# Kolmogorov-Arnold Networks for Reduced-Order Modeling in Unsteady Aerodynamics and Aeroelasticity
## 4 Results and Discussion
### Prediction of Limit Cycle Oscillations  
In this subsection, the nonlinear aerodynamic reduced-order model is coupled with the structural equations of motion to analyze the aeroelastic system. To rigorously assess the predictive capability and precision of the proposed ROM in forecasting limit cycle oscillations (LCO), standard benchmark parameters commonly used in aeroelastic studies [41] are adopted. Specifically, the following parameters are used: the NACA64A010 airfoil, \(Ma=0.8\), \(\alpha_{0}=0^{\circ}\), \(a=-0.6\), \(x_{\alpha}=0.25\), \(r_{\alpha}^{2}=0.75\), \(\omega_{h}/\omega_{\alpha}=0.5\), and \(\mu=75\). The use of these established aeroelastic parameters ensures the ROM is tested under physically meaningful and widely recognized conditions, thus demonstrating its suitability and reliability for predicting nonlinear aeroelastic phenomena.  
Figures 15 and 16 present a detailed comparison of the limit cycle oscillation responses predicted by the proposed nonlinear ROM and those obtained from high-fidelity CFD simulations at \(V_{f}=0.74\) and \(V_{f}=0.8\), respectively. The results demonstrate that the ROM successfully reproduces the amplitude and waveform of the LCO with high fidelity, highlighting its effectiveness in predicting essential nonlinear aeroelastic behavior. In particular, the ROM exhibits excellent agreement with the initial oscillatory response. However, a gradual phase shift becomes apparent as time progresses. This accumulation of phase discrepancy is primarily attributed to slight differences in the dominant oscillation frequency predicted by the ROM relative to the CFD results. Although these frequency deviations are minimal, they tend to accumulate over time, resulting in a visible phase shift after several oscillation cycles. Such behavior is commonly observed in reduced-order models and reflects the challenges in precisely matching long-term temporal dynamics in nonlinear systems.  
Figure 15: Comparison of LCO responses with CFD results at \(V_{f}=0.74\). **(a)** Plunging motion vs. time. **(b)** Pitching motion vs. time.  
Figure 17 further illustrates the comparison of LCO trends across varying flutter speeds, as predicted by the ROM and computed via CFD. The ROM captures the overall trend in LCO amplitude with impressive accuracy, and the predicted results are in close agreement with the CFD data over the entire range. Subsequently, we extend the proposed ROM to the prediction of three-dimensional aeroelastic responses. For validation, the benchmark Active Controls Technology (BACT) model [42] is employed, characterized by an NACA0012 airfoil with parameters \(\alpha_{0}=0^{\circ}\), \(a=0\), \(x_{\alpha}=0.00350\), \(r_{\alpha}^{2}=1.036\), and \(\omega_{h}/\omega_{a}=0.6539\). Following the same training strategy as in the two-dimensional case, the ROM is applied to predict the three-dimensional aeroelastic responses, and its results are evaluated against CFD simulations. Figure 18 presents a comparison of the LCO trends predicted by the ROM and those obtained from CFD over a range of flutter speeds. The results indicate that the ROM predictions exhibit close agreement with the CFD solutions, highlighting the model's ability to generalize effectively to three-dimensional aeroelastic response prediction.  
Figure 16: Comparison of LCO responses with CFD results at \(V_{f}=0.8\). (**a**) Plunging motion vs. time. (**b**) Pitching motion vs. time.  
Figure 17: Comparison of LCO trends with CFD results. (**a**) Plunging amplitude vs. \(V_{f}\). (**b**) Pitching amplitude vs. \(V_{f}\).  
In summary, the KAN-based ROM demonstrates strong predictive capabilities for limit cycle oscillations, providing results that are in close agreement with those of direct CFD simulations. This underscores the capability of the proposed data-driven framework to reliably predict nonlinear aeroelastic responses, demonstrating its potential for broader applications in nonlinear dynamical systems.