## 3 Methodology
### The Importance of Weather in Airport Flight Resilience  
#### 3.2.1 Establishing the Mapping Relationship between Weather and Airport Resilience  
This study introduces an innovative approach by exploring the regression accuracy of Kolmogorov-Arnold Networks (KANs) in terms of weather conditions and flight resilience. KANs leverage the Kolmogorov-Arnold representation theorem for tasks such  
Figure 2: Calculation procedure of operational resilience of flight delays. as time-series prediction and function approximation. Unlike traditional neural networks, KANs utilize learnable univariate spline-based activation functions along the network's edges rather than relying on fixed activation functions at the nodes. This unique architecture enhances the accuracy of predictions under varying weather conditions and offers advantages like faster neural scaling laws and improved interpretability. The methodology based on KAN principles for time-series forecasting is outlined in Figure 3.  
Step 1. Problem Formulation  
We frame the prediction of airline operational resilience as a time-series forecasting task. The goal is to predict the resilience index \(\mathrm{R}_{1}^{k}\), which reflects the operational status of an airline at a future time step \(\Delta t_{i}\), based on historical weather data from the airport and operational conditions. Let the historical weather data in the airport over an operational day (time range \(t_{0}-c\) to \(t_{0}-1\)) be represented by the vector \(x_{t_{0}-c\cdot t_{0}-1}=\left[x_{t_{0}-c},x_{t_{0}-c+1},\cdots,x_{t_{0}-1}\right]\), in which each term corresponds to a set of weather features at time \(\Delta t\). The future resilience values to be predicted are defined by \(R_{t_{0}:T}=\left[R_{t_{0}},R_{t_{0}+1},\cdots,R_{T}\right]\). The objective is to find a function \(f\) such that \(R_{t_{0}:T}\approx f(x_{t_{0}-c\cdot t_{0}-1})\). This function will be approximated using a Kolmogorov-Arnold Network (KAN). Step 2. Kolmogorov-Arnold Network (KAN) Architecture  
The KAN model is based on the Kolmogorov-Arnold representation theorem, which allows for the decomposition of multivariate functions into compositions of univariate functions. Unlike traditional neural networks, in which weights are linear, KANs employ learnable activation functions along the edges, making them better suited for capturing nonlinear relationships in the data. We use a multi-layer KAN architecture in which the inputs are the historical weather data points, and the outputs are the predicted resilience indices. The model layers can be described as follows:  
* Input Layer: The input layer consists of nodes corresponding to the number of weather features over the time range \(t_{0}-c\) to \(t_{0}-1\). * Hidden Layers: The hidden layers employ spline-based activation functions (e.g., B-splines) along the edges. These splines allow the network to capture nonlinear dependencies between weather conditions and airline operational resilience. Each hidden layer applies a transformation \(\Phi_{i}\) to the inputs from the previous layer. Figure 3: Flow chart of KAN algorithm. * Output Layer: The output layer generates the predicted resilience values for the future time steps \(t_{0}\) to \(T\). The final layer combines the outputs of the previous layers using summation operations. We define the shape of a KAN by \([n_{1},\cdots,n_{L+1}]\), where \(L\) denotes the number of layers of the KAN. The general form of the network can be expressed as follows: \[R=KAN(x)=(\Phi_{L}\cdot\Phi_{L-1}\cdot\cdots\cdot\Phi_{1})(x)\] (16) where each \(\Phi_{i}\) is a layer transformation based on the spline-based activation functions. Step 3. Training Procedure The KAN is trained in a supervised manner using historical weather and resilience data. The training process involves optimizing the spline-based activation functions and network parameters using backpropagation. The loss function used for optimization is the Mean Absolute Error (MAE) between the predicted and actual resilience values over the prediction horizon \(T\).