# Kolmogorov-Arnold Networks for Reduced-Order Modeling in Unsteady Aerodynamics and Aeroelasticity
## 2 Reduced-Order Modeling
### Kolmogorov-Arnold Gated Recurrent Networks  
The Kolmogorov-Arnold Gated Recurrent Network (KAGRN) represents an advanced recurrent neural network architecture designed to effectively capture complex temporal dynamics. The network primarily consists of two components: the Kolmogorov-Arnold Recurrent Network (KARN) layers and the gating mechanisms inspired by Gated Recurrent  
Figure 1: The architecture of Kolmogorov–Arnold Networks. Units (GRUs). Specifically, KARN layers utilize learnable activation functions based on the Kolmogorov-Arnold representation theorem to accurately capture complex nonlinearities, while the gating mechanisms selectively control and maintain long-term temporal dependencies. The architecture of Kolmogorov-Arnold Gated Recurrent Networks is shown in Figure 2. Inspired by the Temporal Kolmogorov-Arnold Network (TKAN) [32], we develop this architecture to effectively model intricate temporal relationships inherent in aerodynamic data, enabling the accurate identification and prediction of unsteady aerodynamic and aeroelastic behaviors. To clearly illustrate the proposed architecture, Figure 2 further details the internal structure of the KARN layers, highlighting their mathematical formulation and computational flow. The KARN is designed specifically to model sequential data, capturing context-sensitive dependencies through an integration of Kolmogorov-Arnold Networks and memory-based recurrence mechanisms. KARN modifies each transformation function \(\varphi_{l}\) within the traditional KAN architecture by embedding a temporal dimension. Formally, the input to each recursive sublayer at time \(t\) is expressed by the following relation:  
\[i_{l,t}=W_{l,x}x_{t}+W_{l,n}n_{l,t-1} \tag{3}\]  
where \(W_{l,x}\) and \(W_{l,n}\) denote the weight matrices connecting current inputs \(x_{t}\) and previous memory states \(n_{l,t-1}\), respectively. Given that the dimensions of the input and output in layer \(l\) are \(KAN_{in}\) and \(KAN_{out}\), the associated weight matrices have dimensions \(W_{l,x}\in\mathbb{R}^{(d,KAN_{in})}\) and \(W_{l,n}\in\mathbb{R}^{(KAN_{out},KAN_{in})}\). Each transformed sublayer state \(i_{l,t}\) is then passed through a nonlinear KAN function as  
\[\tilde{p}_{l,t}=\varphi_{l}(i_{l,t}) \tag{4}\]  
where \(\tilde{p}_{l,t}\) represents the nonlinear transformed output at time \(t\). Furthermore, the model incorporates a recurrent memory mechanism update expressed by the following equation:  
\[n_{l,t}=W_{mn}n_{l,t-1}+W_{pn}\tilde{p}_{l,t} \tag{5}\]  
where \(n_{l,t}\) represents the updated hidden memory state at layer \(l\) and time \(t\). Specifically, \(n_{l,t}\) encapsulates the learned latent representations retained and refined across time steps. In this recurrence, \(W_{mn}\) serves to propagate and maintain previous hidden states, while \(W_{pn}\) integrates the current nonlinear outputs into the evolving memory states. This design effectively enables the network to capture temporal dependencies by dynamically adjusting and propagating past latent representations across time steps. Figure 2: The architecture of Kolmogorov–Arnold Gated Recurrent Networks. Gating mechanisms are known to effectively control the propagation of information across time steps, significantly improving a neural network's capability to capture long-term dependencies and mitigating issues such as vanishing or exploding gradients commonly encountered in recurrent structures. Building upon the strengths of the KARN layers, we further incorporate gating mechanisms inspired by Gated Recurrent Units to develop the Kolmogorov-Arnold Gated Recurrent Network. Given an input vector \(x_{t}\in\mathbb{R}^{d}\), the update gate \(z_{t}\) and reset gate \(r_{t}\) are computed at each time step \(t\) as follows:  
\[z_{t}=\sigma(W_{z}x_{t}+U_{z}h_{t-1}+b_{z}),\quad r_{t}=\sigma(W_{r}x_{t}+U_{r }h_{t-1}+b_{r}) \tag{6}\]  
where \(W_{r},W_{z}\) and \(U_{r},U_{z}\) represent corresponding weight matrices, and \(b_{z},b_{r}\) represent corresponding bias vectors.