 Research
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Deep learning for inverse design of lowboom supersonic configurations
Advances in Aerodynamics volume 5, Article number: 13 (2023)
Abstract
Mitigating the sonic boom to an acceptable stage is crucial for the next generation of supersonic transports. The primary way to suppress sonic booms is to develop a low sonic boom aerodynamic shape design. This paper proposes an inverse design approach to optimize the nearfield signature of an aircraft, making it close to the shaped ideal ground signature after propagation in the atmosphere. By introducing the Deep Neural Network (DNN) model for the first time, a predicted input of Augmented Burgers equation is inversely achieved. By the Kfold crossvalidation method, the predicted ground signature closest to the target ground signature is obtained. Then, the corresponding equivalent area distribution is calculated using the classical Whitham’s Ffunction theory from the optimal nearfield signature. The inversion method is validated using the classic example of the C608 vehicle provided by the Third Sonic Boom Prediction Workshop (SBPW3). The results show that the design ground signature is consistent with the target signature. The equivalent area distribution of the design result is smoother than the baseline distribution, and it shrinks significantly in the rear section. Finally, the robustness of this method is verified through the inverse design of sonic boom for the nonphysical ground signature target.
1 Introduction
As a unique acoustic phenomenon during the flight of a supersonic vehicle, a solid sonic boom usually causes a person to be frightened. An enormous overpressure value and a short rise time of the sonic boom may cause permanent damage to the hearing organs, which is also one of the main factors leading to the failure of the supersonic civil aircraft “Concorde” [1]. Therefore, the sonic boom problem is one of the core bottlenecks to be solved in the design of future green supersonic civil aircraft.
Currently, research institutions worldwide have conducted many studies to reduce the intensity of the sonic boom of supersonic civil aircraft. These are mainly divided into the sonic boom suppression technology based on flow control and the aerodynamic layout design method based on different low sonic boom mechanisms. Compared to the sonic boom suppression technology research with a more avantgarde concept, exploring supersonic civil aircraft’s low sonic boom shape design is more practical. NASA confirmed in the SSBD (Shaped Sonic Boom Demonstrator) [2] project that it could effectively change its ground sonic boom signature through the aerodynamic shape of the aircraft, which also provides a solid foundation for the low sonic boom shape design.
Low sonic boom shape design can be divided into two types of positive [3,4,5,6] and inverse designs [7,8,9,10] from a macro design perspective. The design goal is to improve the sonic boom’s ground signature, and the design variable is the vehicle’s aerodynamic shape. The positive design refers to the forward search to reduce the farfield sonic boom signal by using an optimal design method to change the aerodynamic shape of the aircraft. The inverse design refers to artificially giving an ideal farfield ground signature. The sonic boom inversion technique directly finds the corresponding aerodynamic shape. Compared with the positive design, the inverse design method is more efficient in carrying out the low sonic boom shape design. For design proposes, Koziel and PietrenkoDabrowska et al. [11,12,13,14,15,16,17,18] have carried out many optimization designs by inverse surrogate models and response features. The current research is mainly focused on the sonic boom inverse design [19, 20]. The sonic boom inverse design can be divided into two steps from the design process perspective: target nearfield signature matching [21, 22] and target farfield ground signature matching [23, 24]. The target nearfield signature matching method specifies the nearfield overpressure distribution to infer the corresponding aerodynamic profile in reverse. The target ground signature matching method determines the farfield ground signature distribution to inverse the nearfield overpressure distribution. Target nearfield signature matching is easier to achieve because fewer factors affect the link between the vehicle’s aerodynamic shape and the nearfield overpressure distribution. In the process of target farfield ground signature matching, the classical absorption and molecular relaxation effects [25] have a significant influence on the farfield signal, the results obtained by different methods of sonic boom farfield simulation and inverse extrapolation have a considerable difference, and therefore it is the research focus in the sonic boom inverse design. In this paper, the target ground signature matching is studied using the international mainstream highconfidence method for predicting the farfield signals of supersonic aircraft in regular cruises, that is, the farfield signature prediction method based on the Augmented Burgers equation.
In the sonic boom inverse design research, Li et al. [26] proposed a reverse extrapolation method based on the inverse Augmented Burgers equation, that is, by solving the inverse Augmented Burgers equation to reverse the propagation of the sonic boom signal from the ground to the near field. However, the inverse Augmented Burgers equation is inherently pathological, and solving it by introducing regularization will contaminate the real physical solution and bring some other numerical problems. Rallabhandi [27] conducted a study on the inverse equivalent area design of supersonic aircraft. This study uses the discrete concomitant shape optimization method to reduce the sonic boom’s intensity. However, the process did not consider the near and farfield signature distribution and was only applicable to the preliminary design of the aircraft. Zhang et al. [19] proposed a sonic boom inverse design method based on Proper Orthogonal Decomposition (POD), which has certain robustness as an efficient inverse design method. But this method has some limitations because POD initially extracts empirical modes for linear superposition, which can only be described approximately for complex nonlinear systems. Gu et al. [20] used the operator splitting method and the regularized pseudoparabolic equation to solve the inverse Augmented Burgers equation again, which can invert the equivalent area distribution more accurately. However, this method only reverses the midfield sound pressure signal to the near field for the signature inversion problem. Ma et al. [28] compared the sonic boom inverse design method based on the POD and discrete concomitant methods. The results showed that the concomitant equation inversion method has better local excitation signal inversion capability. At the same time, the farfield perceived sound pressure level is more accurate.
Machine learning (ML) has demonstrated remarkable power in recent years for numerous applications [29], like image processing, video and speech recognition, genetics, and disease diagnosis. Deep Neural Networks (DNNs), as a vital component of ML, are suitable for solving illposed inverse problems. Inverse design problems refer to problems in which the desired output is known, but the inputs must be determined based on this output. In these problems, DNNs have proven to be a powerful tool for learning complex relationships between inputs and outputs, and for generating new designs based on a set of constraints. One of the main advantages of using DNNs in inverse design problems is their ability to capture and model complex, nonlinear relationships between inputs and outputs. Unlike traditional approaches, such as rulebased systems or regression models, DNNs can learn these relationships from data and generate new designs based on a set of constraints. This makes them well suited for applications where the relationships between inputs and outputs are not well understood or are too complex to model analytically. Additionally, DNNs have the ability to handle large amounts of data, which is important in many inverse design problems where a large amount of data may be available. This allows them to capture the subtle variations and patterns in the data, leading to more accurate predictions and better designs. In conclusion, the use of DNNs in inverse design problems provides a powerful tool for capturing and modeling complex relationships between inputs and outputs, as well as handling large amounts of data. These advantages make DNNs a valuable tool for solving various inverse design problems in a variety of fields. At present, machinelearning algorithms have some preliminary applications in the aerospace inverse problem. Glaws et al. [30] leveraged emerging invertible neural network tools to enable the rapid inverse design of airfoil shapes for wind turbines. Sekar et al. [31] proposed an approach to perform the inverse design of airfoils using deep convolutional neural networks. Wang et al. [32] developed the framework of nacelle inverse design based on improved Generative Adversarial Networks. Wang et al. [33] proposed an inverse design method for supercritical airfoil based on conditional generative models in deep learning. Ghosh et al. [34] developed a probabilistic framework for inverse aerodynamic design using invertible neural network. However, there is no application of DNNs in sonic boom inverse design. This paper tries to conduct inverse design research based on DNNs to obtain a more accurate nearfield sonic boom signature by inversion.
This paper proposes a nearfield sonic boom signal inversion method based on the Augmented Burgers equation and DNN, and then details the framework of this inverse design algorithm and sample generation. It verifies the inverse design effect of the method using the standard example of C608 aircraft provided by the 3rd Sonic Boom Prediction Workshop (SBPW3, 2020). The results show that the method can accurately invert the farfield ground signature to obtain the nearfield overpressure distribution, which provides technical support for optimizing supersonic aircraft’s low sonic boom aerodynamic profiles.
2 Mathematical model
2.1 Augmented Burgers equation
The Augmented Burgers equation is a physical model to simulate the propagation of sonic booms in a stratified real atmosphere. A comprehensive introduction about the Augmented Burgers equation is provided by Cleveland [35]. The Augmented Burgers equation includes the effects of nonlinearity, classical dissipation, and molecular relaxation phenomena. The dimensionless Burgers equation is shown in Eq. (1). Where the dimensionless pressure, dimensionless time, and dimensionless sound pipe distance are P, τ, σ. And dimensionless gas dissipation parameters, dimensionless relaxation coefficients, and dimensionless relaxation time are Γ, C_{v}, θ_{v}. The sound pipe area, sound velocity, and atmospheric density are S, c_{0}, ρ_{0}.
The five terms on the righthand side of Eq. (1) are the nonlinear effect term, the classical dissipation effect term, the molecular relaxation effect term, the geometric diffusion effect term, and the atmospheric stratification effect term, which represent the effects of the corresponding physical impact on the sound pressure signal, respectively.
2.2 Deep neural network
Deep learning is a specialized branch of machine learning that employs artificial neural networks with multiple hidden layers to tackle complex issues, including image and speech recognition. One widely utilized neural network architecture in deep learning is the Multilayer Perceptron (MLP) network. The MLP network presents several advantages, including its capacity to model nonlinear relationships, process extensive datasets, and produce predictions based on the analyzed data. Furthermore, MLP networks possess the capability to continuously learn and improve through training, rendering them ideal for purposes such as predictive modeling and pattern identification.
MLP is one of the simplest and most effective DNN, consisting of an input layer, hidden layers (intermediate layer), and an output layer. Each layer contains several neurons, as shown in Fig. 1, where each neuron has a different bias and the same activation function. As the base module of the multilayer perceptron, the neuron takes the vector as the input, calculates the weighted sum of the inputs, then adds bias, and finally inputs the activation function to get the generated scalar output. Where x_{0} to x_{n} are the values of the neurons in the previous layer, w_{i} is the weight corresponding to each neuron, b is the bias, and f is the activation function.
Any two nodes between adjacent layers correspond to a weight. All nodes of the previous layer are used as inputs to the subsequent layer. A node in the hidden layers and output layer is connected to all nodes of the last layer, and the result is generated by applying an activation function to the values obtained from these nodes. Sum up the value of each incoming connection multiplied by the weights plus the overall bias, and then apply the activation function. As weights and biases of the whole network, the weight and bias parameters must be adjusted through the training process. The structure of the MLP network used in this paper is shown in Fig. 2, which contains one input layer, one output layer, and six hidden layers for a total of eight layers. Each layer has 400 neurons. Several neurons are omitted in this figure. Because the sonic boom data value domain involves negative numbers, the tanh function is used as the activation function, and the network is trained using the Adam optimizer. During training, both input and output data are normalized so that both input and output are onedimensional vectors with 400 real numbers. The Adam optimization algorithm has been selected for training the deep neural network (DNN), with the learning rate γ fixed at 5 × 10^{− 4}. Additionally, the β_{1} parameter has been set to 0.9, the β_{2} parameter to 0.999, and the θ_{0} parameter to 1 × 10^{− 8}.
3 MLP inverse design framework and sample generation
3.1 Inverse design algorithm framework
This article proposes a highefficiency method based on MLP to find the shape of the aircraft corresponding to a given farfield ground signature. Given the farfield signature, the nearfield signature corresponding to the target farfield signature is obtained through the MLP, and then the shape of the aircraft (here refers to the area distribution) corresponding to the nearfield signature is obtained by the Abel inverse transform. Figure 3 shows the flow diagram of the inverse design.
Figure 4 shows the entire inverse design framework process, where each process step is shown below.

1)
Initialization: Define the baseline nearfield overvoltage distribution as the initial optimization value; generate the ideal ground signature as the optimization target.

2)
Add perturbation: Add perturbation to the baseline nearfield using triangular and sawtooth waves to generate several perturbed nearfield curves.

3)
Calculation of farfield: The farfield signatures corresponding to the nearfield overvoltage distribution after each perturbation is calculated using the Augmented Burgers equation method.

4)
Train DNN: Several obtained data are used as the training samples of the DNN. Specifically, the acquired farfield baseline signature is subtracted from the farfield signature corresponding to the baseline nearfield signature to obtain the input samples of the DNN; the perturbed nearfield signature is removed from the baseline nearfield signature to get the output sample data; finally, the Adam optimization algorithm is used to train the DNN.

5)
Deep learning inversion: After completing the network training, the difference between the optimized target farfield signature and the farfield signature calculated from the baseline nearfield is fed into the DNN to obtain a difference between the inversion result and the baseline nearfield, and the inverse nearfield result is obtained by adding this result with the baseline nearfield.

6)
Calculation of inversion DF (Distance Function): The farfield signatures corresponding to the nearfield results are obtained by the Augmented Burgers equation. Here, DF is a measure suitable for expressing similarity.

7)
Determine DF improvement: If the distance between this inverse design farfield signature and the target farfield signature is improved compared to the previous iteration, then update or replace the baseline nearfield signature by this inverse design result, and jump to step 2, otherwise enter step 8.

8)
Reduce perturbation amplitude: The setting amplitude attenuation rate is 0.9, and the length attenuation rate is 0.9, reducing the amplitude and length of the perturbation, narrowing the sampling range, and jumping to step 2. When the perturbation amplitude drops to the set range, the whole process is stopped.
3.2 Generation of perturbation data
The triangular and sawtooth waves are used to disturb the nearfield overpressure distribution. The three waveforms are symmetric triangular waves, a right sawtooth wave, and a left sawtooth wave, as shown in Fig. 5.
Three waves of different amplitudes and lengths are added to the baseline nearfield overpressure distribution to form the perturbation, and different amplitudes and lengths are shown in Fig. 6.
4 Validation and optimization
4.1 Program verification of sonic boom forwarding propagation equation
In this paper, the study is carried out using the standard example of the C608 aircraft provided by the AIAA Third Sonic Boom Prediction Workshop (SBPW3, 2020) [36], as shown in Fig. 7, with the inflow Mach being 1.4, the angle of attack being 0°, and the flight altitude being 16,215.36 m. The sonic boom nearfield input signal is calculated using the inhouse CFD solver [37, 38], and the farfield ground signature is computed by a selfdeveloped sonic boom program [19, 20, 39]. The SST (ShearStressTransport) turbulence model [40] is adopted for CFD numerical simulation, and the nearfield distance is three times the length of the C608 aircraft (z = 82.29 m). The unstructured hybrid grid provided by SBPW3 is used [36]. The number of nodes of this halfmode grid is about 502.1 million. The spatial grid density NZ = 20,000 and the temporal grid density NT = 50,000 were chosen for the study when calculating the farfield ground signature using the Augmented Burgers equation.
Figures 8 and 9 compare the differences between the currently calculated nearfield and farfield signatures and the published data of SBPW3 [36], respectively, where the red and black lines are the SBPW3 data, and the solid green line is the data calculated by the selfdeveloped program. It can be found that the predicted nearfield and farfield signatures overlap with the SBPW3 data and achieve a reasonable precision, indicating that the accuracy of the calculation of the selfdeveloped CFD program and the sonic boom program meets the requirements. The nearfield overpressure distribution varies at a high frequency, with multiple peaks and valleys. Due to the classical absorption, molecular relaxation effect, etc., the atmosphere has a lowpass filterlike nature and sonic energy, and highfrequency pulsations in the atmospheric propagation process are consumed as internal energy. Thus, the farfield ground waveform is smoother.
4.2 Inverse design example configuration
The C608 vehicle nearfield is used as the baseline nearfield, and the target farfield is inverted using DNN. Based on Plotkin’s proposal for shaping the low sonic boom signature [41], the original ground signature was modified to a sine wavelike shape as the target signature (thereby suppressing the energy in the highfrequency portion of the sonic waves). The original baseline and target ground signature are given in Fig. 10. The perceived noise level (PLdB) represents the ground sonic boom intensity, and the perceived noise level of the shaped waveform is reduced by 1.5061 PLdB.
The L2 norm is used to describe the closeness of the design result and the target waveform, i.e., the objective function of optimization is the similarity between the disturbed farfield and the target farfield, which is calculated as
where DF is the similarity to be calculated, \({P}_{dist}^i\) is the ith point of the disturbed farfield vector, \({P}_{tar}^i\) is the ith point of the target farfield vector, and S is the number of points of the discrete ground signature. The smaller the DF, the higher the similarity.
The baseline nearfield overpressure distribution is perturbed to generate 2400 samples of nearfield data, as shown in Fig. 11. The blue line is the unperturbed nearfield curve, and the perturbed range only contains the part of the aircraft that needs to be reshaped. Considering the magnitude and period of the flow field itself, the maximum magnitude is 0.006 in absolute value, and the maximum length is 100. In this paper, the training of DNN is carried out on a domestic supercomputing platform (“Dongsheng1”), which uses a heterogeneous hardware environment to run the program and can significantly reduce the algorithm’s running time.
4.3 Crossvalidation and optimization results
The importance of obtaining highquality and reliable results cannot be overstated. To that end, Kfold crossvalidation is employed as a technique to assess the performance of the machine learning model under investigation. This step is deemed essential in the validation process of the results obtained from the model. The objective of the Kfold crossvalidation is to confirm the generalizability of the results obtained from the proposed model, rather than being solely specific to the training data. In Kfold crossvalidation, the value of k increases from 3 to 30. Specifically, 2400 samples are randomly divided into k equal parts. The ith of them is selected as the validation set, totaling 2400/k samples. The remaining part is the training set, totaling 2400 × (k1)/k samples. The structure of the training network is kept constant. Finally, the network models trained with different k and i are evaluated. The evaluation method uses this network model to inverse the target farfield signature and then forward the nearfield signature obtained from the inversion to obtain the design farfield signature corresponding to the inversion result, called P_{k, i}. Calculate the similarity DF between P_{t} and P_{k, i}. The results are shown in Table 1. The maximum DF value, minimum DF value, and average DF value for different i at each value of k are shown here. In Table 1, the three numbers marked in red indicate the network models corresponding to the smallest min DF, max DF, and Avg DF, respectively. The networks are trained when k = 20, 11, 16.
The results of plotting all the above data into a heat map are shown in Fig. 12. Where the horizontal coordinates are the k values and the vertical coordinates are the i values. Since i is always less than k, this graph only has importance in the lowerleft region, and the value in the chart is DF.
According to crossverification, the value of the box line can be drawn in different k conditions, as shown in Fig. 13. These include the upper limit, lower limit, median, and two quartiles of the crossvalidation result DF. Connect the two quartiles to draw the box, and then connect the upper and lower limits with the box. The median is in the middle of the box, and the dot represents the abnormal value. The abnormal value is usually eliminated in the process of data analysis.
Finally, the deep neural network trained at k = 16 is selected as the final model.
Figure 14 shows the iterative convergence process of the optimized objective function, showing that the objective function does not decrease after the 12th iteration. Figure 15 shows the mean squared error of the training set and validation set of the 6th iteration of the DNN.
Figure 16 gives the corresponding ground signature with the lowest perceived noise level among all optimization results, from which it can be seen that the optimized ground signature achieves a significant shaping effect. The initially sharp compressional waveform is optimized to a relatively flat waveform with minor fluctuations.
The nearfield signature corresponding to the optimal ground signature is given in Fig. 17. It can be seen that the initially flat waveform is optimized to a repeated oscillation waveform. These sharp compressional and expansion waves are suppressed by the classical atmospheric dissipation and molecular churning effects during the propagation process and eventually become relatively flat as they approach the ground. Improvement of optimization is achieved via shaping the original nearfield signature into wiggles and damping it by atmospheric attenuation.
Once the nearfield overpressure distribution is determined, the equivalent area distribution for the final design can be obtained by the Abel inverse transform [26] as follows.
Note that the supersonic profile must satisfy the slender body assumption [42] to use the above equation.
Based on Eq. (3), the equivalent area distributions of the design and initial shapes can be obtained using numerical integration, and a comparison of the equivalent area distributions is presented in Fig. 18. Here, the equivalent area distribution considers the vehicle’s volume and lift contributions.
As seen in the latter part of Fig. 18, there is a very significant shrinkage at the tail end of the optimized shape, which represents a loss of volume in this part of the airframe (assuming that the lift distribution remains constant before and after the optimization). In addition, the area distribution of the design shape is smoother compared to the equivalent area distribution of the initial shape.
4.4 Discussions on the DNN inverse design method
In this section, the robustness of the present method is examined when nonphysical solutions are set as the target ground signature. The peak and valley of the farfield ground signature are horizontal straight lines. This ground signature is physically nonexistent because there is no horizontal straight line in the actual ground waveform. The DNN models and other parameters are the same as the optimization example in Section 4.2 except for the target ground signature. Figure 19 gives the optimized ground signature against the baseline and target. Figure 20 provides the optimized nearfield signature against the baseline. As shown in Figs. 19 and 20, the optimized ground signature matches the target to a great extent, although the target ground signature is nonphysical. The optimized nearfield and baseline signatures are reversed at multiple peaks and troughs.
5 Conclusions
This paper proposes a deep neural networkbased inversion method for nearfield sonic boom signals of supersonic aircraft, aiming to shape the ground signature into the desired target. And its feasibility in low sonic boom inverse design optimization is verified by the C608 vehicle example. The method can be applied to the optimized design of future supersonic aircraft’s low sonic boom aerodynamic shape. Some conclusions can be drawn as follows.

1)
The forward propagation program of sonic boom based on the Augmented Burgers equation can accurately simulate various physical processes of the sonic boom signal propagation in the natural atmosphere. It can effectively improve the simulation accuracy of the rise time to meet the needs of engineering applications.

2)
A deep neural networkbased sonic boom inversion method is used to inverse the target farfield ground signature to obtain the nearfield overpressure distribution, which is a repeatedly oscillating waveform compared to the original flat baseline nearfield waveform but has little effect on the farfield signature.

3)
Using the sonic boom forward propagation program to calculate the inversion of the nearfield overpressure distribution, the final ground farfield waveform and the target waveform match and achieve a noticeable shape modification effect.

4)
The equivalent area distribution of the final design is smoother than the initial shape. It shrinks significantly in the rear section, which provides technical support for subsequent design optimization based on the equivalent area distribution.

5)
A nonphysical ground signature is set as the target to test the robustness of this inverse design method, and it is shown that this method is robust enough for various inputs. This nature is designerfriendly and does not require extensive sonic boom engineering experience for aircraft designers.
In future endeavors, more sophisticated and appropriate neural networks will be employed to extract the inherent features of the data, with the aim of elevating the accuracy of the network. Furthermore, to enhance the efficiency of the program, optimization of the inversion algorithm will be pursued, in an effort to attain an optimal equilibrium between precision and speed. In addition, the target signature is relevant for inverse design methods, because it is often unclear whether the assumed target characteristics do correspond to the optimum designs in a forward sense. Thus, the target characteristics of sonic booms deserve further research for complex supersonic aircraft.
Availability of data and materials
The datasets generated during the current study are available from the corresponding author upon reasonable request.
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Acknowledgements
The authors thank the anonymous reviewers for their constructive comments.
Funding
This research was supported by the National Key Research and Development Program of China (No. 2020YFB1709500), Natural Science Basic Research Program of Shaanxi province (No. 2021JQ076) and Fundamental Research Funds for the Central Universities.
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Chen, S., Qiu, J., Yang, H. et al. Deep learning for inverse design of lowboom supersonic configurations. Adv. Aerodyn. 5, 13 (2023). https://doi.org/10.1186/s42774023001451
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DOI: https://doi.org/10.1186/s42774023001451
Keywords
 Low boom configuration
 Deep neural network
 Augmented Burgers equation
 Supersonic transports
 Equivalent area