## 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. All authors contributed to the study conception and design. Material preparation, data collection, and data analysis were performed by [PERSON] and [PERSON]. The first draft of the manuscript was written by [PERSON], and all authors commented on previous versions of the manuscript. All authors have read and agreed to the published version of the manuscript. This research was funded by National Natural Science Foundation of China grant number 12202363 and Natural Science Foundation of Chongqing, China grant number 2023 NSCQ-MSX0123. Not applicable. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Dynamic Derivative** & **Reduce Frequency** & **Training Data** & **CFD** & **LSTM** & \(e\) \\ \hline \multirow{4}{*}{\(C_{mix}+C_{mq}\)} & \multirow{4}{*}{0.07} & k = 0.06, 0.08 & \multirow{4}{*}{\(-1.15349\)} & \(-1.14534\) & 0.706\% \\ \cline{3-6}  & & k = 0.05, 0.06, 0.08 & & \(-1.14821\) & 0.457\% \\ \cline{1-1} \cline{3-6}  & & k = 0.04, 0.05, 0.06, 0.08 & & \(-1.14835\) & 0.445\% \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Calculation Results of Dynamic Derivatives. Data Availability Statement: The data presented in this study are available on request from the corresponding author.