High-Efficiency Simulation of Dynamic Stability Derivatives Based on a Particle Swarm Optimization and Long Short-Term Memory Network (PSO-LSTM) Coupling Aerodynamic Model

Abstract

A new high-efficiency method based on a particle swarm optimization and long short-term memory network is proposed in this study to predict the aerodynamic forces in an unsteady state. Based on the predicted aerodynamic forces, the dynamic derivative is further calculated. Using particle swarm optimization to optimize the hyper-parameters of a neural network, the long short-term memory network prediction model can be constructed according to the known simulating aerodynamic data to predict the aerodynamic performance of aircraft in unknown states. By coupling the least-squares method in an aerodynamic derivative model, the dynamic derivative can be quickly obtained. Unsteady motion of NACA 0012 airfoil was taken as the research example to verify this method, and its longitudinal combined dynamic derivatives were predicted and compared with CFD simulation results. The results show that the dynamic derivatives predicted by the PSO-LSTM method have high accuracy, with an error of no more than 1% compared to CFD, and a 70% improvement in efficiency. The method proposed in this study has good generalization ability and can realize fast and accurate prediction of dynamic derivatives with a small number of samples.

1. Introduction

With the development of modern flight control technology, it is possible for aircraft to fly near stall or even post stall. Especially for fighter planes, having post stall capacity enables them to achieve tactical advantages such as rapid emplacement, taking aim first, and effectively avoiding attacks in close-range air combat, which is an important capability for advanced fighter planes. Accurately obtaining the dynamic aerodynamic parameters of the fighter plane has become the key to current advanced fighter plane design. Dynamic derivatives are an important parameter in the dynamic characteristics of aircraft and have a significant impact on the analysis of aircraft stability characteristics, flight control system design, and ballistic analysis. They can provide the data for the analysis, calculation, and design of aircraft dynamic quality, control system, and guidance system [1].

Traditional methods of obtaining dynamic derivatives include wind tunnel testing and flight testing. With the development of computer technology, efficient CFD technology has also become an important means of obtaining dynamic derivatives [2]. Many researchers have conducted research on using high-precision CFD technology to identify dynamic derivatives [3], mainly using time-domain calculation methods, including the perturbation method, coning motion method, forced vibration method, and free vibration method. The basic idea of the perturbation method is that as the vehicle makes a harmonic vibration around the reference point, its unsteady flow field can be decomposed into a linear superposition of the steady flow field at the equilibrium position and the small disturbance flow field. When using first-order approximation, the disturbance flow field can be further simplified as an Euler line equation for solving the amplitude of perturbation. This equation is a linear equation with the same coefficient matrix as the Euler equation for solving steady flow, so the same difference scheme can be used to solve it. Finally, the damping derivative for a specific axis can be obtained using the conversion formula between different axes. The coning motion method replaces the unsteady flow field in inertial frames by solving the steady flow field in noninertial frames, achieving an efficient calculation of dynamic derivatives. The forced vibration method is used to simulate the working conditions of wind tunnel testing and calculate the aerodynamic forces of the unsteady harmonic vibration, then obtain the dynamic derivatives. The free vibration method is also used to simulate the working conditions of wind tunnel testing, which is different from the forced vibration method. This method firstly gives an instantaneous incentive to the aircraft and sets specific mass parameters of the aircraft, then calculates the unsteady aerodynamic force generated by the vibration under the aircraft’s own mass characteristics, and finally identifies the dynamic derivatives based on this [4].

Although the CFD simulation of dynamic derivatives has made great progress, dynamic derivatives are associated with unsteady motion, and their identification requires a lot of unsteady aerodynamic force calculation work, which seriously affects the efficiency of dynamic derivatives identification. For this purpose, researchers are committed to exploring new fast-simulation methods to obtain dynamic derivatives, mainly including the harmonic balance method in the frequency domain and the time spectrum method. Based on the Fourier series expansion, the harmonic balance method transforms the unsteady solution process of the periodic unsteady flow field into the coupled solving process of several steady flow fields, and obtains the unsteady process of the entire flow field through reconstruction. On this basis, a fast-prediction method of dynamic derivatives is established. This method has high computational efficiency, has minimal modifications to existing solving programs, and is easy to program and implement [5]. However, the harmonic balance method has the problem of converting time and frequency domains. Therefore, a time spectrum method (TSM) was further proposed, taking into account the periodic characteristics of the flow. Compared with the harmonic balance method, the operation process of the TSM is carried out in the time domain, avoiding the conversion between the time domain and frequency domain. TSM has been widely applied in various engineering problems of periodic unsteady flow, such as helicopter-rotor-motion and vortex-shedding problems [6].

Overall, the current means of calculating dynamic derivatives with CFD mainly includes the two methods of time domain and frequency domain. The time domain method is simple, but the identification efficiency of dynamic derivatives is low due to a large number of unsteady aerodynamic force calculations. Especially for complex, full aircraft, the huge numbers of grids makes the acquisition of dynamic derivatives seriously lag behind the design of the configuration scheme. Although the computational complexity of frequency domain methods is small, on the one hand, the solver needs to be changed, and on the other hand, the stability of its flow field reconstruction is not high, and the accuracy also needs to be further explored. Overall, there is still a lack of efficient and accurate dynamic derivative simulation methods.

With the development of artificial intelligence, researchers have conducted a lot of research on aerodynamic force identification using neural networks. Among them, ZHANG Ruimin et al. [7] established an unsteady aerodynamic model of aircraft at high angles of attack based on an improved BP neural network method; WANG Chao et al. [8] used neural network methods to model unsteady aerodynamic forces at high angles of attack, and compared the accuracy with dynamic derivative models and polynomial models. The results showed that the aerodynamic modeling method based on neural networks has high accuracy and adaptability. However, the above research methods are all based on Back Propagation (BP) neural networks, which have certain limitations in processing time series data such as unsteady aerodynamics.

For the above problems, we innovatively propose a new method of “Modeling with a small number of samples-dynamic derivatives identification”. We trained a long short-term memory network (LSTM) using a small number of sample data, and built an intelligent model of unsteady aerodynamic force. A particle swarm optimization (PSO) algorithm is used to optimize the hyper-parameters of the neural network to improve the training efficiency and accuracy of the model. By using this model, we predict the unsteady aerodynamic force of unknown states, and the dynamic derivatives are obtained using the least-squares method. By replacing data calculation with models, the effectiveness of obtaining dynamic derivatives is greatly improved. The optimization strategy of the intelligent modeling process also ensures the accuracy of dynamic derivatives prediction. Compared to the method based on the BP neural network mentioned above, the accuracy is improved by 4–5 times, and the required training data are fewer.

2. Dynamic Derivatives Identification Based on the Forced Oscillation Method

At present, there are two commonly used methods to calculate the dynamic derivatives using CFD: the forced oscillation method and the free oscillation method. The forced oscillation method calculates the unsteady aerodynamic force of the aircraft’s motion process by giving it a harmonic motion around the center of gravity, and finally calculates the dynamic derivatives. The free oscillation method is the disturbance motion of an aircraft with a given support, and it further identifies the dynamic derivatives. In comparison, the forced oscillation method has become the mainstream method for identifying dynamic derivatives due to its controllable motion conditions and easy implementation.

Taking the pitch direction as an example, the forced oscillation method is generally given a simple harmonic oscillation, as follows:

$$\alpha ={\alpha }_1+\theta ={\alpha }_1+{\alpha }_{\mathrm{m}}\sin (2\pi ft)$$

(1)

In the formula, ${\alpha }_1$ is the initial angle of attack of oscillation; ${\alpha }_{\mathrm{m}}$ is the oscillation amplitude; $f$ is the oscillation frequency. By numerically solving the time history of the aircraft undergoing forced pitch oscillation, the time history curve of the dynamic pitch moment coefficient $C_{\mathrm{m}}$ and the hysteresis loop $C_{\mathrm{m}}-\alpha$ can be obtained. By integrating the hysteresis loop mentioned above within one cycle, the dynamic derivatives at the angle of attack can be calculated.

$$\left(\frac{\partial C_{\mathrm{m}}}{\partial \stackrel{•}{\theta }}\right)0=\frac{1}{{\alpha }_{\mathrm{m}}\pi }{\displaystyle \underset{t-T}{\overset{t+T}{\int }}\left(C_{\mathrm{m}}-C_{\mathrm{m}0}\right)\cos \left(2\pi ft\right)}dt$$

(2)

In the formula, $C_{\mathrm{m}0}$ is the pitch moment coefficient at the initial angle of attack; $T$ is the oscillation period. The free oscillation method does not limit the rotational degrees of freedom of the aircraft, and the aircraft oscillates freely under its own aerodynamic moment. Taking the pitch direction as an example, the free pitch oscillation formula for a single degree of freedom without mechanical damping can be written as

$$\stackrel{••}{\theta }=\frac{1}{I}{C}_{\mathrm{m}}=\frac{1}{I}\left(C_{\mathrm{m}0}+\left(\frac{\partial C_{\mathrm{m}}}{\partial \theta }\right)0\theta +\left(\frac{\partial C_{\mathrm{m}}}{\partial \stackrel{•}{\theta }}\right)0\stackrel{•}{\theta }\right)$$

(3)

In the formula, $I$ is the dimensionless moment of inertia. After obtaining the time history curve of free pitch oscillation $\theta =\theta \left(t\right)$ and $C_{\mathrm{m}}=C_{\mathrm{m}}\left(t\right)$ through numerical calculation, the frequency of free pitch oscillation $f=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$T$}\right.$ can be determined. Then, by selecting two peak values of the same period, the dynamic derivatives at the equilibrium angle of attack can be obtained:

$$\left(\frac{\partial C_{\mathrm{m}}}{\partial \stackrel{•}{\theta }}\right)0=2If\ln \frac{{\theta }_2}{{\theta }_1}$$

(4)

The above methods take the pitch direction as an example to illustrate the calculation process of pitch dynamic derivatives. The yaw dynamic derivatives and roll dynamic derivatives can be calculated similarly. In addition, the above method obtains the combined derivatives, and it is necessary to add the simulation of translation motion to separate the combined items.

Although the dynamic derivatives identification method based on the forced oscillation method is widely used, it has shortcomings. In general, it is necessary to simulate the steady-state flow field under computational conditions, obtain initial aerodynamic values, and then use dynamic grid technology to solve the unsteady flow field. Although the motion form is a periodic simple harmonic motion, it is affected by the initial inertial effect of the unsteady flow field, and it needs to calculate multiple periods to achieve the stability of the unsteady aerodynamic force, as shown in Figure 1. In the first cycle, peak data of unsteady aerodynamic force cannot be used due to initial effects. The flow field characteristics established in the second cycle begin to coordinate with the motion, at which point the initial effect decreases. The third cycle can basically establish a periodically changing aerodynamic force. So, in order to obtain dynamic derivative values that are as accurate as possible, available aerodynamic force needs to be calculated for at least three cycles, which greatly increases the identification cost of dynamic derivatives. Although time spectrum methods in the frequency domain or harmonic balance methods can be used to reconstruct the flow field through several points, the conversion between the frequency domain and time domain requires significant modifications to the solver, resulting in poor universality. If improving the efficiency of dynamic derivatives identification from the perspective of the time domain, it is necessary to solve the problem of fast and accurate acquisition of unsteady aerodynamic force. Therefore, we propose an aerodynamic modeling method based on PSO-LSTM for fast prediction, which constructs a neural network by training a small number of unsteady aerodynamic sample data. This neural network predicts the aerodynamic force during stable periods in unknown states and then predicts dynamic derivatives.

3. Dynamic Derivatives Identification Technology of Intelligent Aerodynamic Modeling Based on a Particle Swarm Optimization and Long Short-Term Memory Network

Because the identification of dynamic derivatives is based on a large amount of unsteady aerodynamic force, it is very important to obtain unsteady aerodynamic force quickly and accurately. In this paper, we propose a new method to build an unsteady aerodynamic model based on a particle swarm optimization and long short-term memory network. We construct an LSTM network using several sample data and use the PSO algorithm to determine the optimal hyper-parameters of the neural network. Finally, we obtain the unsteady aerodynamic model constructed under a small number of sample data. On this basis, combined with the mathematical model of dynamic derivatives, the least-squares method is used to calculate the dynamic derivatives. The flowchart of the entire method is shown in Figure 2.

3.1. Unsteady Aerodynamic Model Based on the LSTM Network

Recurrent neural networks (RNNs) have been applied in many fields and have achieved remarkable results in recent years [9,10,11]. However, traditional RNNs have two defects: gradient explosion and gradient vanishing [12]. Adding gradient truncation and regularization terms to RNNs can avoid gradient explosion. Gradient vanishing is also referred to as long-term dependencies. This mainly refers to how when the sequence input is long or the network structure is deep, the relevance of the data information in the sequence decreases or even disappears, resulting in the network being unable to learn the important information at the front of the sequence. The structure of RNNs is shown in Figure 3.

LSTM is a derivative of RNNs that can overcome the problem of long-term dependencies. Through extensive research, it has been proven that LSTM has advantages in dealing with time series problems [13]. LSTM adds a structure called a memory cell to the neurons in the hidden layer of an RNN to remember past information [14], and adds three gate structures (input gate, forgetting gate, and output gate) to control the use of historical information. The structure of LSTM is shown in Figure 4.

The input sequence is $\left(x_1,x_2,\cdots ,x_T\right)$ and the hidden layer state is $\left(h_1,h_2,\cdots ,h_T\right)$, at time t:

$$i_t=\sigma \left(W_{hi}{h}_{t-1}+W_{xi}{x}_t+b_i\right)$$

(5)

$$c_t=f_t•c_{t-1}+i_t•g\left(W_{hc}{h}_{t-1}+W_{xc}{x}_t+b_c\right)$$

(6)

$$o_t=\sigma \left(W_{ho}{h}_{t-1}+W_{ox}{x}_t+W_{co}{c}_t+b_o\right)$$

(7)

$$h_t=o_t•g\left(c_t\right)$$

(8)

In the formula, $i_t,f_t,o_t$ are input gate, forgetting gate, and output gate, respectively. $c_t$ is a cell unit. $W_h$ is the weight of a recursive connection. $W_x$ is the weight from the input layer to the hidden layer. $b_i,b_f,b_c,b_o$ are the thresholds for each function. $\sigma \left(•\right)$ and $g\left(•\right)$ are the sigmoid function and tanh function, respectively. $•$ represents the inner product of a vector.

To achieve the prediction purpose of LSTM, a linear regression layer needs to be added:

$$y_t=W_{yo}{h}_t+b_y$$

(9)

In the formula: y_t is the prediction result. b_y is the threshold of the linear regression layer.

Select the Mach number, angle of attack, motion parameters, and corresponding aerodynamic data (lift drag and moment coefficient, etc.) as training data to train the neural network. Use a small number of known aerodynamic data as testing data to conduct real-time testing on the model, ultimately forming a stable and reliable intelligent aerodynamic model.

3.2. Hyper-Parameters Optimization of LSTM Based on Particle Swarm Optimization

Parameters in machine learning algorithms [15] can be divided into model parameters and hyper-parameters. Model parameters are internal variables of the model, such as bias, weight, etc. These parameters do not need to be manually set, but are automatically learned during model training. Gradient-based algorithms, such as Adam, can be used for model parameter optimization. Hyper-parameters need to be manually set before training. The selection of hyper-parameters has a significant impact on the performance of neural network models. Improper hyper-parameters can lead to problems such as nonconvergence, under-fitting, over-fitting, and high computational overhead. Currently, commonly used hyper-parameter tuning methods include grid search [16], random search [17], and Bayesian tuning [18]. Among them, particle swarm optimization is essentially a series of random search-based methods, which combines parallel search and sequential optimization. It allows the use of adaptive hyper-parameters during training. So, particle swarm optimization is selected to optimize the LSTM network.

3.2.1. Principle of PSO

Particle swarm optimization simulates the clustering behavior of insects, beasts, birds, and fish. These groups search for food in a cooperative way. Each member of the group changes its search mode by learning from its own experience and that of other members [19]. The PSO algorithm treats individuals in the population as a particle in a multidimensional search space, with each particle representing a possible solution to the problem. During the optimization process, each particle maintains two vectors, namely, the velocity vector $v_i=\left[{{\displaystyle \mathit{v}}}_i^1,{{\displaystyle \mathit{v}}}_i^2,\cdots ,{{\displaystyle \mathit{v}}}_i^D\right]$ and the position vector $x_i=\left[{{\displaystyle \mathit{x}}}_i^1,{{\displaystyle \mathit{x}}}_i^2,\cdots ,{{\displaystyle \mathit{x}}}_i^D\right]$. In the formula, i is the particle number, and D is the dimension of the problem. The velocity of a particle determines its direction and velocity of motion. Position reflects the position of the solution represented by the particle in the solution space, which is the basis for evaluating the quality of the solution. The algorithm also requires each particle to maintain its own historical optimal position vector and the population to maintain a global optimal vector.

3.2.2. Process of Using PSO to Optimize LSTM

Step 1: Divide experimental data into training samples, validation samples, and prediction samples.

Step 2: Initialize the speed and position of all particles. Number of neurons, batch size, and iteration times of the LSTM model are used as optimization objects.

Step 3: Divide into subgroups.

Step 4: Calculate the fitness value of each particle. Input the validation samples into the trained model for prediction, and use the average absolute percentage error of the model on the validation samples as the particle fitness value.

Fitness function f is defined as:

$$f=\frac{1}{K}{\displaystyle \sum _{i=1}^k\frac{\left|\stackrel{}{{\hat{y}}_i-y_i}\right|}{y_i}}$$

(10)

In the formula, K is the number of validation samples; ${\hat{y}}_i$ is the predicted value of the ith validation sample; y_i is the true value of the ith validation sample.

Step 5: Determine the individual extreme value and the population extreme value based on the fitness values of the initial particles. In the meantime, also consider the best position of each particle as its historically best position.

Step 6: Update the positions of ordinary particles and locally optimal particles.

Step 7: After reaching the maximum number of iterations of the PSO, obtain the optimal hyper-parameters.

Step 8: Construct an LSTM model using optimal hyper-parameters.

The architecture of PSO-LSTM is shown in Figure 5.

3.3. Dynamic Derivatives Prediction Based on Intelligent Aerodynamic Model

We focus on the pitch oscillation of the NACA 0012 airfoil around a 1/4 chord length point, and change the procedure of the angle of attack to the following:

$$\alpha =2°+1°\ast \sin \left(\omega t\right)$$

(11)

In the formula, $\omega$ is the oscillation frequency. Reduction frequency is $k=\frac{\omega C}{2V}$, where C is the reference length and V is the airspeed. Firstly, high-precision CFD technology is used to obtain time series data of moment coefficients varying with angle of attack at different reduction frequencies, then an LSTM network is constructed, with partially reduced frequency data selected as training data and the remaining data as testing data. Specifically, the time series length of the input angle of attack is set to N, and then, the input angle of attack data is an N-dimensional vector $\left[{\alpha }_{t-N+1},{\alpha }_{t-N+2},{\alpha }_{t-N+3},\dots \dots ,{\alpha }_t\right]$. The angle of attack of the first N moments (including the current moment) of the current time t is used as the input, and the moment coefficient of the current time t is used as the output.

The root mean square error $(e_{rms})$ and relative error $(e)$ are used to evaluate the prediction performance of the model. The smaller $(e_{rms})$ and $(e)$, the better the model performance.

$$e_{rms}=\sqrt{\frac{1}{n}{\displaystyle \sum ({\hat{y}}_i-y_i)}^2}$$

(12)

$$e=\frac{\sqrt{\frac{1}{n}{\displaystyle \sum ({\hat{y}}_i-y_i)}^2}}{\sqrt{\frac{1}{n}{\displaystyle \sum (y_i)}^2}}$$

(13)

In the formula, $n$ is the total number of samples, $y_i$ is the true value, and ${\hat{y}}_i$ is the predicted value.

To reduce the impact of magnitude differences between data on the convergence of the model, the data are normalized. Using the min–max normalization method, the formula is as follows:

$$\hat{x}=\frac{x-x_{\min }}{x_{\max }-x_{\min }}$$

(14)

In the formula, $x$ is the raw data, $x_{\max }$ is the maximum value in raw data, $x_{\min }$ is the minimum value in raw data, and $\hat{x}$ is the normalized data.

By using an intelligent model to obtain aerodynamic data of unknown states and combining them with the dynamic derivatives identification method based on forced oscillation, the dynamic derivatives can be quickly obtained.

4. Calculation and Verification of Dynamic Derivatives

4.1. Verification of Basic State

Grid generation is the foundation of flow field calculation. In this case, a structured grid division was performed on the NACA 0012 airfoil, with a chord length of c = 1 m. To avoid boundary interference, the distance from the far-field boundary to the airfoil surface was set to 20 times the chord length, and the airfoil surface was set as a nonslip wall boundary. To improve computational accuracy, local refinement was carried out on the grid near the airfoil, with the height of the first layer grid being 0.0001c, meeting the requirements of $y^{+}$ < 1. The grid division is shown in Figure 6. A fluent pressure-based solver was used to numerically solve two-dimensional incompressible N-S equations. Far field was set as the velocity entry boundary condition, with a corresponding Reynolds number of 1500. At this Reynolds number, the fluid viscosity is relatively high, and a laminar flow model was selected for simulation. Airfoil motion was specified by the User-Defined Function (UDF) file. The SIMPLEC algorithm was used to solve the pressure velocity coupling problem. The convection and viscosity terms in spatial discretization were solved using second-order upwind and second-order central schemes, respectively. The time term was solved using a second-order implicit two-step method. Each pitch period was time-advanced in N time steps, and 25 internal iterations were performed within each time step. To obtain a periodic stable solution, at least 50 pitch periods were calculated for each example. The pressure contour during the calculation process is shown in Figure 7.

After obtaining data using a fluent pressure-based solver, we predicted the aerodynamic force of the NACA 0012 airfoil under simple harmonic vibration using PSO-LSTM. Three sets of data, k = 0.05, 0.06, and 0.08 (equation of motion: $\alpha =2°+1°\ast \sin \left(\omega t\right)$), were used as training data, and k = 0.07 was used as testing data. The time series length N of the input angle of attack was taken as 10. Figure 8 and

Table 1. Error of predicted data (formula of motion: $\alpha =2°+1°\ast \sin \left(\omega t\right)$).

Aerodynamic Data	$${\mathit{e}}_{\mathit{r}\mathit{m}\mathit{s}}$$	$$\mathit{e}$$
Cl	0.049%	0.152%
Cm	0.00079%	2.131%

show the predicted results of aerodynamic data using an intelligent model at k = 0.07, and compare them with the CFD calculation results. The results obtained by the two methods are basically consistent, with a relative error of no more than 2.2%. It can be concluded that the intelligent aerodynamic model constructed in this paper has high accuracy in predicting aerodynamic force.

After obtaining the prediction results of unsteady aerodynamic force, the dynamic derivatives were calculated using the least-squares method. We compared this with the dynamic derivatives calculated using CFD and integration methods, as shown in

Table 2. Calculation Results of Dynamic Derivatives.

Dynamic Derivatives	Reduce Frequency	CFD	LSTM	$\mathit{e}$
$C_{m\dot{\alpha }}+C_{mq}$	0.07	−1.15349	−1.14821	0.457%

. It can be concluded that the dynamic derivatives obtained through PSO-LSTM and CFD are basically consistent. Figure 9 shows the comparison of the timeliness between the dynamic derivatives prediction method proposed in this paper and the conventional CFD method. It can be concluded that the time consumption of the method for obtaining dynamic derivatives through an intelligent model is only 29/100 that of the CFD method.

4.2. Extrapolation Performance of Dynamic Derivatives Prediction Model

For a method that predicts dynamic derivatives through an intelligent model, testing the generalization ability is essential. The LSTM network constructed in this paper has good temporal memory, so it can effectively learn and store the historical features of the flow field contained in the training data. Due to the similarity of flow field characteristics when the nonlinear characteristics of the flow field are weak, this intelligent model has good generalization ability in theory. Next, the extrapolation performance of PSO-LSTM must be tested, using three sets of data, k = 0.05, 0.06, and 0.07, as training data, and k = 0.08 as testing data. The time series length of the input angle of attack N is taken as 10. It can be seen in Figure 10 and

Table 3. Error of predicted data.

Aerodynamic Data	${\mathit{e}}_{\mathit{r}\mathit{m}\mathit{s}}$	$\mathit{e}$
$C_l$	0.217%	1.231%
$C_m$	0.00324%	5.321%

that the predicted aerodynamic coefficients exhibit oscillations with respect to $\alpha$ (angle of attack). Upon analysis, it becomes evident that this phenomenon arises due to the increased reduced frequency at k = 0.08. This increase leads to a higher instantaneous angular velocity in the wing’s motion, resulting in more pronounced variations in the flow field. In certain localized regions, even stronger nonlinear characteristics emerge. It is conceivable that enhancing the model’s performance may be achieved by introducing additional samples to encompass a broader range of motion conditions and nonlinear characteristics.

However, due to the fundamental similarity in the overall flow field characteristics, the relative error in aerodynamic forces during extrapolation remains relatively low.

Furthermore, we used the least-squares method to identify the dynamic derivatives based on the aerodynamic force prediction results of PSO-LSTM, and we used the integration method to calculate the dynamic derivatives based on the CFD simulation results.

Table 4. Calculation Results of Dynamic Derivatives.

Dynamic Derivatives	Reduce Frequency	CFD	LSTM	e
$C_{m\dot{\alpha }}+C_{mq}$	0.08	−1.17025	−1.18166	0.9750%

shows the comparison of dynamic derivative calculation results, with a relative error of 0.9750%. This indicates that the intelligent model of PSO-LSTM has good generalization ability.

4.3. The Influence of Sample Size on the Accuracy of Dynamic Derivatives Prediction

We explored the impact of sample size on the accuracy of dynamic derivatives prediction, with k = 0.06, 0.08, k = 0.05, 0.06, 0.08, and k = 0.04, 0.05, 0.06, 0.08 as training data, and k = 0.07 as testing data. Prediction results of aerodynamic force based on PSO-LSTM are shown in Figure 11, Figure 12 and Figure 13. Prediction error values of aerodynamic data are shown in

Table 5. Error of predicted data.

Training Data	Aerodynamic Data	${\mathit{e}}_{\mathit{r}\mathit{m}\mathit{s}}$	$\mathit{e}$	Modeling Time (s)
k = 0.06, 0.08	$C_l$	0.153%	0.521%	83.9
	$C_m$	0.00237%	4.212%	83.1
k = 0.05, 0.06, 0.08	$C_l$	0.049%	0.152%	126.8
	$C_m$	0.00079%	2.131%	128.7
k = 0.04, 0.05, 0.06, 0.08	$C_l$	0.043%	0.133%	166.4
	$C_m$	0.00075%	2.127%	165.1

. Obviously, as the number of training data increase, the predicted aerodynamic force over time is closer to the CFD calculation results. This is because an increase in the number of training data will improve the learning ability of recurrent neural networks for flow-field features, especially details. But the increase in the number of training data will lead to an increase in training cost of the neural network. For the example in this section, after doubling the number of training data, the prediction accuracy of the pitching moment coefficient increased by 2.085%, the training time increased by 98.67%, and the improvement ratio of accuracy to time is 0.02.

Next, we calculated the dynamic derivatives using the least-squares method based on the pitch moment coefficients obtained from three sets of training data. We used the integration method to solve the dynamic derivatives based on the calculation results of CFD.

Table 6. Calculation Results of Dynamic Derivatives.

Dynamic Derivative	Reduce Frequency	Training Data	CFD	LSTM	$\mathit{e}$
$C_{m\dot{\alpha }}+C_{mq}$	0.07	k = 0.06, 0.08	−1.15349	−1.14534	0.706%
		k = 0.05, 0.06, 0.08		−1.14821	0.457%
		k = 0.04, 0.05, 0.06, 0.08		−1.14835	0.445%

shows the calculation results of dynamic derivatives. It can be concluded that high accuracy has been achieved with the training data of k = 0.05, 0.06, and 0.08. Calculation results of dynamic derivatives under the training data of k = 0.04, 0.05, 0.06, and 0.08 show no significant improvement compared to the identification results under the training data of k = 0.05, 0.06, and 0.08. Because the flow field is generally in a linear or weakly nonlinear state when calculating dynamic derivatives, and the changes in the flow field are relatively mild, a small number of training data can be used for modeling.

5. Conclusions

The rapid identification of dynamic derivatives is one of the key processes in the unsteady aerodynamic design of advanced aircraft. In view of the low efficiency of the identification method of dynamic derivatives, a new method based on a particle swarm optimization and long short-term memory network aerodynamic model was constructed to quickly predict the aerodynamic force of unsteady airfoil motion and combined with the mathematical model of dynamic derivatives to quickly predict the dynamic derivatives. In this paper, validation analysis is conducted using NACA 0012 two-dimensional airfoil dynamic derivatives identification as an example. It is important to note that when there is a change in the airfoil shape, it necessitates the recalibration of Computational Fluid Dynamics (CFD) simulations to obtain training data for the neural network. However, our research indicates that the flow conditions relevant to dynamic derivatives often exhibit linear or weakly nonlinear characteristics. This implies that even in scenarios where data samples are relatively scarce, precise dynamic derivative results can still be obtained. Once the neural network training is completed, we can employ this trained model for the rapid and efficient prediction of the aerodynamic response and dynamic derivatives of the airfoil under unsteady conditions, without relying on the cumbersome CFD simulations. This approach not only reduces the demand for computational resources but also provides designers with faster feedback, enabling them to respond more flexibly to changes and adjustments in airfoil design. It presents an innovative and cost-effective method for the field of aircraft dynamic derivative computation. The conclusions are as follows:

Particle swarm optimization can be used to optimize the hyper-parameters of an LSTM network to ensure the training efficiency and accuracy of the model, ultimately obtaining the unsteady aerodynamic model constructed under a small number of sample data.

The PSO-LSTM model established in this paper can accurately predict the aerodynamic force in unknown motion states and then calculate dynamic derivatives combined with the mathematical model of dynamic derivatives, with an error of no more than 1% compared to CFD and a 70% improvement in efficiency.

Based on the time memory ability of LSTM, the model constructed in this paper has good generalization ability. The relative error of the extrapolated dynamic derivative calculation results is less than 1%.

Although the number of training data has a significant impact on the prediction performance of the model, the flow-field conditions related to dynamic derivatives identification are mostly linear or weakly nonlinear, so using a small number of training data can also obtain accurate dynamic derivative prediction results.