# Kolmogorov-Arnold Networks for Reduced-Order Modeling in Unsteady Aerodynamics and Aeroelasticity
## 5 Conclusions  
In this paper, we propose a reduced-order model for unsteady aerodynamics and aeroelasticity based on Kolmogorov-Arnold Networks are well suited for accurately capturing complex, nonlinear relationships in data, which allows for highly accurate predictions and models across various applications. To capture the time-delay effects, we incorporate a recurrent structure and a gating mechanism, which enable the model to selectively control and preserve long-term temporal dependencies. This design effectively facilitates the modeling of unsteady aerodynamics and aeroelastic responses. The main conclusions are as follows:  
1. The ROM can accurately predict nonlinear unsteady aerodynamic coefficients under different amplitudes and reduced frequencies.
2. The ROM can accurately predict the limit cycle oscillation responses and trends under different flutter speeds.
3. The computational efficiency significantly improved compared to direct CFD computations, and the ROM exhibited better generalization than the previous model. These  
\begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{\(\alpha_{\text{max}}\)} & \multirow{2}{*}{\((h/b)_{\text{max}}\)} & \multirow{2}{*}{\(k\)} & \multicolumn{3}{c}{**Relative Error (ROM1/ROM2)**} \\ \cline{4-5}  & & & & \(RE\) (\(C_{l}\)) & \(RE\) (\(C_{m}\)) \\ \hline \multicolumn{5}{c}{Pitching motion} \\
4.021\({}^{\circ}\) & 0 & 0.058 & 1.89\%/1.94\% & 3.96\%/4.88\% \\
5.157\({}^{\circ}\) & 0 & 0.119 & 1.31\%/2.21\% & 4.02\%/5.15\% \\
5.749\({}^{\circ}\) & 0 & 0.091 & 1.64\%/2.55\% & 7.15\%/6.17\% \\ \multicolumn{5}{c}{Plunging motion} \\
0\({}^{\circ}\) & 0.09 & 0.166 & 1.75\%/1.60\% & 4.24\%/5.75\% \\
0\({}^{\circ}\) & 0.15 & 0.063 & 0.92\%/2.79\% & 3.96\%/6.82\% \\
0\({}^{\circ}\) & 0.21 & 0.107 & 2.01\%/1.77\% & 6.51\%/4.64\% \\ \hline \end{tabular}
\end{table}
Table 2: Comparison of relative error values between different ROMs and CFD simulations.  
Figure 19: Comparison of LCO trends predicted by different ROMs with CFD results. (**a**) Plunging amplitude vs. \(V_{f}\). (**b**) Pitching amplitude vs. \(V_{f}\).  
advantages underscore its potential to reduce computational costs while maintaining high fidelity, making it a promising approach for aerodynamic and aeroelastic analysis.  
In the future, we aim to further develop and refine Kolmogorov-Arnold Network-based models for broader application to complex aerodynamic and aeroelastic phenomena, such as stall flutter, three-dimensional aeroelastic problems, and transonic buffet. The goal is to enhance both predictive accuracy and computational efficiency, thereby extending the practical application of these models across a wider range of scenarios.  
Conceptualization, Y.Z.; methodology, Y.Z. and H.T.; software, Y.Z. and L.W.; validation, Y.Z., H.T. and L.W.; formal analysis, Y.Z.; investigation, H.T.; resources, G.Z.; data curation, G.Z.; writing--original draft preparation, Y.Z. and L.W.; writing--review and editing, G.Z.; visualization, H.T.; supervision, G.Z. and G.Y.; project administration, G.Z.; funding acquisition, G.Y. and G.Z. All authors have read and agreed to the published version of the manuscript.  
This work is supported by the National Natural Science Foundation of China (Grant No. 12372171).  
Not applicable.  
The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding authors.  
The authors give thanks to all colleagues for their generous help in this work.  
The authors have no conflicts to disclose.