Fast Prediction of Airfoil Aerodynamic Characteristics Based on a Combined Autoencoder

Abstract

Aircraft airfoils are classified into two main categories: symmetrical and asymmetrical. Both types of airfoils have a significant impact on the flight performance and safety of the aircraft. The fast prediction of the aerodynamic coefficients and pressure distributions of airfoils is crucial for the design of aircraft. The traditional wind tunnel test and CFD methods have the disadvantages of high test cost and high time consumption. To solve these problems, a combined autoencoder (CAE) network is proposed in this paper, which can achieve the fast prediction of airfoil aerodynamic coefficients and pressure distributions. The network consists of an airfoil shape autoencoder (AE) network and a multilayer perceptron (MLP) network. Firstly, an autoencoder network reflecting the characteristics of the airfoil shape is established, and the effects of different latent variables on the performance of the autoencoder network are investigated. Then, the latent variables obtained from the autoencoder are concatenated with the inflow conditions such as the Reynolds number and the angle of attack to be used as inputs to the MLP network, and the aerodynamic coefficients of different airfoils in different inflow conditions are predicted. The effects of various latent variable inputs, as well as the direct input of the airfoil shape into the MLP network, on the prediction performance of aerodynamic coefficients are compared and analyzed. The optimal aerodynamic coefficient prediction network is then obtained. Finally, the CAE network is also applied to predict the pressure distributions of different airfoils in different inflow conditions and the effects of different latent variables and input conditions on the prediction performance of the pressure distributions are analyzed and compared with the advantages and disadvantages of the CAE network and the conditional variational autoencoder (CVAE) network. The results demonstrate that the proposed method is capable of accurately predicting aerodynamic characteristics in a shorter time, offering a valuable reference for the fast and efficient design of aircraft airfoils.

1. Introduction

Most aircraft are symmetrical with respect to the longitudinal axis in the body frame of reference. Airfoils, one of the most important aerodynamic components of aircraft, are typically classified into two categories: symmetrical and asymmetrical. In general, the vertical and horizontal tails, employ symmetrical airfoils, whereas the wings utilize asymmetrical airfoils. During the preliminary design stage of aircraft, the aerodynamic characteristics of an airfoil, which includes aerodynamic coefficients and pressure distribution, has a direct impact on the flight performance, maneuverability, and stability of the aircraft [1]. Traditionally, the aerodynamic characteristics are derived from computational fluid dynamics (CFD) or wind tunnel tests. However, CFD and wind tunnel tests are time-consuming and resource-intensive.

To reduce the costs, lots of surrogate modelling methods were studied to predict the aerodynamic characteristics of airfoils. These methods can be broadly categorized into two groups: traditional surrogate models and neural network models. Traditional surrogate models, such as polynomial response surface [2], kriging [3], and support vector regression (SVR) [4], typically involve fitting a prediction function between the airfoil shape and the aerodynamic characteristics. However, traditional surrogate models are unable to handle a large volume of training data [5]. One effective approach to address these shortcomings is to utilize neural network models, such as multilayer perceptron (MLP), convolutional neural network (CNN), variational autoencoder (VAE), and generative adversarial network (GAN). These models are trained to learn the mapping relationship between airfoil shape and aerodynamic characteristics. Wallach [6] and Santos [7] used MLP to predict polar curves and drag coefficient of the NACA23012 airfoil at different inflow conditions. Bertrand [8] used CNN and Conditional variational autoencoder (CVAE) to predict the pressure distribution of the RAE2822 airfoil. Notably, their studies concentrate on a single airfoil, thereby limiting their broader applicability.

In order to predict the aerodynamic characteristics of different airfoils, the airfoil shape information needs to be used as an input to the neural network. Currently, the airfoil shape can be input into the neural network mainly by airfoil parameterization, airfoil images, and airfoil coordinates.

For airfoil parameterization, common methods include the NACA airfoil definition [9], which uses a series of digits (4 to 7) to describe the airfoil shape; the Joukowski transformation [10], which maps a circle to an airfoil shape; the PARSEC method [11], which parameterizes the airfoil surface using a set of polynomial coefficients; Hicks-Henne bump functions [12], which modify the airfoil surface with local shape changes; class shape transformation (CST) [13], which represents the airfoil shape using a combination of basic shapes and polynomial terms; and free-form deformation (FFD) [14], which allows flexible manipulation of the airfoil geometry using a lattice of control points; singular value decomposition (SVD), which uses proper orthogonal decomposition to derive a set of ordered, orthogonal basis modes from a set of predetermined training airfoils. Masters [15] compared these methods by considering the geometric shape recovery of over 2000 airfoils. From the result, they concluded that as the number of design variables increases, the performance capability of all these methods increases, and when the number of design variables reaches 30, the vast majority of methods are able to fully represent the UIUC airfoils, but when the number of design parameters is less than 30, there is a large difference in the performance capability among the methods, with the SVD method having the best performance capability. Subsequently, based on these airfoil parameterization methods, the researchers carried out aerodynamic performance prediction studies for different airfoils. Bouhlel [16] adopted the SVD method to parameterize airfoil, and used ANN to predict the lift, drag, and moment coefficients of arbitrary airfoils, found that the ANN model outperformed the mixture of kriging models. Li [17] also used SVD method to parameterize airfoil shape, and used surrogate models provide fast aerodynamic analysis and gradient computation. Du [18] used B-Spline to parameterize the airfoil and built a GAN based on B-Spline, realized the rapid airfoil aerodynamic prediction and design optimization. Liu [19] modelled the nonlinear relationship between airfoil shape and aerodynamic characteristics using GP regression, and obtained the lift, drag, and moment coefficients of NACA four-digit airfoils at a fixed inflow condition. Qian [20] used support vector machine (SVM) and deep neural network (DNN) to model the relationship between the aerodynamic characteristics and the CST parameters at a fixed inflow condition. Yonekura [21] used the S-CVAE and N-CVAE to realize the inverse design of the NACA and Joukowski airfoils. These airfoil parameterization methods essentially control the airfoil shape through a set of control parameters or mathematical equations, which can give a generalized representation of the airfoil shape, but the accuracy of the representation, especially for those airfoils with strange shapes, is low. Additionally, these approaches also have limited geometric freedom due to the involvement of a restricted number of design variables [5].

Therefore, some researchers transformed the shape of airfoils into binary images and used CNN to extract features from these images to predict the aerodynamic characteristics of different airfoils. The binary image is capable of characterizing all airfoil shapes with proper resolution. Sekar [22] used CNN to extract the geometric parameters from airfoil shapes. Then, the extracted geometric parameters along with Reynolds number and angle of attack are fed as input to the MLP network to obtain an approximate model to predict the flow field. Bublik [23] and Bhatnagar [24] presented CNN to quickly predict the pressure fields around the airfoil, taking the initial flow fields as input. However, since the input size is large, their work cannot handle large flow fields due to memory limitations. In addition, their work did not consider the influence of inflow conditions. These methods described above mainly use CNN networks to extract airfoil shape features and then predict the flow field, essentially creating surrogate models instead of CFD simulations that can indirectly obtain the airfoil aerodynamic characteristics. Yilmaz [25] studied a classifier based on CNN to predict airfoil performance. Yu [26] used an improved CNN network to predict airfoil lift coefficient. Zhang [27] used the MLP and CNN to predict the lift coefficients of different airfoils at a fixed inflow condition. Chen [28] also adopted CNN to regress airfoil coefficients (lift, drag, and moment) of different airfoils at a fixed inflow condition. Nevertheless, Chen and Zhang’s work only examined the correlation between airfoils and aerodynamic coefficients under fixed inflow conditions, which has significant limitations. Zhao [29] established a mapping relationship between iced airfoils and aerodynamic coefficients using a deep operator network that employs CNN to extract the shape features of the iced airfoils. Although CNN is adept at image processing and can handle irregular airfoil shapes, the thickness of the airfoil’s contour line and the image resolution can influence the representation of the airfoil, potentially compromising the accuracy of the aerodynamic characteristic prediction model. In addition, there is a lack of further verification as to whether the airfoil image features extracted by CNN can effectively represent the airfoil shape.

Other researchers have used airfoil coordinates to represent its shape, which can accurately depict the airfoil by controlling the number of points. Using coordinate points to represent airfoil shapes offers precise geometric accuracy, versatility for complex shapes, and simplicity in data format. This method allows direct use in numerical simulations such as CFD without the need for geometric conversion and ensures broad compatibility with most simulation software and CAD tools, making it a highly efficient and reliable choice for engineering applications. Wang [30] used the VAE network to extract latent features of pressure distribution and established a mapping relationship between airfoil coordinates and the latent features using the MLP network. Wang [1] proposed a method for predicting the pressure distributions of an airfoil using conditional generative adversarial network (cGAN) with 199 points to present the airfoil. Ma [31] used airfoil coordinates (include 301 points) concatenated with inflow conditions to predict the corresponding pressure distribution based on MLP. Employing coordinates allows for highly accurate representations. However, the number of parameters representing the airfoil shape among the input parameters of the neural network utilized by the aforementioned researchers is considerably greater than the number of parameters representing the inflow conditions. This may hinder neural networks from identifying the primary relationship between airfoil shape, inflow conditions and aerodynamic characteristics. As a result, their networks will tend to learn more about the influence of the airfoil shape on the aerodynamic coefficients and weaken the influence of the inflow conditions on the aerodynamic coefficients, thus decreasing the prediction accuracy.

In terms of aerodynamic coefficients prediction, previous studies have mainly focused on predicting the coefficients for individual airfoil under varying inflow conditions or for different airfoils at a fixed inflow condition. However, few studies have investigated the combined effects of changes in airfoil shape and inflow conditions. Additionally, accurately representing the airfoil shape may require a large number of points, which can make it difficult for a neural network to capture the main relationship between the airfoil shape and the aerodynamic characteristics. For predicting pressure distribution, previous studies have commonly used neural networks that take the airfoil shape concatenated with inflow conditions (including angle of attack, Mach number, and Reynolds number) as inputs, and the pressure distribution as output. For a specific airfoil, with only three inflow condition parameters as inputs, and the number of output parameters reaching hundreds, this could potentially lead to generate new information during prediction, which will result in inaccurate prediction.

In this paper, a combined autoencoder network (CAE) is designed to predict aerodynamic coefficients and pressure distribution. First, we establish a database encompassing the lift coefficient, drag coefficient, moment coefficient, and pressure distribution for 1499 airfoils from the UIUC airfoil database at various Reynolds numbers and angles of attack. Secondly, an autoencoder (AE) is used to extract the principal features of airfoil coordinates, thereby reducing the number of parameters needed to represent the airfoil shape. Then, two MLPs are used to establish the relationship between the extracted features, inflow conditions, and the aerodynamic characteristics.

This study aims to conserve computational resources and accelerate the aircraft design process. Our novelty lies in three aspects. First, we introduce a novel network capable of predicting the aerodynamic coefficients and pressure distribution for various airfoils under different inflow conditions. Second, to accurately represent the airfoil shape while reducing the number of input parameters, an AE is employed to extract the principal features of airfoil coordinates, which has better airfoil feature extraction capabilities than CNN. Finally, the lift, drag and moment coefficients rather than the inflow conditions, are used as inputs to predict the pressure distribution. This approach captures a more intrinsic and comprehensive relationship, resulting in more precise predictions.

Our paper is organized as follows. Section 2 introduces the methodology in detail. Section 3 presents and analyzes the experimental results. Finally, we conclude our work in Section 4.

2. Methodology

In this paper, we propose a CAE network to predict airfoil coefficients and pressure distributions. Our method has two main components: data preparation and model training. During the data preparation stage, we filter the UIUC airfoils to create the UIUC airfoil database. Subsequently, we employ Xfoil to calculate the aerodynamic coefficients and pressure distributions of the airfoils, resulting in an airfoil aerodynamics dataset for subsequent training and testing of the model. During the model training stage, we train and test the CAE which consists of an airfoil shape autoencoder (AE) network and two multilayer perceptron (MLP) networks. The AE is used to extract the airfoil shape features, which are then concatenated with the inflow conditions and fed into an MLP to predict airfoil coefficients. Alternatively, these extracted features can be concatenated with the airfoil coefficients and fed into another MLP to predict the pressure distribution. The research flowchart of this paper is shown in Figure 1.

2.1. Data Preparation

2.1.1. Airfoil Shape Dataset

This study uses the University of Illinois UIUC airfoil database [32], which contains a total of 1499 airfoils. Each airfoil is depicted by a total of 199 points distributed on its upper and lower surfaces. Notably, the density of these points is incremented near the critical regions of the leading and trailing edges. These points are arranged in the following order: trailing edge, lower surface, leading edge, upper surface, and trailing edge. To achieve consistency across the dataset, a spline interpolation method is implemented, ensuring that all airfoils share identical x-coordinate values. Consequently, this approach allows for the utilization of only the y-coordinates in the training process. This not only reduces the complexity of the input and output dimensions for the neural network but also significantly enhances the efficiency of the training procedure.

2.1.2. Airfoil Aerodynamic Characteristics Dataset

In this study, Xfoil [33] is utilized to determine aerodynamic coefficients and pressure distribution at various Reynolds numbers and angles of attack. Approximately 130,000 samples were obtained by calculating the aerodynamic characteristics of 1499 airfoils at 4 Reynolds numbers and 26 angles of attack, excluding anomalous data. Equation (1) shows the detail of Reynolds numbers and angles of attack. Figure 2 displays the UIUC airfoils and their corresponding pressure distributions. In the figure, ‘x/c’ represents the normalized chord-wise position along the airfoil, where ‘x’ is the position and ‘c’ is the chord length, ‘y’ is the vertical displacement from the chord line, and ‘$C_p$’ is the pressure coefficient.

$$\left\{\begin{array}{c}Re\in \{{10}^4,{10}^5,{10}^6,{10}^7\}\\ \alpha \in \left\{-10°,-9°,-8°,-7°,-6°,-5°,-4°,-3°,-2°,-1°,0°,1°,2°,3°,4°,5°,6°,7°,8°,9°,10°,11°,12°,13°,14°,15°\right\}\end{array}\right.$$

(1)

After obtaining the dataset, we use min-max normalization to scale the inflow conditions, the aerodynamic coefficients (lift, drag, and moment coefficients) and pressure distributions to fall within the range of 0 to 1. The normalization preprocessing enhances the prediction performance and accelerates the training process. Specifically, the inflow conditions, the aerodynamic coefficients and the pressure distribution values are transformed using the min-max normalization formula:

$$X_{normalized}=\frac{X-X_{min}}{X_{max}-X_{min}}$$

(2)

where $X$ is the original value, $X_{min}$ is the minimum value in the dataset, and $X_{max}$ is the maximum value in the dataset.

2.2. The Combined Autoencoder (CAE) Network

This paper proposes a novel neural network, termed the combined autoencoder (CAE) network, designed to predict both aerodynamic coefficients and pressure distribution. The architecture of the CAE network consists of two main components: an AE and two MLPs.

2.2.1. Autoencoder (AE)

The AE is utilized for dimensionality reduction and feature extraction of the airfoil shapes. It comprises an encoder and a decoder. The encoder compresses the high-dimensional input data (199 y-coordinates of the airfoil points) into a low-dimensional latent space, capturing the principal features of the airfoil shape. The decoder reconstructs the original airfoil shape from these latent features.

Encoder: The encoder includes two hidden layers with 100 and 50 neurons, respectively. It transforms the input y-coordinates into a lower-dimensional latent space.

Latent Dimension: The latent dimension can be varied (e.g., 5, 10, 20, 40) to analyze the effect on the performance of the network.

Decoder: The decoder mirrors the encoder’s structure with two hidden layers of 50 and 100 neurons, respectively, and reconstructs the 199 y-coordinates of the airfoil shape.

2.2.2. Multilayer Perceptrons (MLPs)

The CAE network uses two separate MLPs for predicting aerodynamic coefficients and pressure distributions. The extracted features from the AE, along with the inflow conditions (Reynolds number and angle of attack) or aerodynamic coefficients, are used as inputs to these MLPs.

With regard to the MLP for aerodynamic coefficients prediction, this MLP predicts the lift, drag, and moment coefficients based on the latent features and inflow conditions. It consist of:

Input Layer: The number of neurons in the input layer varies depending on the latent dimension of AE (e.g., 7, 12, 22, 42).

Hidden Layers: Nine hidden layers, each containing 200 neurons, capture the complex relationships between the inputs and the aerodynamic coefficients.

Output Layer: Three neurons correspond to the lift, drag, and moment coefficients.

With regard to the MLP for pressure distribution prediction, this MLP predicts the pressure distribution over the airfoil surface using the latent features and the predicted aerodynamic coefficients. It also consists of:

Input Layer: The number of neurons in the input layer also varies based on the latent dimension of AE (e.g., 8, 13, 23, 43).

Hidden Layers: Nine hidden layers, each with 400 neurons, model the intricate dependencies between the inputs and the pressure distribution.

Output Layer: 199 neurons represent the pressure distribution across the airfoil surface points.

The architecture of the CAE network is illustrated in Figure 3.

The loss function of the CAE network consists of three components: the reconstruction loss of the airfoil shape, the prediction loss of pressure distribution, and the prediction loss of aerodynamic coefficients. We use Mean Squared Error (MSE) to measure the loss between the predicted values and the true values. Equation (3) shows the entire loss function.

$$\begin{array}{cc}\hfill Loss& =MSE\left(\widehat{y},y\right)+MSE\left({\widehat{C}}_p,C_p\right)+MSE\left(\widehat{C},C\right)\hfill \\ & =\frac{1}{N \times 199}{\displaystyle {\displaystyle \sum }_{i=1}^N}{\displaystyle {\displaystyle \sum }_{j=1}^{199}}[{\left({\widehat{y}}_{ij}-y_{ij}\right)}^2+{\left({\widehat{C}}_{P_{ij}}-C_{p_{ij}}\right)}^2]+\frac{1}{N}{\displaystyle {\displaystyle \sum }_{i=1}^N}{\displaystyle {\displaystyle \sum }_{j=1}^3}{\left({\widehat{C}}_{ij}-C_{ij}\right)}^2\hfill \end{array}$$

(3)

where $\widehat{y}$ is the predicted value of airfoil coordinates, and $y$ is the true value of airfoil coordinates, and $\widehat{C}$ is the predicted value of aerodynamic coefficients, $C$ is the true value of aerodynamic coefficients, ${\widehat{C}}_p$ is the predicted value of pressure coefficient, and $C_p$ is the true value of pressure coefficient. Meanwhile, $N$ is the number of airfoils, 199 is the number of points on an airfoil or pressure distribution, and 3 indicates the number of aerodynamic coefficients.

3. Results and Discussion

The CAE network is implemented by PyTorch. We randomly select 90% of the dataset as the training set, and the remaining 10% as the test set. To ensure consistent division of the training and test sets for each iteration, we set the same random seed. The loss function, as presented in Equation (3), is optimized using the Adam optimizer with an initial learning rate of 0.001. To accelerate the training process, the learning rate is multiplied by 0.5 every 50 epochs. The activation function used is ReLU. The network is trained with a fixed batch size of 256.

3.1. Validation of Aerodynamic Characteristics

Firstly, the fidelity of the aerodynamic characteristics data generated by Xfoil is analyzed. Figure 4 shows a comparison of experimental data obtained from [34] and Xfoil data for NACA 0012 at $Re=1.5\times {10}^6$ and $Ma=0.126$. In the figure, ‘alpha’ is the angle of attack, ‘Cl’ and ‘Cd’ are the lift and drag coefficients of airfoil, respectively. The figure shows that the Xfoil data closely corresponds to the experimental data for most angles of attack in terms of aerodynamic coefficients, except for some angles of attack where the airfoil stalls. Furthermore, the pressure distribution also aligns with the experimental data at $\alpha =5°$.

3.2. Airfoil Shape Feature Extraction

Figure 5 shows the convergence histories of the loss function and Figure 6 shows the MSE distribution statistics for different latent dimensions after ${10}^5$ epochs. In Figure 5, the horizontal coordinate is the epochs of training, and the vertical coordinate is the training Mean Square Error (MSE). In Figure 6, ‘count’ denotes the number of airfoils with different MSE errors. The MSE of the training and testing dataset of each latent variable network after training are shown in

Table 1. The MSE of the training and test sets.

	Latent 5	Latent 10	Latent 20	Latent 40
Train MSE	$3.16079\times {10}^{-6}$	$3.18427\times {10}^{-7}$	$1.26265\times {10}^{-7}$	$1.32642\times {10}^{-7}$
Test MSE	$3.27149\times {10}^{-6}$	$3.29071\times {10}^{-7}$	$1.27878\times {10}^{-7}$	$1.47454\times {10}^{-7}$

. It is clear that the loss for each latent dimension decreases throughout training, stabilizing below a specific threshold. The MSE reaches its minimum when the latent dimension is set to 20, indicating that the AE exhibits the least error in reconstructing the airfoil shape at this dimension. In this case, the histograms tend to cluster around the y-axis, with the majority of MSE falling below $2\times {10}^{-7}$. Figure 7 presents a comparison between the reconstructed airfoils and their original counterparts, using a random subset of 16 airfoils from the test set. Since most of the aerodynamic lift comes from the first quarter of the chordwise location, the leading edges of the airfoils were zoomed in to show local details between the reconstructed airfoils and their original. The figure shows that the AE, with a latent dimension of 20, effectively captures the shape features of the airfoils, and has good reconstruction ability.

3.3. Prediction of Aerodynamic Coefficients

To verify the performance of the CAE, we built another MLP network. The MLP inputs comprise the airfoil y-coordinates without extraction, the Reynolds number, and angle of attack, and consist of 201 neurons. The remaining parameter settings are the same as those of the MLP part of the CAE to predict the aerodynamic coefficients.

To reduce the impact of random initial values of the network on the prediction results, the preceding training set was used to train the CAE 10 times independently. Additionally, we conducted a comparative analysis of the training processes for CAEs with different latent dimensions, which included settings of 5, 10, 20, and 40 dimensions, respectively. In subsequent sections, CAE5, CAE10, CAE20 and CAE40 are employed to refer to them respectively.

Figure 8 shows the boxplots of MSE for both the training set and the test set after 10 training sessions. Based on the results, we can compare the effect of different latent dimensions of the AE on the aerodynamic coefficients. The figure shows that as the latent dimension increases up to 20, both the training and test MSEs generally decrease. The CAE20 exhibits the lowest training and test errors, in contrast, the CAE40 presents the highest training and test errors among the compared models. Furthermore, both the CAE5 and CAE40 have higher test errors compared to the MLP. This could be due to the fact that when the latent dimension is small, the airfoil shape features is difficult to be represented comprehensively. On the other hand, when the latent dimension is large, such as in CAE40, the network has a high capacity to capture a wide range of features. However, this increased capacity also means that the model can start to capture noise and irrelevant patterns in the data, leading to the extraction of spurious correlations. These spurious correlations do not represent meaningful relationships within the airfoil data but rather are artifacts of the increased dimensionality and noise in the dataset. Moreover, when the latent dimension is considerably larger than other input parameters (such as inflow conditions), the networks will tend to learn more about the influence of airfoil shape on aerodynamic coefficients and to weaken the influence of other input parameters on aerodynamic coefficients [1]. Consequently, the model’s predictions become less accurate, as evidenced by the higher prediction errors. Additionally, the figure shows that the prediction results of the CAE network exhibit greater stability across multiple trainings, indicating that the results are more consistent and have a smaller variance. In contrast, the results of the MLP network trained for 10 times shows a large variance, suggesting a lack of training stability. To sum up, the CAE network with 20 latent dimension (CAE20) presents superior performance in predicting aerodynamic coefficients.

Subsequently, we focus on evaluating the predictive accuracy of various models for specific aerodynamic coefficients. For each model, we select the training results that are closest to the mean of the test set and compare their effects on the three aerodynamic coefficients: the lift, drag, and moment coefficients. Figure 9 illustrates the statistical histograms of the prediction errors for each aerodynamic coefficient of the different models in the test set.

Table 2. Test set MSE for $C_L,C_D,C_M$ in different aerodynamic coefficient prediction models (shown as percentages relative to CAE5).

	CAE5	CAE10	CAE20	CAE40	MLP
$C_L$	100	77.27	70.83	106.06	89.39
$C_D$	100	82.78	70.15	78.21	77.77
$C_M$	100	78.42	66.33	110.48	86.89

presents the MSE for $C_L,C_D,C_M$ in different aerodynamic coefficient prediction models. The MSE values are shown as a percentage relative to the CAE5 model, which is set as 100%. In the figure, each row presents $C_L,C_D,C_M$ respectively, while each column presents different models. Each subfigure presents the distribution statistics of the models in predicting the respective aerodynamic coefficients. From the figure, it can be observed that the histogram for CAE20 exhibits a concentration of MSE values at the lower end of the scale, with the highest peaks occurring close to zero MSE for all three coefficients compared to the other models. This concentration and the peaks close to zero indicate that CAE20 consistently produces lower MSE values, demonstrating better prediction accuracy. It is clear from the figure and table that the CAE20 has the lowest prediction error among the three aerodynamic coefficients.

3.4. Prediction of Airfoil Pressure Distribution

To verify the performance of CAE for predicting the pressure distribution, we employed a CVAE model based on the work of [30]. The CVAE model takes the pressure distribution as its input and output, and the conditions of CVAE are airfoil shape features, extracted from AE, lift, drag, moment coefficient or Reynold number, angle of attack. The AE part of both CVAE and CAE share the same structure and parameters. The terms CVAE5, CVAE10, CVAE20, and CVAE40 indicate that the AE part of CVAE has latent dimension of 5, 10, 20, and 40, respectively. The CVAE consists of an encoder and a decoder, each with a single hidden layer containing 100 neurons. The CVAE has a latent dimension of 40. The remaining part of the network are identical to those of the CAE. Figure 10 illustrates the architecture of the CVAE network.

The CAE models and the CVAE models were trained from random initialization for 10 times to evaluate the performance. Figure 11 shows the boxplots of MSE for both the training and test sets for the CAE and the CVAE models. We can see from the figure that when the input conditions of the network are airfoil shape features and three aerodynamic coefficients (Input 1), the prediction accuracy of the CAE network is approximately 1 order of magnitude higher than that of the CVAE network. In addition, for the CAE network, setting the latent dimension to 20 provides optimal accuracy in both the training and test sets. When the input conditions consist of airfoil shape features, the Reynolds number, and angle of attack (Input 2), the CAE network still demonstrates higher prediction accuracy. For the same network, CAE or CVAE, Input 1 has greater precision and less dispersion than Input 2. One possible reason is that there is an integral relationship between aerodynamic coefficients and pressure distribution that the network has learned, and this relationship has better robustness than the mapping relationship between inflow conditions and pressure distribution.

The MSE distributions of CAE20 and CVAE10 with two types of inputs on the test set are shown in Figure 12. The figure shows that the CAE network produces less error in predicting the pressure distribution than the CVAE network after reconstruction. In addition, Input1 produces less error than Input2. Therefore, the pressure distribution prediction network exhibits optimal predictive capabilities when using a 20 hidden dimension network with extracted airfoil features and three aerodynamic coefficients as inputs. Figure 13 shows a comparison of the prediction performance of different models for 12 randomly selected pressure distribution results from the test set under different input conditions. From the figure, CAE20 has the most accurate prediction of the true value for the airfoil pressure distribution when the input is Input1, making it the most effective model.

4. Conclusions

This paper proposes a CAE network composed of an AE and two MLPs to predict the aerodynamic coefficients and pressure distribution of airfoils. To accomplish this, the AE is used to extract the shape features of the airfoils, and MLPs are used to establish the relationship between the extracted features, and the aerodynamic characteristics. Next, we evaluate the performance of the aerodynamic coefficient predictions between CAE and MLP. Finally, we conduct a comparative analysis of the predictive results of the pressure distribution between the CAE and the CVAE under various input conditions. Based on these evaluations, the following conclusions can be drawn.

Insufficient latent dimension of the AE part may fail to capture the comprehensive features of an airfoil, whereas an excessive number may introduce spurious correlations, both scenarios contributing to an augmented reconstruction error of the airfoil shape. Therefore, it is necessary to determine an appropriate number of latent dimensions to represent the shape of the airfoil. Within the context of this study, the selection of 20 latent dimension yields the minimal reconstruction error, suggesting an effective balance in capturing the essential features of the airfoil shape without introducing unnecessary complexity.

In predicting airfoil aerodynamic coefficients, using an autoencoder to reduce the dimensionality of the airfoil coordinates can improve prediction accuracy to some extent. The highest prediction accuracy is achieved when the latent dimension of AE is 20.

In predicting the pressure distribution, this paper designs CAE networks incorporating two types of inputs: one consisting of the inflow conditions and the airfoil shape features, and the other comprising the aerodynamic coefficients and the airfoil shape features, as well as CVAE networks with both types of inputs. Overall, under the same input conditions, the CAE network exhibits superior accuracy in predicting pressure distribution compared to the CVAE network. Moreover, when aerodynamic coefficients and airfoil shape features are used as inputs, the prediction accuracy of the pressure distribution is significantly improved compared to using the inflow conditions and the airfoil features as inputs.