# Kolmogorov-Arnold Networks for Reduced-Order Modeling in Unsteady Aerodynamics and Aeroelasticity
## 4 Results and Discussion
### Prediction of Unsteady Aerodynamics  
In this subsection, we primarily introduce the application of the proposed KAN-based reduced-order model for predicting nonlinear unsteady aerodynamic forces. To validate the effectiveness and accuracy of the proposed ROM in capturing complex unsteady aerodynamic behaviors, we applied it to predict the aerodynamic coefficients of an NACA 64A010 airfoil undergoing single-degree-of-freedom forced pitching and plunging harmonic motions. The airfoil motions were centered at the 0.2 chord position, operating at a Mach number of 0.8, which is characterized by the formation of strong shock waves on the airfoil surface. The goal was to assess the ROM's capability in accurately capturing the nonlinear aerodynamic characteristics across various amplitudes and reduced frequencies \(k\). Predicted aerodynamic coefficients from the KAN-based ROM were compared against high-fidelity CFD simulations to quantitatively evaluate the accuracy and reliability of the proposed approach. Figures 9-11 present predictions of aerodynamic coefficients for the airfoil undergoing single-degree-of-freedom harmonic pitching motions. Similarly, Figures 12-14 illustrate predictions for harmonic plunging motions. These figures notably highlight the hysteresis loops of the lift and moment coefficients, demonstrating the model's ability to accurately replicate the complex unsteady aerodynamic phenomena. The relative error (RE) between the predicted ROM results and CFD simulation data was quantified using the formula provided in Equation (14). A comprehensive summary of test case parameters, along with calculated relative errors for each scenario, is presented in Table 1. The results consistently demonstrate that the proposed KAN-based ROM achieves high precision across both pitching and plunging motions for various amplitudes and reduced frequencies, aligning closely with the high-fidelity CFD results. Figure 9: Comparison of predicted aerodynamic coefficients with CFD results for case 1. (**a**) Hysteresis loop of lift coefficient. (**b**) Hysteresis loop of moment coefficient. \[RE=\frac{\|y_{\text{ROM}}-y_{\text{CFD}}\|_{F}}{\|y_{\text{CFD}}\|_{F}}\times 100\% \tag{14}\]  
In particular, lift coefficient predictions exhibit consistently minimal deviations, underscoring the robustness of the proposed ROM in capturing the aerodynamic lift accurately. However, the prediction accuracy for the moment coefficient, which is more sensitive to nonlinear aerodynamic effects, is comparatively lower than that of the lift coefficient. This observation further supports the notion that the moment coefficient inherently presents greater challenges for accurate prediction due to its pronounced nonlinear behavior. Figure 11: Comparison of predicted aerodynamic coefficients with CFD results for case 3. (**a**) Hysteresis loop of lift coefficient. (**b**) Hysteresis loop of moment coefficient. Figure 12: Comparison of predicted aerodynamic coefficients with CFD results for case 4. (**a**) Hysteresis loop of lift coefficient. (**b**) Hysteresis loop of moment coefficient. Figure 10: Comparison of predicted aerodynamic coefficients with CFD results for case 2. (**a**) Hysteresis loop of lift coefficient. (**b**) Hysteresis loop of moment coefficient. It is important to note that the accuracy of the moment coefficient prediction for case 5 in this study is notably lower. Given the data-driven nature of the proposed ROM, its performance is intrinsically linked to the quality and representativeness of the input training data [40]. Consequently, even KAN-based models may encounter limitations in certain complex scenarios. Therefore, the reduced accuracy observed in this case likely stems from the model's inability to sufficiently extract necessary information from the provided input data. Nonetheless, the overall performance of the KAN-based ROM remains reliable, demonstrating a high degree of precision in predicting nonlinear unsteady aerodynamic forces.