit{z}\right)$ is the output from the generator, which is referred to as fake data.

Although many successful applications of GAN have been reported [35], an instability in training GAN was also reported [36,37]. WGAN-gp [29] is a solution for improving the stability. WGAN-gp uses the earth-mover’s (EM) distance instead of the Jensen–Shannon (JS) divergence in GAN. EM distance $W(·,·)$ is a distance between two probability distributions defined by

$$\begin{array}{cc}\hfill & W(p_r,p_g)=\underset{\gamma \sim \Pi (p_r,p_g)}{inf}{\mathbb{E}}_{\mathit{x},{\mathit{x}}^{\prime }\sim \gamma }[\parallel \mathit{x}-{\mathit{x}}^{\prime }\parallel ],\hfill \end{array}$$

(3)

where $p_r$ and $p_g$ are probability distributions of train data and generated data, respectively. Then, the loss function of WGAN-gp is

$$\begin{array}{c}\hfill {\mathcal{L}}_{WGANgp}(f_w,g_{\theta })=W(p_r,p_g)+\lambda {\mathbb{E}}_{\mathit{x}\sim p_r}\left[{\left(\parallel {\nabla }_{\mathit{x}}{f}_w\left(\mathit{x}\right)\parallel -1\right)}^2\right],\end{array}$$

(4)

where $f_w$ and $g_{\theta }$ represent the generator and discriminator, respectively.

2.2. Conditional VAE

The VAE [31] is an encoder–decoder-type deep neural network. The VAE model is shown in Figure 2. The encoder embeds features into the latent space, and the decoder reconstructs the data from the latent space:

$$\begin{array}{cc}\hfill & f_{enc}\left(\mathit{x}\right)={\left(\mu ,{\sigma }^{\mathbf{2}}\right)}^{\top },\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & \mathit{z}\sim N(\mu ,{\sigma }^{\mathbf{2}}),\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & f_{dec}\left(\mathit{z}\right)={\mathit{x}}^{\prime },\hfill \end{array}$$

(7)

where $\mathit{\mu }$ and ${\mathit{\sigma }}^{\mathbf{2}}$ are the mean and variance vectors of the latent space. The loss function of the VAE model is

$$\begin{array}{cc}\hfill {\mathcal{L}}_{VAE}& ={\mathcal{L}}_{\mathrm{llike}}+{\mathcal{L}}_{\mathrm{prior}}\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & ={\mathbb{E}}_{\mathit{x}}\left(\parallel \mathit{x}-{\mathit{x}}^{\prime }{\parallel }^2\right)+D_{\mathrm{KL}}\left(q\left(\mathit{z}\right|\mathit{x})\right|\left|p\left(\mathit{z}\right)\right),\hfill \end{array}$$

(9)

where $D_{KL}\left(q\left(\mathit{z}\right|\mathit{x})\right|\left|p\left(\mathit{z}\right)\right)$ is the Kullback–Leibler (KL) divergence between the distribution of samples in latent space $q\left(\mathit{z}\right|\mathit{x})$ and the prior $p\left(\mathit{z}\right)$. The standard normal distribution is used as prior $p\left(\mathit{z}\right)$.

2.3. Conditional VAEGAN

A VAEGAN couples the VAE and GAN models, and it consists of an encoder, decoder, and discriminator, as illustrated in Figure 3. The encoder extracts features from the data and embeds them into the latent space, whereas the decoder generates data from latent vectors. When generating new data in the VAEGAN, latent vectors are specified, and the decoder is processed from the latent vectors. The loss function of the VAEGAN model is

$$\begin{array}{c}\hfill {\mathcal{L}}_{\mathrm{VAEGAN}}={\mathcal{L}}_{\mathrm{llike}}^{\mathrm{Dis}}+{\mathcal{L}}_{\mathrm{GAN}}+{\mathcal{L}}_{\mathrm{prior}}.\end{array}$$

(10)

${\mathcal{L}}_{\mathrm{llike}}^{\mathrm{Dis}}$ denotes the reconstruction loss represented by the discriminator. ${\mathcal{L}}_{\mathrm{GAN}}$ and ${\mathcal{L}}_{\mathrm{prior}}$ are the GAN loss and prior loss defined in the GAN and VAE models, respectively.

3. CVAE -WGAN-gp Model for Airfoil Generation

3.1. Conditional VAE-WGAN-gp

VAEGAN is coupled with WGAN-gp to improve stability and is trained in a conditional manner. The loss function of VAE-WGAN-gp is expressed as

$$\begin{array}{c}\hfill {\mathcal{L}}_{\mathrm{VAE}\mathrm{WGAN}\mathrm{gp}}={\mathcal{L}}_{\mathrm{llike}}^{\mathrm{Dis}}+{\mathcal{L}}_{\mathrm{WGAN}\mathrm{gp}}+{\mathcal{L}}_{\mathrm{prior}}.\end{array}$$

(11)

The architecture of the model is the same as that in Figure 3. The training dataset was fed to the encoder with labels, and the data were embedded into the latent space. The prior of the latent space was employed as the standard normal distribution. The decoder reconstructs data from the latent vector and labels. The generated fake data, ${\mathit{x}}^{\prime }$, were fed into the discriminator with the label and training data. The discriminator distinguishes between real data and fake data. When generating new data, the latent vector is inputted into the decoder, and the decoder outputs new data.

The network architectures are shown in Figure 4. The encoder consisted of five layers with 512, 256, 128, and 64 nodes. The latent dimension was fixed to 4 because [1] reported that the sensitivity of the latent dimensionality is low for this airfoil design task. The decoder also consisted of five layers with 64, 128, 256, 512, and 496 nodes. The discriminator consisted of three layers with 512, 256, and 1 node. A leaky rectified linear unit (leaky ReLU) [38] was employed as the activation function. The dimension of the latent space was grid-searched, and 4 was chosen. The Adam optimizer was used, the learning rate was set as $0.0001$, and the model was trained for 50,000 epochs.

3.2. Airfoil Generation Framework

First, a training dataset was prepared. The dataset consisted of shape data and the lift coefficients $C_{\mathrm{L}}$, the corresponding performance index, is calculated. The data preparation procedure is explained in Section 4.1. The CVAE-WGAN-gp model was trained using this dataset. The lift coefficient $C_{\mathrm{L}}$ was used as label to be input into the model. Once the training was completed, the decoder was used to generate new data. The decoder inputs latent vectors and labels and outputs new data. When generating the data, latent vectors were sampled using a standard normal distribution. The aerodynamic performance of the generated data was evaluated. In the numerical experiments, XFoil [22] was used to evaluate airfoils. The aim of the inverse design is to obtain shapes that satisfy the required labels. However, the loss function of the CVAE-WGAN-gp model does not consider the reproduction of aerodynamic performance. Therefore, the reproduction of the aerodynamic performance is not guaranteed by the model. To evaluate the model, the error in aerodynamic performance must be evaluated. Adversarial attacks on the machine learning models are an important topic [39]. However, in this study, we assume that the dataset is under full control by a designer, and we do not consider adversarial attacks.

4. Numerical Experiments

4.1. Dataset

The airfoil dataset was constructed using the NACA 4-digit airfoil dataset [40]. The 4-digit airfoil is defined by three parameters: maximum camber, position of maximum camber, and maximum thickness normalized by chord length. Totally, 10,000 airfoil shapes are generated by the definition. The lift coefficients $C_{\mathrm{L}}$ are calculated using XFoil version 6.97 [22]. Among 10,000 shapes, the XFoil computation converges for only 3709 airfoils. Some airfoils indicate too high camber, which leads to unconvergence. The airfoil shape is represented by a set of points, where the number of points is 248 because the XFoil calculation requires more than 120 points. x and y coordinates of the points were assembled into one vector as $\mathit{x}=\left(x_1,x_2,\cdots ,x_{248},y_1,y_2,\cdots ,y_{248}\right)$ (Figure 5).

The histogram of the lift coefficients is shown in Figure 6. The histogram is close to the uniform distribution except that the frequency increases around $C_{\mathrm{L}}$∼0.5 and decreases at $C_{\mathrm{L}}$∼1.5. The coordinates $\mathit{x}$ and the label $C_{\mathrm{L}}$ are standardized before feeding into the neural networks.

4.2. Airfoil Generation

The computation is carried out on a machine manufactured by FUJITSU Client Computing Limited, Tokyo, Japan, equipped with Intel Core i7 CPU (2.8 GHz) with 32 GB memory. The wall clock time for training 50,000 epochs was 16,650 s, and less than 1 s for generating 100 shapes. The learning curve is shown in Figure 7. The proposed CVAE-WGAN-gp and conventional cWGAN-gp models were trained, and the generated shapes are shown in Figure 8. The red figures indicate that the XFoil calculations for the shapes did not converge, whereas the blue figures indicate otherwise.

Some of the generated shapes are similar to NACA airfoils. The examples of airfoils that are generated under the condition of $C_{\mathrm{L}}=0.5$ are shown in Figure 9. The figure shows a reasonable solution with respect to the aeronautical point of view, except that some shapes indicate zig-zag curves. Such zig-zag curves lead to failure of flow analysis. Some shapes are weird from an aeronautical point of view; e.g., some figures in the bottom of Figure 8d have protrusions that prevent smooth flow.

To evaluate the accuracy of shape generation quantitatively, ref. [3] used following indices. In this study, we also use the same indices:

• ${\varphi }_{\mathrm{mean}}$: Smoothness index.
• MSE: Mean squared error of $C_{\mathrm{L}}$.
• $\mu$: Index of variety of generated shapes.

The smoothness index ${\varphi }_{\mathrm{mean}}$ is the mean of the smoothness indices $\varphi$ defined for each shape. $\varphi$ is defined as

$$\begin{array}{cc}\hfill & {\mathit{v}}_k={\left({\mathit{x}}_{k+1}-{\mathit{x}}_k,{\mathit{y}}_{k+1}-{\mathit{y}}_k\right)}^{\top }\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & \varphi =\sum _{k=1}^{N}arccos\left({\displaystyle \frac{{{\mathit{v}}_k}^{\top }{\mathit{v}}_{k+1}}{\parallel {\mathit{v}}_k\parallel \parallel {\mathit{v}}_{k+1}\parallel }}\right)\hfill \end{array}$$

(13)

If a shape is a complete circle, $\varphi =2\pi$, and if a shape contains zig-zag lines, $\varphi$ indicates a larger value than $2\pi$. ${\varphi }_{\mathrm{mean}}$ indicates the smoothness of the generated shapes, and a smaller value indicates better smoothness. ${\varphi }_{\mathrm{mean}}$ of the training dataset is ${\varphi }_{\mathrm{mean}}=2.14\pi$. The number is slightly larger than $2\pi$ because of the camber line of the airfoils; the centerline of many airfoils is an upward convex curve.

The MSE is a quantitative index of the lift coefficient error. The MSE is defined as the mean of $\parallel C_{\mathrm{L}}^{\mathrm{r}}-C_{\mathrm{L}}^{\mathrm{l}}{\parallel }^2$ for all shapes whose XFoil calculation converges. This MSE is not included in the loss function because the reconstruction error in the loss function is the distance between the output shape and the training shape, whereas this MSE is the distance between the lift coefficients. $\mu$ is defined as the mean deviation of shapes from the mean shape, that is, $\parallel {\mathit{g}}_i-{\mathit{g}}_{\mathrm{mean}}\parallel$, where ${\mathit{g}}_i$ is a vector of a shape and ${\mathit{g}}_{\mathrm{mean}}$ is the mean of ${\mathit{g}}_i$.

The indices of the proposed CVAE-WGAN-gp model and other models are compared in

Table 1. Success rates and errors of generated shapes.

	${\mathit{\varphi }}_{\mathbf{mean}}\downarrow$	MSE ↓	$\mathit{\mu }\uparrow$
cGAN $(d=3)$	4.91$\pi$	0.047	0.152
cWGAN-gp $(d=3)$	$3.46\pi$	0.047	0.320
$\mathcal{N}$-CVAE [1]	3.95$\pi$	0.027	0.226
VAE-WGAN-gp $(d=4)$	3.50$\pi$	0.028	0.243

. cWGAN-gp and CVAE-WGAN-gp achieved better scores than the others with respect to ${\varphi }_{\mathrm{mean}}$. cWGAN-gp and CVAE indicated better scores only for one or two indices, that is, cWGAN-gp was better in ${\varphi }_{\mathrm{mean}}$ and $\mu$, and CVAE was better in MSE. However, CVAE-WGAN-gp showed good scores for all the three indices.

It is reasonable that the MSE of CVAE-WGAN-gp and CVAE is almost similar because both models use the loss between the training data and generated data, that is, ${\mathcal{L}}_{\mathrm{llike}}$ and ${\mathcal{L}}_{\mathrm{llike}}^{\mathrm{Dis}}$, which does not contain the loss of cWGAN-gp. Because loss ${\mathcal{L}}_{\mathrm{llike}}$ measures the direct distance between the training and generated data, the model can generate data similar to the training data, which leads to a lower MSE.

${\varphi }_{\mathrm{mean}}$ was smaller in the cWGAN-gp and CVAE-WGAN-gp models than in the CVAE model. A smaller ${\varphi }_{\mathrm{mean}}$ was obtained owing to the Wasserstein distance, as shown in the difference between cGAN and cWGAN-gp. The discriminator measures the distance between the training data and generated data using the Wasserstein distance.

$\mu$ was significantly large in cWGAN-gp, and CVAE-WGAN-gp followed cWGAN-gp. In cWGAN-gp, some of the generated shapes were wired, as shown in Figure 8. The lift coefficients cannot be calculated for these wired shapes, as illustrated in red in Figure 8, which leads to a larger MSE value. CVAE-WGAN-gp indicated a larger $\mu$ than CVAE, whereas the MSEs of both models were almost equal.

It has been already reported that VAEGAN outperforms the GAN model [34], which is consistent with this study. CVAE-WGAN-gp exhibited good properties of both CVAE and cWGAN-gp, owing to its architecture and loss functions.

4.3. Latent Distribution

The latent distributions of cWGAN-gp and CVAE-WGAN-gp are shown in Figure 10. When discussing the latent space of GAN, Achour et al. [26] used the t-SNE [41] to reduce the latent dimension so that the latent space is visualized. We used the same t-SNE to visualize the latent space of CVAE-WGAN-gp. We note that t-SNE has some hyperparameters, and by changing the hyperparameters, the visualization result sometimes drastically changes. In this manuscript, the latent space with t-SNE is provided to convey the image of the difference in latent space. In the latent space of WGAN-gp, data with both large and small $C_{\mathrm{L}}$s are randomly distributed throughout the space. On the other hand, the latent space of CVAE-WGAN-gp has a clumped distribution of those with small $C_{\mathrm{L}}$s, and the small and large data are separated. This difference is due to the presence of the encoder. Let $\mathit{x}$ and ${\mathit{x}}^{\prime }$ be latent variables that satisfy $\parallel \mathit{x}-{\mathit{x}}^{\prime }\parallel \le \epsilon \}$, c and $c^{\prime }$ be the labels, and functions f and $f_{\mathrm{VAEGAN}}$ be the generator of WGAN-gp and CVAE-WGAN-gp, respectively. Then, the generated shapes are written as $f(\mathit{x},c)$ and $f({\mathit{x}}^{\prime },c^{\prime })$. In the case of WGAN-gp, if we focus on a point in the latent space, both data with large CL and data with small CL exist around that point, which are generally significantly different. It implies that $\parallel f(\mathit{x},c)-f({\mathit{x}}^{\prime },c^{\prime })\parallel$ is significantly large. In the case of CVAE-WGAN-gp, $C_{\mathrm{L}}$ of the generated shapes from nearby two latent variables is close. It implies that $\parallel f_{\mathrm{VAEGAN}}(\mathit{x},c)-f_{\mathrm{VAEGAN}}({\mathit{x}}^{\prime },c^{\prime })\parallel$ is expected to be smaller than $\parallel f(\mathit{x},c)-f({\mathit{x}}^{\prime },c^{\prime })\parallel$. Hence,

$$\begin{array}{c}\hfill {\displaystyle \frac{\parallel f(\mathit{x},c)-f({\mathit{x}}^{\prime },c)\parallel }{\parallel \mathit{x}-{\mathit{x}}^{\prime }\parallel }}={\displaystyle \frac{\parallel f(\mathit{x},c)-f({\mathit{x}}^{\prime },c^{\prime })\parallel }{\parallel \mathit{x}-{\mathit{x}}^{\prime }\parallel }}{\displaystyle \frac{\parallel f(\mathit{x},c)-f({\mathit{x}}^{\prime },c)\parallel }{\parallel f(\mathit{x},c)-f({\mathit{x}}^{\prime },c^{\prime })\parallel }}\le 1,\end{array}$$

holds, and consequently,

$$\begin{array}{c}\hfill {\displaystyle \frac{\parallel f(\mathit{x},c)-f({\mathit{x}}^{\prime },c^{\prime })\parallel }{\parallel \mathit{x}-{\mathit{x}}^{\prime }\parallel }}\le {\displaystyle \frac{\parallel f(\mathit{x},c)-f({\mathit{x}}^{\prime },c^{\prime })\parallel }{\parallel f(\mathit{x},c)-f({\mathit{x}}^{\prime },c)\parallel }}\end{array}$$

holds. The latent space distributions are quite different between WGAN-gp and CVAE-WGAN-gp. This difference arises from the encoder part of CVAE-WGAN-gp.

The generator or decoder generates new data from latent data. In cWGAN-gp, the generator generates data with both high $C_{\mathrm{L}}$ and low $C_{\mathrm{L}}$ from a similar latent variable. Because the label is also an input to the generator as well as the latent variable, the generator can output different shapes from the same latent variable. However, it is desirable if different data are located in different areas of the latent space. In the CVAE-WGAN-gp model, the decoder can generate different data from different latent variables, which results in a lower MSE than cWGAN-gp.

5. Conclusions

This study proposed CVAE-WGAN-gp, a model that generates an airfoil that satisfies required lift coefficients. The proposed model showed a better performance in terms of smoothness, reproduction of lift coefficients, and shape varieties, whereas CVAE and cWGAN-gp exhibited some but not all these three desired properties. The proposed model has properties of both the CVAE and cWGAN-gp models. The latent space was compared with both the proposed and cWGAN-gp models, and it was shown that the encoder of the CVAE-WGAN-gp model embeds data in the latent space, and the latent distributions are different from that of WGAN-gp.A better MSE is explained by the neatness of the latent space. The fact that different data are embedded in different latent areas helps the decoder to generate accurate data from the latent space. The hyper parameters have to be properly chosen to obtain better results.