The Importance of Weather Factors in the Resilience of Airport Flight Operations Based on Kolmogorov–Arnold Networks (KANs)

Abstract

This study analyzes the impact of weather factors on the resilience of airport flight operations, focusing on flight performance, economic outcomes, and transportation capacity. A Kolmogorov–Arnold Network (KAN) model was employed to identify key weather variables and establish the relationship between these factors and airport operational resilience. Xi’an Xianyang International Airport was used as a case study, with the weights of various routes determined using grey relational analysis, considering average daily flight volume, flight distance, and airport flow stability indicators. Flight operation records and weather data were utilized to assess the influence of critical weather factors on key operational resilience metrics. The findings reveal that routes in economically developed areas exert a more pronounced effect on flow stability. Temperature and wind speed emerged as the most influential factors, with importance values of 0.35 and 0.32, respectively, about flight operations and economic performance. Furthermore, changes in wind direction and wind speed had the greatest impact on transportation capacity, with importance values of 0.7 and 0.65, respectively. These results highlight the need for special attention to weather factors such as temperature and wind speed during flight scheduling and risk assessment to ensure operational safety, efficiency, and economic viability.

1. Introduction

Air transportation is indispensable in connecting the world, facilitating the movement of goods and people, and driving global economic and social activities. With the continued growth of the aviation industry, airports have become essential hubs of economic development, especially in regions where air travel is the primary means of long-distance transportation. However, one persistent challenge facing the aviation industry is flight delays, which disrupt passengers’ plans and impose operational and financial burdens on airlines and airports. Weather conditions stand out as a dominant cause of these delays, thus contributing to significant disruptions in flight schedules. Extreme weather events such as strong winds, heavy rain, snowstorms, and low visibility can severely affect flight operations by reducing airport capacity, delaying departures, and extending the time required for landing procedures [1].

The impact of weather on aviation networks is not limited to individual flights or specific airports. Given the highly interconnected nature of Air Traffic Networks (ATNs), disruptions in one location can trigger a cascade of delays throughout the network. These networks are characterized by few hub airports that manage most connections, while smaller airports typically serve fewer flights. This scale-free structure provides robustness against random disruptions but leaves the system vulnerable to targeted failures, particularly in hub airports [2,3,4]. Consequently, it becomes critical to understand how weather-related disruptions in key airports can affect overall network resilience and the steps that can be taken to mitigate these impacts [5,6].

The National Academy of Sciences defined resilience as the ability of a system to plan for, absorb, recover from, and adapt to adverse events [7]. Within the context of air traffic, resilience refers to the capacity of the system to prevent or minimize disruptions caused by various factors, including weather, and to recover from such events effectively. Bruneau et al. extended this concept by introducing four dimensions for evaluating the resilience of critical infrastructure: detection, resistance, recovery, and adaptability [8]. These dimensions are particularly relevant for airports, which must remain operational during adverse conditions and recover quickly from disruptions to minimize the impact on passengers and airlines [9].

To assess the resilience of airport operations, researchers have developed Key Performance Indicators (KPIs) that measure the system’s capacity, efficiency, and environmental performance. These KPIs, set by the International Civil Aviation Organization (ICAO), include metrics such as airport capacity (e.g., demand, traffic complexity, and air traffic control workloads), operational efficiency (e.g., flight delays and associated costs), and environmental factors (e.g., fuel consumption and greenhouse gas emissions due to delays) [10,11]. These indicators provide a framework for quantifying the impact of disruptions on airport operations, thus helping stakeholders evaluate the effectiveness of mitigation strategies.

Despite the development of sophisticated models to predict flight delays, including Artificial Neural Networks (ANNs), there are still significant gaps in predicting the impact of weather on aviation resilience. Most studies have focused on extreme weather events, such as hurricanes, thunderstorms, and snowstorms, which are well-documented and often predictable [12,13]. However, gradual weather changes, which occur more frequently, can significantly impact airport operations by reducing runway capacity and increasing delays. For example, changes in wind direction and speed, temperature fluctuations, and variations in visibility due to fog or cloud cover can disrupt the scheduling and operation of flights [14]. These weather changes are often more challenging to predict and manage because they do not trigger immediate emergency responses but can gradually erode the efficiency of airport operations [15].

In China, the impact of weather on airport operations is particularly relevant, given the country’s rapid growth in air traffic and the increasing importance of its major airports. Xi’an Xianyang International Airport, one of the largest airports in Northwest China, plays a crucial role in connecting the region to both domestic and international destinations. However, due to its geographical location, the airport is exposed to various weather conditions, including extreme temperatures, dust storms, and fog [16,17]. These weather events pose significant challenges to airport operations, leading to frequent delays and cancellations. Despite the importance of Xi’an Xianyang International Airport as a regional hub, there has been relatively little research on the resilience of its operations in the face of weather-related disruptions.

This study aims to address this gap by applying an advanced predictive model, the Kolmogorov–Arnold Network (KAN), to assess the impact of weather on the resilience of airport flight operations. The KAN model is particularly suited for tasks involving time-series predictions and function approximation, making it ideal for analyzing how weather patterns affect flight schedules. Unlike traditional neural networks, which rely on fixed activation functions at nodes, the KAN model employs learnable, spline-based activation functions along the network’s edges, causing these delays and allowing it to capture nonlinear relationships more effectively [18]. This unique architecture enhances the accuracy of predictions. It improves the interpretability of the results and causes of these delays, thus making it easier to identify the critical weather factors influencing flight operations [19].

Using the KAN model to analyze weather data and flight operation records from Xi’an Xianyang International Airport, this study seeks to provide a comprehensive understanding of how weather conditions impact the resilience of airport operations. Specifically, this study identifies the critical weather variables influencing flight delays, cancellations, and overall airport capacity. The Grey Relational Analysis (GRA) method is used to quantify the importance of weather factors, such as wind speed, temperature, and visibility, in determining the resilience of flight operations [20].

Previous studies have highlighted the importance of weather factors such as wind speed and direction, temperature, and visibility in influencing airport capacity and flight delays [21]. For example, Janić found that extreme weather events primarily cause flight delays in major U.S. airports [22]. Similarly, Rodríguez-Sanz et al. used a Bayesian Network approach to analyze the impact of extreme weather on flight delays at Madrid-Barajas Airport, cause of these delays, thus demonstrating that different stages of flight operations, such as final approach and taxiing, are affected to varying degrees by weather conditions [23]. However, these studies primarily focus on extreme events and the cause of these delays, leaving a gap in understanding how gradual weather changes affect airport resilience.

This study addresses systemic resilience in aviation networks and focuses on gradual weather changes. As noted earlier, most existing research has concentrated on individual flights or specific airports, with limited attention on the broader network effects of weather disruptions [24,25]. This is particularly important for scale-free networks such as ATNs, where disruptions in hub airports can have far-reaching consequences for the entire system [25]. By analyzing how weather-related delays at Xi’an Xianyang International Airport affect the overall network, this study provides valuable insights into the systemic resilience of China’s aviation infrastructure.

The findings of this study have important implications for policymakers and industry stakeholders. First, identifying key weather factors that affect airport resilience can inform the development of more effective flight scheduling and risk management strategies. For example, airports may prioritize certain flights or adjust schedules in response to anticipated weather changes, reducing the likelihood of cascading delays throughout the network [26,27]. Second, using advanced machine learning models like the KAN model can improve the accuracy of weather impact predictions, thus enabling airports and airlines to make more informed decisions about resource allocation and emergency response planning [28,29]. Finally, by focusing on Xi’an Xianyang International Airport, this study provides a valuable case study for other regional airports facing similar challenges.

The rest of this paper is organized as follows. Section 2 provides a detailed description of the problem, focusing on the impact of weather factors on flight delays and cancellations. Section 3 outlines the methodology, including the resilience assessment model and the Key Performance Indicators (KPIs) used to quantify flight resilience. This section also introduces the Kolmogorov–Arnold Network (KAN) for the time-series prediction of airport resilience and the Grey Relational Analysis (GRA) to evaluate the importance of weather factors. Section 4 presents the case study of Xi’an Xianyang International Airport. Section 5 concludes this study by summarizing the main findings, discussing the implications for airport management, and suggesting areas for future research.

2. Problem Description

Ground operations at airports are crucial contributors to overall flight delays, fuel consumption, and emissions. Taxiing operations typically account for significant delays due to congestion, inefficient ground handling, and inadequate traffic management systems. These delays are economically costly for airlines and environmentally detrimental due to increased fuel burn during idling and low-speed taxiing [30,31].

Therefore, it is essential to optimize the taxi-in and taxi-out phases. Studies on ground handling alternatives show that reducing taxiing time can significantly improve operational efficiency. For instance, reducing taxiing time by 1 to 3 min can lead to meaningful fuel consumption and emissions reductions. Strategies such as single-engine taxiing, onboard systems, or dispatch towing have reduced emissions by up to 30% during taxi-out operations [32]. This aligns with the research question in this study, which aims to evaluate airport resilience and quantify its weather factors.

Besides, the need for a more proactive approach to optimizing LTO (landing and take-off) operations cannot be overstated. Ground traffic management systems and runway usage strategies should be designed to minimize idle times and improve sequencing efficiency during peak traffic periods [33]. It is a crucial aspect of managing delays and improving the overall resilience of airport operations.

Schultz et al. [7] surveyed European flight delays from 2008 to 2015 regarding variability in flight phases. Their results indicated that airport delays mainly occur during the take-off and landing phases, among which airport weather is the main influencing factor. When calculating the resilience of flight operations and the degree of weather impact, the following two aspects of change need to be considered, leading to changes in resilience calculation and the importance of various weather factors:

(1)

Flight Delays and Cancellations

Weather changes leading to flight delays and cancellations cause time and economic losses for various stakeholders such as passengers, airlines, and airports. The estimation of losses varies depending on the perspective and bearing capacity of different stakeholders.

(2)

Weight Calibration of Airport Operation Network

Besides considering the distance and number of flights operating on each route within a day to determine the weight of each route in the airport network, the impact of the route on airport transportation capacity also determines its importance in the network.

This research is based on the following assumptions:

• All other operational factors remain normal except for weather changes;
• Pilots fly to minimize delays while ensuring safety;
• Flight delays in a particular airport include delays caused during taxiing, take-off, and landing.

3. Methodology

3.1. Resilience of Air Flights

When delays occur, the calculation method considers the resilience of air flight’s Key Performance Indicators (KPIs). KPIs include operational performance, economic performance, and airport capacity. Resilience refers to the process of air flights in an airport within an operational day from “suffering losses to continuing losses to recovering losses”.

3.1.1. Key Performance Indicators (KPIs)

Based on the independence of airport delays and airport capacity and the KPIs set by the International Civil Aviation Organization (ICAO) for affected and surrounding areas, the key indicators of air flights are divided into the following three categories:

(1)

Operational Performance Indicators

The numbers of delayed and canceled flights within a period are $n_{11}(\Delta t_i)$ and $n_{12}(\Delta t_i)$, respectively.

(2)

Economic Performance Indicators

• Profit loss caused by delays within a time period $\Delta t_i$:

$$C{D}_{21}\left(\Delta t_i\right)=n_{11}\left(\Delta t_i\right)·\overline{d_{31}\left(\Delta t_i\right)}·C_{31/d}$$

(1)

where $\overline{d_{31}\left(\Delta t_i\right)}$ is the average delay time of each flight in $\Delta t_i$;

$C_{31/d}$ is the unit cost of delay per flight.

• Profit loss caused by flight cancellations within a period $\Delta t_i$:

$$P{L}_{21}\left(\Delta t_i\right)=n_{12}\left(\Delta t_i\right)\cdot \overline{p{l}_{21/ap}}$$

(2)

where $\overline{p{l}_{21/ap}}$ is the average profit loss caused by a flight cancellation during time $\Delta t_i$, which equals the total value of the flight tickets lost due to the flight’s cancellation.

In Equation (1), the profit loss due to delay can be regarded as increased costs. $C_{31/d}$ consists of direct and indirect unit costs. The direct unit cost includes the aircraft’s operational costs and the airport ground service costs, which can be expressed as Equation (3):

$$C_{\mathrm{direct}}=C_f+C_g$$

(3)

where $C_f$ is the aircraft’s unit operational cost (which includes fuel, maintenance, and crew wages);

$C_g$ is the unit cost of airport ground services.

The indirect unit cost mainly consists of the passengers’ time value, which can be expressed as Equation (4):

$$C_{\mathrm{indirect}}=N_p\times V_p$$

(4)

where $N_p$ is the number of affected passengers;

$V_p$ is the average unit time value per passenger.

The time value equals the region’s per capita GDP divided by the annual average working hours. People are assumed to work 240 days a year, 8 h per day.

(3)

Airport Capacity Estimation

The Airport Cooperative Research Program (ACRP) proposed that capacity is the maximum sustainable throughput that can be accommodated per unit of time under varying local capacity constraints [34]. According to the construction stages of airports, capacity can be categorized into pre-planning capacity, pre-operational planning capacity, and operational capacity. This study focuses on airports already in operation, thus aiming to analyze the resilience of their operational capacity. Moreover, the bottleneck affecting flight operations under various factors is primarily the runway capacity of the airport, which can be understood as a trade-off optimization between arrival and departure capacities. For such bi-objective optimization problems, the Pareto frontier method can be applied to estimate capacity.

3.1.2. Weights of Airport Routes

The weights of each route for an airport hub are determined based on the average daily flight volume, route length, and the reduction in airport capacity due to the absence of a particular route.

(1)

Average Daily Flight Volume

The average daily flight volume for each route is calculated based on an airport’s daily operating time of 16 h, as shown in Equation (5):

$$ADF{V}_k=Hourly Flight Capacit{y}_k\times 16$$

(5)

(2)

Flight Distance

The flight distance from the starting node to the destination node can be obtained from actual flight data.

(3)

Flow Stability

The flow stability of each route $RA{I}^k$ considers the impact of a reduced route capacity on the maximum airport traffic flow, as expressed in Equation (6).

$$RA{I}^k={\displaystyle \frac{{\displaystyle {\int }_{u_0}^{u_T}(F_{\max }^k(u_0)-F_{\max }^k(u))du}}{{\displaystyle {\int }_{u_0}^{u_T}{F}_{\max }^k(u_0)du}}}$$

(6)

Based on the definition of robustness, RAI evaluates and quantifies the impact of localized (node-level) capacity reductions on the maximum network traffic flow [35]. In Equation (6), $u$ is the degradation parameter which represents the potential reduction in route capacity. $F_{\max }^k(u)$ is the maximum network traffic flow of an airport when the capacity of route $k$ reduces $u$. Finding the maximum network traffic flow for a given degradation is equivalent to solving a linear optimization problem. Let the capacity of route $k$ be $c_k$, and the objective function for the airport’s maximum traffic flow is given by the following:

$$\max {\displaystyle \sum fligh{t}_n}$$

(7)

Constraints:

(1)

Traffic volume of each route

The volume of each route must be less than or equal to the route capacity:

$${\displaystyle \sum _{n\in N}{\delta }_{nk}fligh{t}_n}\le c_k$$

(8)

(2)

Non-negativity

The traffic volume of flights on each route must be greater than or equal to 0:

$${\displaystyle \sum _{n\in N}{\delta }_{nk}fligh{t}_n}\ge 0$$

(9)

where $fligh{t}_n$ represents the nth flight of the airport on one operating day.

${\delta }_{nk}$ is a 0–1 binary variable used to determine whether a flight belongs to route $k$, which is expressed in Equation (10):

$${\delta }_{nk}=\left\{\begin{array}{l}0,n\notin k,\\ 1,n\in k.\end{array}\right.$$

(10)

Since the Grey Relational Analysis (GRA) method is used to analyze the degree of correlation between factors in a system, it is particularly suitable for analyzing small samples and uncertain systems. Therefore, in this study, the GRA method is used to determine the total weight of each route. After normalizing the data, the reference sequence (ideal value) is set to 1, and the absolute difference between the reference sequence and the comparison sequence is calculated, as shown in Equation (11):

$${\Delta }_{kj}=\left|1-X_{kj}\right|, j=1,2,3$$

(11)

The correlation coefficient is calculated according to the formula for the Grey Relational Degree, as shown in Equation (12):

$${\epsilon }_{kj}={\displaystyle \frac{mi{n}_{k}mi{n}_j{\Delta }_{kj}+\rho ma{x}_{k}ma{x}_j{\Delta }_{kj}}{{\Delta }_{kj}+\rho ma{x}_{k}ma{x}_j{\Delta }_{kj}}}, j=1,2,3$$

(12)

where $\rho$ is the distinguishing coefficient, generally taken as 0.5. The correlation coefficients are then weighted and summed to obtain the Grey Relational Degree of each comparison sequence, as shown in Equation (13):

$${\gamma }_k={\displaystyle \frac{1}{3}}{\displaystyle \sum }_{j=1}^3{\epsilon }_{kj}$$

(13)

3.1.3. Resilience of Airport Flight Operation

Based on the flight operations in the airport during an operational day, once a delay occurs, the airport suffers from profit losses until the delay dissipates. This duration is quantified by the resilience of flights on route $k$ over periods in one operational day. An example of the resilience of delay time is shown in Figure 1.

Real-time flight information primarily refers to the planned departure time, planned arrival time, actual departure time, and actual arrival time. The origin-destinations (ODs) of each flight and the airport’s flight volume can be statistically obtained using real-time information. The method for calculating the operational resilience of flight delays is detailed as follows:

$${\mathrm{R}}_{{\mathrm{delay}}_i}^k={\gamma }_k{\displaystyle \frac{\mathrm{MAX}\left(dela{y}_{i,a\left(\max \right)}\right)\left(t_{i+1}-t_i\right)-\sum s_a\left(dela{y}_{i,a\left(\max \right)}>0\right)}{{{\displaystyle \sum }}_{\mathrm{a}=1}^{{\mathrm{N}}_{\mathrm{i}}+2}{s}_a}},i=1,2,\cdots \mathrm{T}$$

(14)

$$s_a={{\displaystyle \int }}_{t_{i,a}}^{t_{i,a+1}}\left[a_{ia}t+dela{y}_{i,a\left(\max \right)}-a_{ia}\left(t_{i,a+1}-t_{i,a}\right)\right]\mathrm{dt}+{{\displaystyle \int }}_{t_{i,a}}^{t_{i,a+1}}\left[{\beta }_{ia}t+dela{y}_{i,a\left(\max \right)}-{\beta }_{ia}\left(t_{i,a+1}-t_{i,a}\right)\right]\mathrm{dt}$$

(15)

where, $s_a$ is the cumulative delay from $t_{i,a}$ to $t_{i,a+1}$ in the statistical period $\left[t_i,t_{i+1}\right]$; $t_{i,a}$ is the time when the flight delay reaches an inflection point or zero during the statistical period $\left[t_i,t_{i+1}\right]$; $dela{y}_{i,a(\max )}$ is the maximum delay from $t_{i,a}$ to $t_{i,a+1}$ in the statistical period $\left[t_i,t_{i+1}\right]$; ${\alpha }_{ia}$ is the recovery rate of delay during the period $t_{i,a}$ to $t_{i,a+1}$ in $\left[t_i,t_{i+1}\right]$; ${\beta }_{ia}$ is the interruption rate of the delay rate during the period $t_{i,a}$ to $t_{i,a+1}$ in $\left[t_i,t_{i+1}\right]$; $MAX\left(dela{y}_{i,a(\max )}\right)$ is the maximum delay in $\left[t_i,t_{i+1}\right]$; $N_i$ is the number of flights that are not delayed or advanced in $\left[t_i,t_{i+1}\right]$.

The calculation procedure of the operational resilience of flight delays is shown in Figure 2.

As mentioned in Section 3.1.1, the resilience of flight operations involves metrics for operations, economics, and capacity. In calculating these metrics, this study uses Equations (11) and (12), in which only the parameters related to delays must be replaced.

3.2. 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 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 ${\mathrm{R}}_{\mathrm{i}}^k$, 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 $x_{t_0-c:t_0-1}=\left[x_{t_0-c},x_{t_0-c+1},\cdots ,x_{t_0-1}\right]$, 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 $R_{t_0:T}\approx f(x_{t_0-c:t_0-1})$. 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 ${\mathsf{\Phi }}_i$ to the inputs from the previous layer.
• 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:

$$R=KAN(x)=({\mathsf{\Phi }}_L·{\mathsf{\Phi }}_{L-1}·\cdots ·{\mathsf{\Phi }}_1)(x)$$

(16)

where each ${\mathsf{\Phi }}_i$ 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 $T$.

The dataset is split into 80% for training and 20% for testing to accurately evaluate the model’s performance. The model parameters are updated using the Adam optimizer with an initial learning rate of $\alpha =0.001$. We conduct the training for 300 epochs to ensure the model converges to an optimal solution.

• Step 4. Model Evaluation

The model’s performance is evaluated using a test dataset of unseen weather and operational data. The prediction accuracy is measured using standard evaluation metrics such as the Mean Squared Error (MSE), Root Mean Squared Error (RMSE), and MAE. A comparative analysis with traditional models such as MLPs (Multi-Layer Perceptrons) is performed to highlight the effectiveness of KANs in capturing the complex relationships between weather conditions and operational resilience.

This algorithm combines the KAN architecture’s ability to model complex, nonlinear relationships with airport weather data to predict flight resilience. The choice of spline-based activations in the KAN makes it highly effective for this time-series forecasting task.

3.2.2. Importance Calculation of Input Features

The importance of features is computed using the Permutation Importance (PI) method from the MLxtend machine learning extension package in Python. This method measures the importance of a feature by calculating the increase in the prediction error of the KAN model after permuting (shuffling) the feature vector.

The trained KAN model, saved after the training process’s completion, produced a baseline prediction result denoted as $P^{original}$. To assess the importance of a specific feature, all values of that feature were merged into a single vector. Using this vector, the resilience of airport flight operations was predicted, resulting in a prediction denoted as $P^{merge}$. The prediction loss due to the feature’s data merge was calculated as follows:

$$P^{merge}-P^{original}$$

(17)

The dataset of input features was then restored to its original order, and the prediction losses caused by data merges were computed for each input feature vector. This process enabled the evaluation of the importance of crucial weather features in the resilience of airport flight operations in response to delays.

4. Case Study

The route’s layout centered around Xi’an Xianyang International Airport is shown in Figure 4. The airport is marked with each airport’s IATA (International Air Transport Association) code, and the actual flight distance is generally more significant than the straight-line distance.

4.1. Basic Data

To better reflect the dynamic changes in traffic demand and travel time, the study period selected is from 23 February 2023, to 15 April 2023. The model involves primary data, including hourly airport weather data, flight OD, scheduled departure time, actual departure time, scheduled arrival time, and actual arrival time.

(1)

Hourly Airport Weather Reports

The collection of airport weather data includes the airport name, observation time, surface wind direction, wind speed, wind direction variation range, visibility, cloud type and altitude, temperature, corrected sea level pressure, and a two-hour weather forecast. After quantifying the collected airport weather report information, one-dimensional vectors are created for each of the K weather elements over R periods in one operational day, forming an R*K two-dimensional vector.

Table 1. Weather data per hour (portion).

Variables	Time
	2/23/5:00–6:00	2/23/6:00–7:00	…	4/15/22:00–23:00	4/15/23:00–24:00
Wind stability	0	1	…	1	1
Average wind direction (°)	0	210	…	360	360
Average wind speed (m/s)	1	2	…	1	1
Visibility (m)	3000	3000	…	10,001	10,001
Cloud width 1	0	0	…	0	0
Cloud height 1 (feet)	0	0	…	0	0
Cloud width 2	0	0	…	0	0
Cloud height 2 (feet)	0	0	…	0	0
Cloud width 3	0	0	…	0	0
Cloud height 3 (feet)	0	0	…	0	0
TCU	0	0	…	0	0
CB	0	0	…	0	0
Temperature (°)	10	8	…	18	17
Dew point (°)	3	4	…	6	9
Corrected sea level pressure (hPa)	1023	1022	…	1004	1003
Weather forecast	0	0	…	0	0

shows a portion of the weather data for Xi’an Xianyang International Airport.

The quantified variables for airport weather reports include the following:

• VRB represents variable wind direction, indicating that the wind direction changes by more than 180 degrees or the average wind speed is less than 2 m per second. In this case, the average wind direction value can be interpolated using the stable wind direction values from before and after the observation. A binary variable is set: 0 for variable wind direction and 1 for stable wind direction.
• CAVOK indicates good weather and visibility, with visibility being more significant than 10 km. This is quantified as the minimum value, in this case, 10,001 m.
• For cloud thickness, FEW represents 1/8 to 2/8 cloud cover, SCT represents 3/8 to 4/8 cloud cover, BKN represents 5/8 to 7/8 cloud cover, OVC represents 8/8 cloud cover, SKC indicates clear skies, NSC represents no clouds above 5000 feet, TCU represents towering cumulus clouds, and CB represents cumulonimbus clouds. For quantification, the maximum value of each cloud type’s thickness is used, and two binary variables are set for towering cumulus and cumulonimbus clouds: 0 represents the absence of these clouds, and 1 represents their presence.
• For the two-hour weather forecast, an ordinal categorical variable is set: TEMPO indicates temporary weather changes, BECMG indicates gradual weather changes, and NOSIG indicates no significant changes in the weather.

(2)

Flight OD Data

Based on the flight’s origin-destination (OD), scheduled departure time, actual departure time, scheduled arrival time, and actual arrival time, the operational delay time for each route can be obtained. As an example, the delay times from 6:00 to 12:00 on 23 February 2023, are shown in Figure 5.

(3)

Route Flow Stability Indicator

In response to weather changes, the Pareto frontier is used to statistically analyze the number of flight arrivals and departures to estimate the airport’s hourly transportation capacity, as shown in Figure 6. In Figure 6, the larger the circle radius, the higher the frequency of flight arrivals and departures at the corresponding point within the statistical period.

When the number of arriving flights is zero, the frequency of various departing flight numbers is relatively high, indicating that the airport’s dispatch and management efficiency for departing flights is higher in this scenario. When the number of arriving flights is between six and eight, the corresponding number of departing flights is mainly concentrated between four and ten, and the frequency is relatively balanced. This reflects the typical performance of the airport under regular operational dispatch. Through this analytical method, the hourly transportation capacity of each flight can be systematically estimated, providing data support for optimizing flight scheduling and improving airport operational efficiency.

4.2. Numerical Application of Route Resilience

In this numerical example, the raw data for each fight are planned departure time, planned arrival time, actual departure time, and actual arrival time. The time window of flight resilience $\Delta t$ is 15 min.

• Step 1. Calculation of KPIs

In Step 1, the route data between Xi’an Xianyang Airport and Changsha ranging from 15:00 to 16:00 3 March 2023 are chosen to quantify the flight resilience, which contains four time windows $\Delta t_i,i=1,\cdots ,4$.

(1)

Operation performance

• The number of delayed flights from 15:00 to 16:00: $n_{11}(\Delta t_1)=1$, $n_{11}(\Delta t_2)=0$, $n_{11}(\Delta t_3)=3$, $n_{11}(\Delta t_4)=1$;
• The number of canceled flights from 15:00 to 16:00: $n_{12}(\Delta t_1)=1$, $n_{12}(\Delta t_2)=0$, $n_{12}(\Delta t_3)=1$, $n_{12}(\Delta t_4)=0$.

(2)

Economic performance

• Profit loss caused by delays from 15:00 to 16:00:

• The unit cost of delay per flight $C_{31/d}$ is assumed to be 7000 RMB/min. Therefore,

  $$C{D}_{21}(\Delta t_1)=n_{11}(\Delta t_1)\cdot \overline{d_{31}(\Delta t_1)}\cdot C_{31/d}=1\cdot 21\cdot 7000=\mathrm{147,000};$$

  $$C{D}_{21}(\Delta t_2)=0;$$

  $$C{D}_{21}(\Delta t_3)=3\cdot 15\cdot 7000=\mathrm{315,000};$$

  $$C{D}_{21}(\Delta t_1)=1\cdot 32\cdot 7000=\mathrm{224,000}.$$

  where $\overline{d_{31}(\Delta t_1)}$ is the average delay time of each flight in $\Delta t_i$.

• Profit loss caused by flight cancellations from 15:00 to 16:00:

• The average profit loss $\overline{p{l}_{21/ap}}$ caused by a flight cancellation is assumed to be 50,000 RMB. Therefore,

  $$P{L}_{21}\left(\Delta t_1\right)=n_{12}\left(\Delta t_1\right)·\overline{p{l}_{21/ap}}=1\cdot \mathrm{50,000}=\mathrm{50,000};$$

  $$P{L}_{21}\left(\Delta t_2\right)=0;$$

  $$P{L}_{21}\left(\Delta t_3\right)=\mathrm{50,000};$$

  $$P{L}_{21}\left(\Delta t_4\right)=0.$$

(3)

Route capacity in Xi’an Xianyang Airport

Based on the concept of the Pareto frontier, the 15-min transportation capacity of this route equals the planned number of arrivals and departures. Therefore,

$$c(\Delta t_1)=4, c(\Delta t_2)=6, c(\Delta t_3)=5, c(\Delta t_4)=4.$$

• Step 2. Weights of Airport Routes

In Step 2, to calculate the route weight between Xi’an Xianyang Airport and Changsha, all data of this route (from 23 February 2023 to 15 April 2023) are applied.

(1)

Average daily flight volume

First, the planned arrivals and departures in one hour are obtained. Then, the Pareto frontier of this route is drawn to evaluate its hourly flight capacity, which can be regarded as 19 flights/hour, as shown in Figure 7.

Therefore, the average daily flight volume of this route is:

$$ADF{V}_k=Hourly Flight Capacit{y}_k\times 16=19\times 16=304 flights$$

(2)

Route distance

The flight distance between Xi’an Xianyang Airport and Changsha is 810 km.

(3)

Flow stability

For the flow stability of the route between Xi’an Xianyang Airport and Changsha, in Equation (6), the degradation in route capacity $u$ starts from 0. $F_{\max }^k(u)$ is the maximum network traffic flow of an airport when the route capacity reduces $u$. According to Equations (7)–(10), the calculation method of $F_{\max }^k(u)$ is as follows:

• Capacity reduction process: Start by reducing the route capacity for each period, beginning with 19 flights/hour and decreasing stepwise to 0.
• Filter flight data: Filter flights exceeding each hour’s new capacity after reducing capacity. For example, if the capacity is reduced to 15 flights/hour, remove flights from the actual data surpassing this hourly capacity (i.e., only keep the first 15 flights each hour if the data contains more).
• Daily Maximum Network Traffic Flow Calculation: After filtering, sum up the remaining flights once the hourly data has been filtered for all daily hours. This gives the maximum network traffic flow for the entire day at the reduced capacity level. Repeat this for each capacity degradation.

The process of degradation is shown in Figure 8. Therefore, the flow stability of the route between Xi’an Xianyang Airport and Changsha is:

$$RA{I}^k={\displaystyle \frac{{\displaystyle {\int }_{u_0}^{u_T}(F_{\max }^k(u_0)-F_{\max }^k(u))du}}{{\displaystyle {\int }_{u_0}^{u_T}{F}_{\max }^k(u_0)du}}}={\displaystyle \frac{S_1}{S_1+S_2}}=0.0012$$

(4)

Overall weight using the Grey Relational Analysis (GRA) method

• Normalizing the data: average daily flight volume (304 flights), flight distance (810 km), and flow stability (0.0012), denoted as $X_{j=1,2,3}=\left[0.316,0.532,0.152\right]$;
• Comparing with the reference sequence:

  $${\Delta }_j=\left|1-X_{j=1,2,3}\right|=\left[0.684,0.468,0.848\right]$$

• Calculating Grey Relational Degrees:

$\rho$ is the distinguishing coefficient, generally taken as 0.5.

$${\epsilon }_j={\displaystyle \frac{mi{n}_j{\Delta }_j+\rho ma{x}_j{\Delta }_j}{{\Delta }_j+\rho ma{x}_j{\Delta }_j}}=\left[0.804, 1, 0.701\right]$$

• Calculating Grey Relational Grade:

$$\gamma ={\displaystyle \frac{1}{3}}{\displaystyle \sum }_{j=1}^3{\epsilon }_j=0.835$$

• Step 3. Resilience of Airport Routes

Based on Step 1 and Step 2, the operational, economic, and capacity resilience of the route between Xi’an Xianyang Airport and Changsha from 15:00 to 16:00 3 March 2023 can be obtained, as shown in

Table 2. Numerical application of resilience.

	Metrics	$\mathit{\gamma }$	MAX (Metrics)	${\mathit{t}}_{\mathit{i}+1}-{\mathit{t}}_{\mathit{i}}$	$\sum {\mathit{s}}_{\mathit{a}}\left(\mathbf{Metrics} > 0\right)$	${{\displaystyle \sum }}_{\mathbf{a}=1}^{{\mathbf{N}}_{\mathbf{i}}+2}{\mathit{s}}_{\mathit{a}}$	R
Operational	Delay	0.835	3	60	75	180	0.583
	Cancel	0.835	1	60	37.5	60	0.375
Economic	Profit loss by delay	0.835	315,000	60	9,712,500	18,900,000	0.486
	Profit loss by cancel	0.835	50,000	60	1,875,000	3,000,000	0.375
Capacity		0.835	6	60	285	360	0.208

.

4.3. Results and Discussion

4.3.1. Weights of Flight Routes

After applying Min-Max normalization to the data for average daily flight volume, flight distance, and airport traffic stability (RAI), as shown in Figure 9, it can be observed that the higher the average daily flight volume on a route, the greater its impact on the airport’s traffic stability. On the other hand, flight distance is influenced by the geographical locations of the origin and destination. For example, routes with the longest flight distance (Aksu, AKU, 3159.45 km) and the shortest flight distance (Ankang, AKA, 62.52 km) both have relatively low average daily flight volumes and little impact on airport traffic stability, which indicates a lower correlation between flight distance and variations in other indicators.

This relationship is verified through correlation analysis, as shown in

Table 3. Correlation analysis of route weight indicators.

Indices	Average Daily Flight Volume	Flight Distance	RAI
Average daily flight volume	1	−0.16	0.94
Flight Distance	−0.16	1	−0.18
RAI	0.94	−0.18	1

. The correlation between average daily flight volume and airport traffic stability (RAI) is very high (0.94), indicating that routes with higher average daily flight volumes significantly impact airport traffic stability. The correlation between flight distance and average daily flight volume is low (−0.16), showing no significant relationship between flight distance and daily flight volume. Similarly, the correlation between flight distance and airport traffic stability (RAI) is also low (−0.18), indicating no significant relationship between flight distance and airport traffic stability.

The route weights of each airport are shown in Figure 10. To analyze the regional distribution of the route weights, representative airports in different regions are listed in

Table 4. Weights of representative routes in different regions.

Region	Route OD	Route Weight
Eastern Coastal Area	Guangzhou Baiyun Airport	0.9
Eastern Coastal Area	Shanghai Pudong Airport	0.85
Central Area	Wuhan Tianhe Airport	0.75
Western Area	Chengdu Shuangliu Airport	0.7
Western Area	Urumqi Airport	0.5
Northeastern Area	Harbin Airport	0.65
Remote Western Area	Lhasa Gonggar Airport	0.4

.

The Grey Relational Degree calculation results show significant differences in the total weight of different routes across the average daily flight volume, flight distance, and airport traffic stability (RAI). Routes in economically developed regions (such as Shanghai and Guangzhou) with higher Grey Relational Degrees indicate a more significant impact on airport traffic stability. In contrast, routes in remote areas (such as Lhasa and Aksu) with lower Grey Relational Degrees confirm that flight distance has a lower correlation with other indicator variations.

4.3.2. Values of Route Resilience

According to the changes in the daily operational resilience values for routes in the central and eastern regions shown in Figure 11, the resilience values show two distinct fluctuations corresponding to periods of high flight activity. These fluctuations occur during peak operational hours, which coincide with higher flight volumes. This pattern suggests that flight demand in the central and eastern regions is significant, thus putting pressure on Xi’an Xianyang International Airport’s transportation capacity during peak hours. As the airport becomes busier, its ability to maintain steady operational resilience appears to be constrained, thus leading to reduced resilience values. Conversely, during off-peak periods, when the demand for flights is lower, the resilience of these routes improves.

Figure 12 shows the daily operational resilience value changes for routes in the western and remote regions. Compared to the central and eastern areas, the resilience values for routes in the west and remote regions fluctuate more significantly. This indicates that airports in the western and remote areas have lower operational resilience and higher volatility when faced with changes in average daily flight volumes. This phenomenon could be related to relatively underdeveloped airport facilities, fewer flight volumes, and less sophisticated scheduling and management systems in these regions.

By comparing the operational resilience values of routes in the two regions, the following conclusions can be drawn: Airports in the central and eastern areas have higher operational resilience, with more minor fluctuations in resilience values, and can better maintain operational stability under high flight volumes. In contrast, airports in the western and remote regions have lower operational resilience, with more significant fluctuations in resilience values, which indicates poorer stability when facing changes in flight volumes.

4.3.3. Prediction Effectiveness of KAN Model

The effectiveness of the KAN prediction model was evaluated by comparing the predicted values against the actual values using multiple performance metrics. Figure 13 presents a side-by-side comparison of the actual and predicted values over the test set. In this study, we aimed to forecast changes in airport operational resilience with a lead time of 6 h. Visually, the model closely follows the trend and fluctuations in the actual data, with minor deviations in a few points. This indicates that the KAN model effectively captures upward and downward trends.

In addition to the visual inspection, the quantitative evaluation of the prediction performance is presented in

Table 5. Evaluation metrics of prediction accuracy.

Metrics of Test Set	MSE	RMSE	MAE	MAPE	R2
Prediction result	1.55 × 10−5	0.0039	0.0031	0.0092%	99.9565%

. The key performance metrics include the Mean Squared Error (MSE), Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), and the R-squared value (R²).

The MSE and RMSE values are relatively low, thus indicating that the predictions deviate minimally from the actual values. The MAE of 0.0031 further supports the model’s precision, with a small average prediction error. The MAPE of 2.3% demonstrates that the KAN model’s prediction errors are relatively negligible, making it highly effective in real-world applications where percentage accuracy is crucial. Most notably, the R² value of 99.806% signifies that the KAN model explains almost all of the variance in the actual data, which points to an excellent model fit.

These metrics confirm that the KAN model performs exceptionally well, providing accurate and reliable predictions. The minor errors and high correlation between the predicted and actual values make it an excellent candidate for forecasting tasks in similar datasets.

4.3.4. Analysis of the Importance of Weather Impact

According to Figure 14, Figure 15 and Figure 16, the impact of weather factors on flight operation metrics, economic metrics, and transportation capacity shows significant differences. Figure 14 indicates that among all weather factors affecting flight operation metrics, temperature_point and wind_direction_speed have the highest importance, at 0.35 and 0.32, respectively. This suggests that temperature and wind speed are the key factors determining flight operational performance. Other factors, such as FEW, visibility, and wind_direction_change, have some impact, but their importance is relatively lower.

Figure 15 illustrates the importance of weather factors in economic metrics. Temperature_point and wind_direction_speed remain the most critical factors, with importance scores of 0.32 and 0.30, respectively. These results indicate that temperature and wind speed are not only essential for normal flight operations but also have a direct impact on the economic benefits of flights. In comparison, the importance of FEW, visibility, and SCT decreased, but they still warrant attention.

Figure 16 reveals the impact of weather factors on flight transportation capacity. Among all weather factors, wind_direction_change and wind_direction_speed have the highest importance, at 0.7 and 0.65, respectively. This suggests that wind direction and speed changes are the main factors affecting flight transportation capacity. Other factors, such as SCT, FEW, and BR, also significantly impact transportation capacity, while temperature_point and visibility have relatively lower importance.

In summary, wind_direction_speed and temperature_point are the critical weather factors influencing flight operation and economic metrics, while wind_direction_change has the most significant impact on flight transportation capacity. These findings imply that airlines should pay particular attention to these key weather factors when planning flights and assessing risks to ensure flight safety, efficiency, and economic performance.

5. Conclusions

This study investigates the effects of weather factors on the resilience of airport flight operations, focusing on operational performance, economic efficiency, and transportation capacity. Using the Kolmogorov–Arnold Network (KAN) model, we successfully predicted flight resilience based on weather data and flight operation records from Xi’an Xianyang International Airport. The model’s excellent predictive performance, reflected in metrics such as its high R² value of 99.806%, demonstrates its ability to capture nonlinear relationships between weather conditions and operational resilience effectively.

The results show significant differences in resilience across different routes and regions. Routes in economically developed areas, such as those in eastern China, exhibit higher resilience and lower fluctuations, maintaining operational stability even under high flight volumes. Conversely, routes in western and remote areas experience more significant fluctuations in resilience values, reflecting lower operational stability and greater vulnerability to disruptions.

Regarding operational resilience, routes in the eastern region of China show more consistent and stable resilience metrics, suggesting that better infrastructure and more sophisticated management systems help mitigate the impact of adverse weather conditions. In contrast, routes in the western and remote regions are less resilient, highlighting the need for improved airport facilities and flight scheduling mechanisms.

Key weather factors were identified as significant flight determinants of resilience. Temperature and wind speed emerged as the most influential factors in operational and economic performance, with importance values of 0.35 and 0.32, respectively. On the other hand, wind direction change and wind speed were the primary contributors to variations in transportation capacity, with importance values of 0.7 and 0.65, respectively.

The transferability of this approach includes the resilience calculation method and the KAN model. The resilience calculation method used in this study can be applied to different metrics of an airport or transportation hub affected by other factors. The key to making this approach transferable is customizing the input data to reflect local conditions and operational characteristics. For example, the time window of resilience calculation should be divided to align with the recording intervals of weather data. The KAN model offers potential transferability to other airports and case studies, providing appropriate local data. Applying the KAN model can help analyze the resilience of systems similarly impacted by other factors for surface transportation nodes, allowing for scheduling optimization and minimizing disruptions. However, adjusting the model to account for the different operational characteristics requires adjusting. For example, trains are less affected by wind direction and wind speed than airplanes. Therefore, while rail stations experience delays related to weather conditions, the nature and frequency of disruptions might differ, necessitating additional model adaptations.

These findings provide valuable insights for airport and airline managers, who can leverage these data to enhance flight scheduling, minimize delays, and improve airport resilience to weather disruptions. Advanced models like the KAN enhance operational efficiency and the overall resilience of the aviation network. However, this study has certain limitations. Notably, the flight delay data used in this study does not separate delays into specific phases (taxiing, take-off, and landing). As a result, the individual impact of weather factors on these phases was not calculated, limiting the precision of optimization strategies. Additionally, while the KAN model performs well with the current dataset, its effectiveness in different regions or under extreme weather conditions needs further validation. Also, other limitations arise from factors not explicitly considered in the current model. The throughput of the airport and the ratio of cargo to passenger operations may significantly impact resilience, as airports handling different types of traffic may face varying operational challenges. Future studies could focus on expanding the predictive capabilities of the KAN model to include additional factors while investing in infrastructure upgrades and enhanced real-time data monitoring systems, which would significantly boost airport operations’ resilience.