# Kolmogorov-Arnold Networks for Reduced-Order Modeling in Unsteady Aerodynamics and Aeroelasticity
## 1 Introduction  
In the field of aerospace engineering, accurately predicting aerodynamic and aeroelastic responses is vital for numerous tasks such as system control [1], shape optimization [2], and multifield coupling [3]. High-fidelity computational fluid dynamics (CFD) simulations have become indispensable tools in capturing complex aerodynamic interactions, particularly in scenarios involving flutter [4], transonic buffet [5], and other research in flight mechanics on controllability and stability [6]. However, these CFD techniques require substantial computational resources and prolonged processing durations, restricting their practical utility for iterative design analyses and real-time applications. To address these computational challenges, researchers have increasingly turned to reduced-order models (ROMs), which employ system identification methods [7] based on measured input-output data from the system of interest. Using a concise mathematical expression within a black-box framework, these ROMs capture the essential system behavior while significantly reducing computational costs, thus facilitating real-time or iterative analyses. However, traditional reduced-order modeling methods frequently exhibit limitations when dealing with strongly nonlinear flow behaviors or intricate transient aerodynamic phenomena. Linear approaches, including AutoRegressive with eXogenous inputs (ARX) [8],AutoRegressive Moving Average (ARMA) [9], and AutoRegressive Integrated Moving Average (ARIMA) [10], have demonstrated notable efficiency in predicting aerodynamic responses but typically fail to accurately represent the complexities associated with nonlinear flow interactions. Although nonlinear polynomial-based methods, such as Wiener and Volterra series expansions [11; 12], provide enhanced approximation capabilities, their accuracy and reliability degrade as the system's nonlinearity and complexity increase. These limitations underscore a fundamental challenge in aerodynamic modeling: the increasing inadequacy of conventional reduced-order methods when applied to systems with strong nonlinearity, high dimensionality, or unsteady characteristics. This issue becomes particularly pronounced in transonic flow regimes, where shock-boundary layer interactions [13], shock oscillations [14], and other nonlinear wave phenomena [15] introduce substantial complexity. Under such conditions, flow behaviors can exhibit abrupt changes and sensitive dependence on initial or boundary conditions, further straining the capabilities of linear or low-order nonlinear models. With many modern aerodynamic applications operating near transonic speeds, there is a pressing need for more adaptable and robust modeling frameworks that can accurately capture the associated dynamics. To overcome these limitations, recent advancements have turned towards leveraging machine learning techniques, particularly neural networks, to develop more accurate and efficient reduced-order models. Neural networks have shown great potential in capturing complex, nonlinear relationships inherent in fluid dynamics that traditional ROMs may miss. By training on large datasets generated from high-fidelity CFD simulations, these neural network-based models can learn the intricate patterns and dependencies within the data, enabling them to predict flow behaviors with high precision [16; 17]. This approach is bolstered by the ability of data-driven models to generalize across different flow conditions, making them particularly valuable for tasks such as flow control and optimization. In addition, advances in computational resources and algorithmic techniques have facilitated the training of more sophisticated models, further enhancing their predictive capabilities [18]. Accurate prediction of unsteady aerodynamic forces is crucial for both unsteady aerodynamics and aeroelasticity modeling. In essence, aeroelasticity modeling can be regarded as one specific application of unsteady aerodynamic reduced-order modeling.