# Kolmogorov-Arnold Networks for Reduced-Order Modeling in Unsteady Aerodynamics and Aeroelasticity
## 1 Introduction
At present, machine learning-based approaches for constructing such models primarily include methods based on radial basis function neural networks (RBFNNs) [19; 20], neurofuzzy models [21; 22], residual neural networks (ResNets) [23], and deep neural networks (DNNs) [24; 25]. [PERSON] et al. [19] proposed an efficient method for constructing a nonlinear aerodynamic reduced-order model by introducing a recursive structure into the radial basis function neural network (RBFNN), enabling the prediction of limit cycle oscillations. [PERSON] et al. [22] developed a nonlinear system identification strategy by combining a recurrent local linear neurofuzzy model with a multilayer perceptron, thereby enabling accurate and efficient reduced-order modeling of unsteady aerodynamic forces. [PERSON] et al. [23] proposed a reduced-order model based on a residual convolutional neural network within a Nonlinear AutoRegressive model with an eXogenous input (NARX) framework, successfully predicting the strongly nonlinear aerodynamic hysteresis behavior of an airfoil undergoing pitching motion. [PERSON] et al. [24] developed a deep neural network-based ROM capable of capturing both unsteady aerodynamic forces and aeroelastic responses over a range of Mach numbers. Despite these advances, existing machine learning-based ROMs still encounter several challenges. In particular, issues related to prediction accuracy under complex nonlinear flow conditions, training stability during model development, and generalization to unseen flow regimes remain unresolved. These limitations underscore the need for further improvements in network architectures to enhance the overall robustness and applicability of data-driven models. Recently, Kolmogorov-Arnold Networks (KANs) have attracted increasing attention for their ability to learn complex nonlinear mappings with enhanced interpretability and flexibility. Inspired by the Kolmogorov-Arnold representation theorem, KANs replace fixed activation functions with learnable univariate transformations along each edge, enabling the dynamic construction of adaptive basis functions from data [26; 27]. This architectural innovation has led to a series of theoretical and algorithmic advancements. Structural improvements include spline-based parameterizations, constrained polynomial representations, and differentiable function modules designed to improve smoothness and numerical stability. Concurrently, researchers have developed gradient regularization techniques, sparsity-promoting constraints, and dynamic pruning methods to reduce overfitting and enhance generalization performance [28]. These innovations have led to strong empirical performance across diverse applications. In symbolic regression, structural refinements, such as the use of monotonic cubic Hermite splines in place of B-splines in KANs, can significantly improve interpretability [29]. In scientific machine learning, KANs have shown strong potential as an alternative to traditional neural architectures by serving as the backbone of neural ODE frameworks. They have demonstrated superior performance in modeling time-dependent and grid-sensitive dynamical systems, achieving higher accuracy, better generalizability, and more efficient parameterization than standard neural networks [30]. In control theory, KANs have been applied to optimal control problems involving continuous-time dynamics, where their integration within the KAN-based framework has demonstrated improved accuracy and efficiency compared to classical MLPs [31]. Moreover, KANs are increasingly being explored for time-series prediction tasks [32; 33; 34], where capturing temporal dependencies and nonlinear evolution patterns is critical. Recent studies have incorporated KANs into recurrent frameworks, enabling accurate forecasting across a range of sequential data problems. These advancements underscore the suitability of KANs for modeling complex dynamical systems characterized by nonlinear and time-delayed behaviors.