# Kolmogorov-Arnold Networks for Reduced-Order Modeling in Unsteady Aerodynamics and Aeroelasticity
## 4 Results and Discussion
### Training of Reduced-Order Model  
The choice of input signal significantly influences the accuracy and generalizability of the reduced-order model, necessitating a random excitation that encompasses a broad frequency spectrum and amplitude variations [18]. In this study, we employ filtered white Gaussian noise (FWGN) signals as inputs for both the training and validation datasets, as illustrated in Figure 7, where \(T\) represents discrete time steps. Specifically, the training dataset comprises 5000 time steps with relatively large amplitudes, whereas the validation dataset consists of 2000 time steps featuring smaller amplitudes. The smaller-amplitude input utilized in the validation set is intentionally chosen to rigorously assess and enhance the model's generalization capabilities across varying input conditions. Additionally, the spectral content of these signals is examined using Fast Fourier Transform (FFT) analysis to confirm adequate frequency coverage, as presented in Figure 8.  
Subsequently, the generated input signals at each discrete sampling point are processed through the CFD solver to obtain the corresponding aerodynamic coefficients, thereby establishing comprehensive input-output (I/O) pairs. These resulting I/O datasets serve as the fundamental building blocks for developing the training and validation datasets, thus providing a robust basis for constructing and evaluating the reduced-order model described previously.  
The sliding window technique is a prevalent preprocessing strategy for sequential data, particularly beneficial in time-series modeling applications [38]. This method involves segmenting the original sequential dataset into multiple overlapping subsequences of fixed length, thereby explicitly capturing temporal dependencies inherent within the data. [PERSON] et al. [39] successfully employed the sliding window approach to preprocess input data for gust response modeling, significantly improving predictive accuracy. Inspired by their results, we also adopt this technique to systematically organize our sequential input data. This structured preprocessing effectively preserves the temporal dynamics inherent in the original dataset, enabling the proposed KAN-based reduced-order model to capture temporal dependencies more accurately, thereby significantly enhancing predictive accuracy and overall model performance.  
In this work, the model was trained using the backpropagation-through-time (BPTT) algorithm, a widely adopted method for optimizing sequential neural networks. To enhance training efficiency and reduce the likelihood of convergence to local minima, the Adam optimizer, which adaptively adjusts learning rates during the optimization process, was  
Figure 8: Fourier analysis of training and validation cases in pitching motion.  
Figure 7: Input signal for training and validation. (**a**) Training input signal with 5000 time steps. (**b**) Validation input signal with 2000 time steps.  
employed to minimize the mean-squared error (MSE) loss between predicted and actual aerodynamic coefficients. Furthermore, a validation set was explicitly established during training to regularly assess the model's predictive performance on unseen data, thereby improving the generalization capability and robustness of the model. The trained model comprises two hidden layers, each with 32 units.