\section{Experiments}\label{main:exp}
\subsection{Dataset and Experimental Setup}
\paragraph{Datasets} To evaluate the performance of FlightPatchNet, we conduct extensive experiments on ADS-B data provided by OpenSky \footnote[1]{https://opensky-network.org/datasets/states/} from 2020 to 2022. In this paper, six key attributes are extracted from the original data, including longitude, latitude, altitude, and velocity in \textit{x, y, z} dimensions. The dataset is chronologically divided into three parts for training, validation, and testing with a ratio of 8:1:1. 
\paragraph{Baselines and Setup} We compare our model with ten competitive models, including five IMS-based models: \textbf{LSTM} \citep{LSTM8489734}, \textbf{CNN-LSTM} \citep{CNN-LSTM9145522}, \textbf{Bi-LSTM} \citep{Sahadevan}, \textbf{FlightBERT} \citep{Guo2023FlightBERT}, \textbf{WTFTP} \citep{Zhang2023FlightTP};  five DMS-based model: \textbf{FlightBERT++} \citep{Guo2023FlightBERT++}, \textbf{TimeMixer} \citep{wang2024timemixer}, \textbf{TimesNet} \citep{wu2022timesnet}, \textbf{MICN} \citep{wang2023micn}, \textbf{Pathformer} \citep{chen2024pathformer}. These models have covered mainstream deep learning architectures, including Transformer (FlightBERT, FlightBERT++, Pathformer), CNN (CNN-LSTM, TimesNet, MICN), RNN (LSTM, Bi-LSTM, CNN-LSTM, WTFTP) and MLP (TimeMixer), which helps to provide a comprehensive comparison. For fairness, all the models follow the same experimental setup with lookback window $L=60$ and prediction horizon  $T\in\{1,3,9,15\}$.  Our model is trained with MSE loss, using the Adam optimizer. We adopt the Mean Absolute Error (MAE) and Root Mean Squared Error (RMSE)  as evaluation metrics. More details about the dataset, baselines, implementation and hyper-parameters are shown in Appendix~\ref{app:exp}.
\subsection{Main Results}
Comprehensive flight prediction results are demonstrated in Table~\ref{table:exp_main_res} (see Appendix~\ref{error_bar} for error bar). FlightPatchNet achieves the most outstanding performance across various prediction lengths for longitude and latitude in terms of both MAE and RMSE, while it does not achieve the optimal results for altitude compared with other strong baselines such as FlightBERT++. For simplification, we consider prediction horizon $T=15$ and compare our model with the second best. FlightPatchNet achieves an overall 18\textit{.}62\% reduction on MAE and 41\textit{.}29\% reduction on RMSE for longitude, and 35\textit{.}31\% reduction on MAE and 44\textit{.}80\% reduction on RMSE for latitude. For the prediction performance of altitude, FlightBERT++ outperforms our model by 45\textit{.}51 meters reduction on MAE but has a large  RMSE which may caused by high-bit errors in the prediction. FlightPatchNet obtains the smallest RMSE results for all variables, indicating that our model can provide a more robust and stable prediction. Compared with the most promising multi-scale modeling MTS prediction methods, including the pure MLP-based model TimeMixer, the CNN-based methods TimesNet and MICN and the Transformer-based method Pathformer, FlightPatchNet achieves superior prediction performance. These existing multivariate time-series forecasting methods typically decompose time series into seasonal and trend components, and primarily focus on periodic modeling. However, short-term flight trajectories do not exhibit obvious periodic patterns. The trend features in altitude and the temporal variations in longitude and latitude deserve more attention. Furthermore, as the prediction horizon increases, IMS-based models suffer from serious performance degradation due to error accumulation. 

\begin{table*}[ht]
\centering
\captionsetup{font=small}
\caption{Flight trajectory prediction results.  A lower MAE or RMSE represents a better prediction. The prediction horizon \textit{$T \in \left\{1,3,9,15\right\}$} and look-back window size $L = 60$ for all experiments. The best results are highlighted in \textbf{bold }and the second best are \underline{underlined}. Note that $0.00001^\circ$ is about 1m.}
\label{table:exp_main_res}

\setlength{\tabcolsep}{1mm}{
\scalebox{0.7}{
    \begin{tabular}{c|c|c|cccc|cccc|cccc}
\toprule
\multirow{2}{*}{}     & \multirow{2}{*}{Model}                                                           & \multirow{2}{*}{Metric} & \multicolumn{4}{c|}{Lon($0.00001^\circ$)}                                                                                                                                                 & \multicolumn{4}{c|}{Lat($0.00001^\circ$)}                                                                                                                                                & \multicolumn{4}{c}{Alt(m)}                                                                                                                                                                           \\ \cmidrule(l){4-15} 
                      &                                                                                  &                         & \multicolumn{1}{c|}{1}                           & \multicolumn{1}{c|}{3}                            & \multicolumn{1}{c|}{9}                             & 15                            & \multicolumn{1}{c|}{1}                           & \multicolumn{1}{c|}{3}                            & \multicolumn{1}{c|}{9}                            & 15                            & \multicolumn{1}{c|}{1}                               & \multicolumn{1}{c|}{3}                               & \multicolumn{1}{c|}{9}                               & 15                              \\ \midrule
\multirow{10}{*}{IMS} & \multirow{2}{*}{LSTM}                                                            & MAE                     & \multicolumn{1}{c|}{56} & \multicolumn{1}{c|}{427}                          & \multicolumn{1}{c|}{1747}                          & 3132                          & \multicolumn{1}{c|}{49} & \multicolumn{1}{c|}{493}                          & \multicolumn{1}{c|}{2116}                         & 3717                          & \multicolumn{1}{c|}{92.27}                           & \multicolumn{1}{c|}{159.30}                          & \multicolumn{1}{c|}{549.55}                          & 882.86                          \\ \cmidrule(l){3-15} 
                      &                                                                                  & RMSE                    & \multicolumn{1}{c|}{95} & \multicolumn{1}{c|}{691}                          & \multicolumn{1}{c|}{2597}                          & 4578                          & \multicolumn{1}{c|}{89} & \multicolumn{1}{c|}{740}                          & \multicolumn{1}{c|}{2956}                         & 5143                          & \multicolumn{1}{c|}{142.05}                          & \multicolumn{1}{c|}{233.39}                          & \multicolumn{1}{c|}{763.84}                          & 768.45                          \\ \cmidrule(l){2-15} 
                      & \multirow{2}{*}{Bi-LSTM}                                                         & MAE                     & \multicolumn{1}{c|}{155}                         & \multicolumn{1}{c|}{747}                          & \multicolumn{1}{c|}{2319}                          & 3890                          & \multicolumn{1}{c|}{137}                         & \multicolumn{1}{c|}{824}                          & \multicolumn{1}{c|}{2711}                         & 4404                          & \multicolumn{1}{c|}{432.50}                          & \multicolumn{1}{c|}{761.50}                          & \multicolumn{1}{c|}{1648.68}                         & 2006.21                         \\ \cmidrule(l){3-15} 
                      &                                                                                  & RMSE                    & \multicolumn{1}{c|}{202}                         & \multicolumn{1}{c|}{1124}                         & \multicolumn{1}{c|}{3387}                          & 5532                          & \multicolumn{1}{c|}{181}                         & \multicolumn{1}{c|}{1142}                         & \multicolumn{1}{c|}{3639}                         & 5982                          & \multicolumn{1}{c|}{563.74}                          & \multicolumn{1}{c|}{953.37}                          & \multicolumn{1}{c|}{2132.91}                         & 2420.74                         \\ \cmidrule(l){2-15} 
                      & \multirow{2}{*}{CNN-LSTM}                                                        & MAE                     & \multicolumn{1}{c|}{139}                         & \multicolumn{1}{c|}{700}                          & \multicolumn{1}{c|}{2282}                          & 4149                          & \multicolumn{1}{c|}{131}                         & \multicolumn{1}{c|}{801}                          & \multicolumn{1}{c|}{2623}                         & 5139                          & \multicolumn{1}{c|}{520.03}                          & \multicolumn{1}{c|}{746.67}                          & \multicolumn{1}{c|}{1569.68}                         & 1136.80                         \\ \cmidrule(l){3-15} 
                      &                                                                                  & RMSE                    & \multicolumn{1}{c|}{240}                         & \multicolumn{1}{c|}{1033}                         & \multicolumn{1}{c|}{3263}                          & 5981                          & \multicolumn{1}{c|}{212}                         & \multicolumn{1}{c|}{1130}                         & \multicolumn{1}{c|}{3559}                         & 7353                          & \multicolumn{1}{c|}{1176.96}                         & \multicolumn{1}{c|}{926.40}                          & \multicolumn{1}{c|}{1936.63}                         & 1658.53                         \\ \cmidrule(l){2-15} 
                      & \multirow{2}{*}{WTFTP}                                                           & MAE                     & \multicolumn{1}{c|}{175}                         & \multicolumn{1}{c|}{1484}                         & \multicolumn{1}{c|}{2002}                          & 2657                          & \multicolumn{1}{c|}{112}                         & \multicolumn{1}{c|}{1169}                         & \multicolumn{1}{c|}{1586}                         & 2110                          & \multicolumn{1}{c|}{145.02}                          & \multicolumn{1}{c|}{230.49}                          & \multicolumn{1}{c|}{588.44}                          & 957.41                          \\ \cmidrule(l){3-15} 
                      &                                                                                  & RMSE                    & \multicolumn{1}{c|}{218}                         & \multicolumn{1}{c|}{1905}                         & \multicolumn{1}{c|}{2606}                          & 3531                          & \multicolumn{1}{c|}{171}                         & \multicolumn{1}{c|}{1739}                         & \multicolumn{1}{c|}{2328}                         & 3124                          & \multicolumn{1}{c|}{415.13}                          & \multicolumn{1}{c|}{497.24}                          & \multicolumn{1}{c|}{1021.52}                         & 1583.38                         \\ \cmidrule(l){2-15} 
                      & \multirow{2}{*}{FlightBERT}                                                      & MAE                     & \multicolumn{1}{c|}{123}                         & \multicolumn{1}{c|}{241} & \multicolumn{1}{c|}{1162}                          & 2407                          & \multicolumn{1}{c|}{88}                          & \multicolumn{1}{c|}{\underline{158}} & \multicolumn{1}{c|}{963}                          & 1238                          & \multicolumn{1}{c|}{24.67}                           & \multicolumn{1}{c|}{35.67}                           & \multicolumn{1}{c|}{78.58}                           & 134.29                          \\ \cmidrule(l){3-15} 
                      &                                                                                  & RMSE                    & \multicolumn{1}{c|}{241}                         & \multicolumn{1}{c|}{526} & \multicolumn{1}{c|}{2189}                          & 3969                          & \multicolumn{1}{c|}{154}                         & \multicolumn{1}{c|}{\underline{286}} & \multicolumn{1}{c|}{1904}                         & 3093                          & \multicolumn{1}{c|}{234.17}                          & \multicolumn{1}{c|}{272.59}                          & \multicolumn{1}{c|}{384.22}                          & 462.28                          \\ \midrule
\multirow{12}{*}{DMS} & \multirow{2}{*}{FlightBERT++}                                                    & MAE                     & \multicolumn{1}{c|}{173}                         & \multicolumn{1}{c|}{317}                          & \multicolumn{1}{c|}{\underline{871}}  & \underline{1187} & \multicolumn{1}{c|}{85}                          & \multicolumn{1}{c|}{210}                          & \multicolumn{1}{c|}{\underline{612}} & \underline{1048} & \multicolumn{1}{c|}{\textbf{9.39}}                   & \multicolumn{1}{c|}{\textbf{21.89}}                  & \multicolumn{1}{c|}{\textbf{47.84}}                  & \textbf{78.46}                  \\ \cmidrule(l){3-15} 
                      &                                                                                  & RMSE                    & \multicolumn{1}{c|}{360}                         & \multicolumn{1}{c|}{659}                          & \multicolumn{1}{c|}{\underline{1846}} & 3131 & \multicolumn{1}{c|}{148}                         & \multicolumn{1}{c|}{425}                          & \multicolumn{1}{c|}{\underline{959}} & \underline{2127} & \multicolumn{1}{c|}{175.29} & \multicolumn{1}{c|}{167.16} & \multicolumn{1}{c|}{327.93} & 384.18 \\ \cmidrule(l){2-15} 
                      & \multirow{2}{*}{TimeMixer}                                                       & MAE                     & \multicolumn{1}{c|}{67}                          & \multicolumn{1}{c|}{765}                          & \multicolumn{1}{c|}{3100}                          & 5581                          & \multicolumn{1}{c|}{42}                          & \multicolumn{1}{c|}{422}                          & \multicolumn{1}{c|}{1679}                         & 3028                          & \multicolumn{1}{c|}{21.18}                           & \multicolumn{1}{c|}{50.02}                           & \multicolumn{1}{c|}{119.80}                          & 157.47                          \\ \cmidrule(l){3-15} 
                      &                                                                                  & RMSE                    & \multicolumn{1}{c|}{115}                         & \multicolumn{1}{c|}{1466}                         & \multicolumn{1}{c|}{5517}                          & 9698                          & \multicolumn{1}{c|}{76}                          & \multicolumn{1}{c|}{789}                          & \multicolumn{1}{c|}{2976}                         & 5318                          & \multicolumn{1}{c|}{\textbf{109.57}}                          & \multicolumn{1}{c|}{145.55}                          & \multicolumn{1}{c|}{\underline{281.31}}                          & \underline{374.82}                          \\ \cmidrule(l){2-15} 
                      & \multirow{2}{*}{TimesNet}                                                        & MAE                     & \multicolumn{1}{c|}{73}                          & \multicolumn{1}{c|}{1383}                         & \multicolumn{1}{c|}{5459}                          & 9421                          & \multicolumn{1}{c|}{46}                          & \multicolumn{1}{c|}{753}                          & \multicolumn{1}{c|}{2981}                         & 5086                          & \multicolumn{1}{c|}{36.41}                           & \multicolumn{1}{c|}{79.42}                           & \multicolumn{1}{c|}{178.49}                          & 257.39                          \\ \cmidrule(l){3-15} 
                      &                                                                                  & RMSE                    & \multicolumn{1}{c|}{124}                         & \multicolumn{1}{c|}{2281}                         & \multicolumn{1}{c|}{8377}                          & 14420                         & \multicolumn{1}{c|}{83}                          & \multicolumn{1}{c|}{1237}                         & \multicolumn{1}{c|}{4557}                         & 7796                          & \multicolumn{1}{c|}{154.37}                          & \multicolumn{1}{c|}{184.36}                          & \multicolumn{1}{c|}{392.23}                          & 543.37                          \\ \cmidrule(l){2-15} 
                      & \multirow{2}{*}{MICN}                                                            & MAE                     & \multicolumn{1}{c|}{69}                          & \multicolumn{1}{c|}{680}                          & \multicolumn{1}{c|}{2235}                          & 3912                          & \multicolumn{1}{c|}{\underline{40}}                          & \multicolumn{1}{c|}{384}                          & \multicolumn{1}{c|}{1296}                         & 2256                          & \multicolumn{1}{c|}{48.28}                           & \multicolumn{1}{c|}{544.35}                          & \multicolumn{1}{c|}{1992.86}                         & 3431.19                         \\ \cmidrule(l){3-15} 
                      &                                                                                  & RMSE                    & \multicolumn{1}{c|}{116}                         & \multicolumn{1}{c|}{1133}                         & \multicolumn{1}{c|}{3831}                          & 6598                          & \multicolumn{1}{c|}{\underline{73}}                          & \multicolumn{1}{c|}{633}                          & \multicolumn{1}{c|}{2167}                         & 3704                          & \multicolumn{1}{c|}{\underline{112.31}}                          & \multicolumn{1}{c|}{945.57}                          & \multicolumn{1}{c|}{3270.77}                         & 5612.64                         \\ \cmidrule(l){2-15} 
                      & \multirow{2}{*}{Pathformer}                                                      & MAE                     & \multicolumn{1}{c|}{\underline{52}}                          & \multicolumn{1}{c|}{\underline{232}}                          & \multicolumn{1}{c|}{1374}                          & 1914                          & \multicolumn{1}{c|}{45}                          & \multicolumn{1}{c|}{232}                          & \multicolumn{1}{c|}{863}                          & 1806                          & \multicolumn{1}{c|}{42.03}                           & \multicolumn{1}{c|}{47.65}                           & \multicolumn{1}{c|}{141.08}                          & 259.53                          \\ \cmidrule(l){3-15} 
                      &                                                                                  & RMSE                    & \multicolumn{1}{c|}{\underline{89}}                          & \multicolumn{1}{c|}{\underline{373}}                          & \multicolumn{1}{c|}{2227}                          & \underline{2686}                          & \multicolumn{1}{c|}{83}                          & \multicolumn{1}{c|}{373}                          & \multicolumn{1}{c|}{1346}                         & 2570                          & \multicolumn{1}{c|}{114.37}                          & \multicolumn{1}{c|}{\underline{136.59}}                          & \multicolumn{1}{c|}{406.37}                          & 645.02                          \\ \cmidrule(l){2-15} 
                      & \multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}FlightPatchNet\\ (Ours)\end{tabular}} & MAE                     & \multicolumn{1}{c|}{\textbf{48}}                 & \multicolumn{1}{c|}{\textbf{153}}                 & \multicolumn{1}{c|}{\textbf{546}}                  & \textbf{966}                  & \multicolumn{1}{c|}{\textbf{32}}                 & \multicolumn{1}{c|}{\textbf{105}}                 & \multicolumn{1}{c|}{\textbf{381}}                 & \textbf{678}                  & \multicolumn{1}{c|}{\underline{13.34}}  & \multicolumn{1}{c|}{\underline{32.65}}  & \multicolumn{1}{c|}{\underline{78.57}}  & \underline{123.97} \\ \cmidrule(l){3-15} 
                      &                                                                                  & RMSE                    & \multicolumn{1}{c|}{\textbf{87}}                 & \multicolumn{1}{c|}{\textbf{233}}                 & \multicolumn{1}{c|}{\textbf{885}}                  & \textbf{1577}                 & \multicolumn{1}{c|}{\textbf{64}}                 & \multicolumn{1}{c|}{\textbf{175}}                 & \multicolumn{1}{c|}{\textbf{652}}                 & \textbf{1174}                 & \multicolumn{1}{c|}{123.78}                 & \multicolumn{1}{c|}{\textbf{121.48}}                 & \multicolumn{1}{c|}{\textbf{174.63}}                 & \textbf{244.34}                 \\ \bottomrule
\end{tabular}
}
}
\end{table*}

\paragraph{Visualization of FlightPatchNet Predictions}  
Figure~\ref{fig:flightPatchNet_pred} shows that FlightPatchNet can comprehensively capture the temporal variations of longitude and latitude, while it fails to fully reveal the temporal patterns from original altitude series. 
\begin{figure}[h]
\centering
\subfigure{
    \centering
    \includegraphics[width=0.9\linewidth]{figure/figure_1.pdf}
}\\

\subfigure{
\centering
    \includegraphics[width=0.9\linewidth]{figure/figure_400.pdf}
}\\
\subfigure{
    \centering
    \includegraphics[width=0.9\linewidth]{figure/figure_600.pdf}
}
   
    \captionsetup{font=small}
    \caption{Visualization of the ground truth and predictions of FlightPatchNet when the prediction horizon $T=15$ and look-back window size $L=60$. The series of altitude are in original data while those of longitude and latitude are in differential data, all denoted by meters.}
    \label{fig:flightPatchNet_pred}
\end{figure}
\paragraph{Visualization of FlightPatchNet Predictions for Altitude} 
\begin{figure}[h]
\centering
    \subfigure[]{
     \centering
        \includegraphics[width=0.8\linewidth]{figure/altitudec.pdf}
        \label{altitude_a}
    }
    \subfigure[]{
        \centering
        \includegraphics[width=0.8\linewidth]{figure/altitudeb.pdf}
         \label{altitude_b}
    }\\
    \subfigure[]{
     \centering
        \includegraphics[width=0.8\linewidth]{figure/altitudea.pdf}
         \label{altitude_c}
    }
    \captionsetup{font=small}

    \caption{Visualization of ground truth for altitude and predictions of FlightPatchNet when the prediction horizon $T=15$ and look-back window size $L=60$, all denoted by meters.}
    \label{altitude_pred_gt}
\end{figure}
We present the visualization of FlightPatchNet predictions and ground truth for the altitude in Figure~\ref{altitude_pred_gt}. As shown in Figure~\ref{altitude_a}, when the series of altitudes are relatively smooth and stationary with obvious global trends, FlightPatchNet can effectively capture these trends and make accurate predictions. When the series suffers from many change points caused by frequent abrupt fluctuations, as depicted in Figure~\ref{altitude_b} and ~\ref{altitude_c}, FlightPatchNet tends to focus more on the irregular change points during prediction, leading to a large deviation from the ground truth. As a result, FlightPatchNet struggles to capture the real temporal variations in altitude and fails to provide accurate predictions. 

\paragraph{Effectiveness of Differential Coding}
The results in Table~\ref{table:exp_res_diff} show that using differential coding for longitude and latitude can significantly improve their prediction performance but slightly decrease the accuracy of altitude. The differential coding can reveal the temporal variations of longitude and latitude, which helps the temporal modeling in flight trajectories. However, the variations of altitude in the original series may come from unexpected noise. FlightPatchNet has a strong modeling capacity for temporal variations and tends to focus more on the noise points during altitude prediction, leading to a large bias towards the ground truth.

\begin{table}[h]
\centering
\captionsetup{font=small}
\caption{Flight trajectory prediction results for longitude and latitude in original data and differential data when prediction horizon $T=15$. The best results are highlighted in \textbf{bold}. Note that altitude and velocities are always in original data.}
\label{table:exp_res_diff}

\setlength{\tabcolsep}{1mm}{
    \scalebox{0.78}{
        \begin{tabular}{@{}cccccc@{}}
\toprule
Models                                                                                                & Diff                                               & Metric & Lon($^\circ$)      & Lat($^\circ$)      & Alt(m)       \\ \midrule
\multicolumn{1}{c|}{\multirow{4}{*}{LSTM}}                                                            & \multicolumn{1}{c|}{\multirow{2}{*}{$\checkmark$}} & MAE    & 0.03132          & 0.03717          & 883          \\ \cmidrule(l){3-6} 
\multicolumn{1}{c|}{}                                                                                 & \multicolumn{1}{c|}{}                              & RMSE   & 0.04578          & 0.05143          & 1206         \\ \cmidrule(l){2-6} 
\multicolumn{1}{c|}{}                                                                                 & \multicolumn{1}{c|}{\multirow{2}{*}{$\times$}}     & MAE    & 0.82230          & 0.12008          & 769          \\ \cmidrule(l){3-6} 
\multicolumn{1}{c|}{}                                                                                 & \multicolumn{1}{c|}{}                              & RMSE   & 1.20424          & 2.44136          & 1053         \\ \midrule
\multicolumn{1}{c|}{\multirow{4}{*}{Bi-LSTM}}                                                         & \multicolumn{1}{c|}{\multirow{2}{*}{$\checkmark$}} & MAE    & 0.03890          & 0.04404          & 2006         \\ \cmidrule(l){3-6} 
\multicolumn{1}{c|}{}                                                                                 & \multicolumn{1}{c|}{}                              & RMSE   & 0.05532          & 0.05982          & 2421         \\ \cmidrule(l){2-6} 
\multicolumn{1}{c|}{}                                                                                 & \multicolumn{1}{c|}{\multirow{2}{*}{$\times$}}     & MAE    & 1.71433          & 0.19014          & 2091         \\ \cmidrule(l){3-6} 
\multicolumn{1}{c|}{}                                                                                 & \multicolumn{1}{c|}{}                              & RMSE   & 2.43607          & 0.27621          & 2666         \\ \midrule
\multicolumn{1}{c|}{\multirow{4}{*}{CNN-LSTM}}                                                        & \multicolumn{1}{c|}{\multirow{2}{*}{$\checkmark$}} & MAE    & 0.04149          & 0.05139          & 1137         \\ \cmidrule(l){3-6} 
\multicolumn{1}{c|}{}                                                                                 & \multicolumn{1}{c|}{}                              & RMSE   & 0.05981          & 0.07353          & 1659         \\ \cmidrule(l){2-6} 
\multicolumn{1}{c|}{}                                                                                 & \multicolumn{1}{c|}{\multirow{2}{*}{$\times$}}     & MAE    & 8.59512          & 1.95957          & 1638         \\ \cmidrule(l){3-6} 
\multicolumn{1}{c|}{}                                                                                 & \multicolumn{1}{c|}{}                              & RMSE   & 23.07600         & 8.15418          & 2114         \\ \midrule
\multicolumn{1}{c|}{\multirow{4}{*}{\begin{tabular}[c]{@{}c@{}}FlightPatchNet\\ (Ours)\end{tabular}}} & \multicolumn{1}{c|}{\multirow{2}{*}{$\checkmark$}} & MAE    & \textbf{0.00966} & \textbf{0.00678} & 124          \\ \cmidrule(l){3-6} 
\multicolumn{1}{c|}{}                                                                                 & \multicolumn{1}{c|}{}                              & RMSE   & \textbf{0.01577} & \textbf{0.01174} & 244          \\ \cmidrule(l){2-6} 
\multicolumn{1}{c|}{}                                                                                 & \multicolumn{1}{c|}{\multirow{2}{*}{$\times$}}     & MAE    & 0.19348          & 0.05385          & \textbf{61}  \\ \cmidrule(l){3-6} 
\multicolumn{1}{c|}{}                                                                                 & \multicolumn{1}{c|}{}                              & RMSE   & 0.26243          & 0.07457          & \textbf{170} \\ \bottomrule
\end{tabular}
    }
}
\end{table}
\paragraph{Effectiveness of Multi Scales} 

\begin{figure}[h]
     \centering
        \includegraphics[width=0.8\linewidth]{figure/patch_size_lon.pdf}
        

    \captionsetup{font=small}
    \caption{MAE and RMSE of FlightPatchNet for longitude with single scale and multi scales ($L=60$).}
    \label{fig:effect_multi_scale_lon}
\end{figure}
\begin{figure}[h]
     \centering
        \includegraphics[width=0.8\linewidth]{figure/patch_size_lat.pdf}
        
    % \includegraphics[width=1\linewidth]{figure_1-50.pdf}
    \captionsetup{font=small}
    \caption{MAE and RMSE of FlightPatchNet for latitude with single scale and multi scales ($L=60$).}
    \label{fig:effect_multi_scale_lat}
\end{figure}
\begin{figure}[h]
     \centering
        \includegraphics[width=0.8\linewidth]{figure/patch_size_alt.pdf}
    \captionsetup{font=small}
    \caption{MAE and RMSE of FlightPatchNet for altitude with single scale and multi scales ($L=60$).}
     \label{fig:effect_multi_scale_alt}
\end{figure}
To investigate the effect of multi-scale modeling, we conduct experiments on single scale for \{2,6,10,20,30\}. The results in Figure~\ref{fig:effect_multi_scale_lon}, \ref{fig:effect_multi_scale_lat} and \ref{fig:effect_multi_scale_alt} illustrate the critical contribution of multi scales to our model. We observe that different variables prefer distinct time scales. For instance, a patch size of 10 obtains the second-best prediction performance on longitude and latitude but the worst performance on altitude when prediction horizon $T=15$. This indicates that longitude, latitude and altitude exhibit distinct temporal patterns, and different scales can extract diverse complementary features, which can be effectively leveraged to obtain competitive and robust prediction performance. 

\subsection{Ablation Study}
We conduct ablation studies by removing corresponding modules from FlightPatchNet. Specifically, \textbf{w/o global temporal attention} does not capture the correlations between time steps. \textbf{w/o scale fusion} considers each time scale of equal importance. \textbf{w/o channel fusion} does not explore the relationships between variables. 
Table~\ref{tab:ablation_block} shows the contribution of each component.
\begin{table}[h]
\centering
\captionsetup{font=small}
\caption{Performance comparisons on ablative variants. The best results are highlighted in {\textbf{bold}}. Hor represents the prediction horizon $T\in\{1,3,9,15\}$.}
\label{tab:ablation_block}
\setlength{\tabcolsep}{1mm}{
    \scalebox{0.72}{
       \begin{tabular}{@{}c|c|cccccc@{}}
\toprule
\multirow{2}{*}{Case}                                                                         & \multirow{2}{*}{Hor} & \multicolumn{2}{c|}{\begin{tabular}[c]{@{}c@{}}Lon\\ ($0.00001^ \circ$)\end{tabular}} & \multicolumn{2}{c|}{\begin{tabular}[c]{@{}c@{}}Lat\\ ($0.00001^\circ$)\end{tabular}} & \multicolumn{2}{c}{\begin{tabular}[c]{@{}c@{}}Alt\\ (m)\end{tabular}} \\ \cmidrule(l){3-8} 
                                                                                              &                      & \multicolumn{1}{c|}{MAE}                  & \multicolumn{1}{c|}{RMSE}                 & \multicolumn{1}{c|}{MAE}                 & \multicolumn{1}{c|}{RMSE}                 & \multicolumn{1}{c|}{MAE}                & RMSE                        \\ \midrule 
\multirow{4}{*}{\begin{tabular}[c]{@{}c@{}}w/o \\ global\\ temporal\\ attention\end{tabular}} & 1                    & 51                                        & 90                                        & 34                                       & 66                                        & 21                                      & 137                         \\
                                                                                              & 3                    & 190                                       & 308                                       & 132                                      & 222                                       & \textbf{28}                             & 112                         \\
                                                                                              & 9                    & 667                                       & 1085                                      & 466                                      & 791                                       & \textbf{67}                             & \textbf{164}                \\
                                                                                              & 15                   & 1232                                      & 2005                                      & 876                                      & 1486                                      & \textbf{112}                            & \textbf{221}                \\ \midrule
\multirow{4}{*}{\begin{tabular}[c]{@{}c@{}}w/o\\ scale\\ fusion\end{tabular}}                 & 1                    & 53                                        & 92                                        & 35                                       & 67                                        & 24                                      & 130                         \\
                                                                                              & 3                    & 169                                       & 268                                       & 114                                      & 188                                       & 33                                      & \textbf{100}                \\
                                                                                              & 9                    & 609                                       & 975                                       & 409                                      & 688                                       & 91                                      & 194                         \\
                                                                                              & 15                   & 1112                                      & 1787                                      & 759                                      & 1280                                      & 162                                     & 282                         \\ \midrule
\multirow{4}{*}{\begin{tabular}[c]{@{}c@{}}w/o\\ channel\\ fusion\end{tabular}}               & 1                    & 50                                        & 89                                        & 34                                       & 65                                        & 20                                      & 160                         \\
                                                                                              & 3                    & 166                                       & 265                                       & 112                                      & 187                                       & 29                                      & 122                         \\
                                                                                              & 9                    & 573                                       & 924                                       & 398                                      & 667                                       & 73                                      & 174                         \\
                                                                                              & 15                   & 1059                                      & 1707                                      & 727                                      & 1240                                      & 132                                     & 250                         \\ \midrule
\multirow{4}{*}{FlightPatchNet}                                                               & 1                    & \textbf{48}                               & \textbf{87}                               & \textbf{32}                              & \textbf{64}                               & \textbf{13}                             & \textbf{124}                \\
                                                                                              & 3                    & \textbf{153}                              & \textbf{233}                              & \textbf{105}                             & \textbf{175}                              & 33                                      & 122                         \\
                                                                                              & 9                    & \textbf{546}                              & \textbf{885}                              & \textbf{381}                             & \textbf{652}                              & 79                                      & 175                         \\
                                                                                              & 15                   & \textbf{966}                              & \textbf{1577}                             & \textbf{678}                             & \textbf{1174}                             & 124                                     & 244                         \\ \bottomrule
\end{tabular}
    }
}
\end{table}
Removing the global temporal attention dramatically decreases the multi-step prediction performance, demonstrating the necessary of capturing the correlations between different time steps. Scale fusion can effectively improve the prediction accuracy, indicating that different time scales of trajectory series contain rich and diverse temporal variation information. Channel fusion also improves the model performance, suggesting the importance of exploring relationships between different variables in complex temporal modeling. 

\subsection{Model Complexity}
\begin{table}[h]
    \centering
    \captionsetup{font=small}
\caption{Model Complexity Comparisons. The look-back window size $L = 60$ and the prediction horizon $T = 15$ for all models.}
\label{table:modelcomplexity}
\setlength{\tabcolsep}{1mm}{
    \scalebox{0.72}{
     \begin{tabular}{@{}l|c|c|c|c@{}}
     \toprule

Type                 & Models               & \begin{tabular}[c]{@{}c@{}}Parameters\\ (MB)\end{tabular} & \begin{tabular}[c]{@{}c@{}}FLOPs\\ (M)\end{tabular} & \begin{tabular}[c]{@{}c@{}}Running Time\\ (s/iter)\end{tabular} \\ \midrule 
\multirow{2}{*}{DMS} & FlightPatchNet & 5.69                                                      & 64.38                                               & 0.0069                                                          \\ \cmidrule(l){2-5} 
                     & FlightBERT++         & 44.26                                                     & 3000.00                                                & 0.0112 \\ \cmidrule(l){2-5} 
                     & Pathformer         & 2.64                                                     & 258.55                                               & 0.02174  \\ \cmidrule(l){2-5} 
                     & TimeMixer         & 0.45                                                    & 3115.00                                                & 0.0037    \\ \cmidrule(l){2-5} 
                     & TimesNet         & 37.50                                                    & 196159.87                                              & 0.0534    \\   \cmidrule(l){2-5}                            
                     & MICN         & 1.28                                                    & 1235.83                                              & 0.0012
                     \\ \midrule
\multirow{4}{*}{IMS}  &FlightBERT           & 25.31                                                     & 1620.00                                                & 0.2406                                                          \\ \cmidrule(l){2-5} 
                      & LSTM                 & 0.03                                                      & 1.67                                                & 0.0583                                                          \\ \cmidrule(l){2-5} 
                     & Bi-LSTM              & 0.51                                                      & 31.15                                               & 0.1241                                                          \\ \cmidrule(l){2-5} 
                     & CNN-LSTM             & 0.04                                                      & 1.22                                                & 0.0429  
                      \\ \cmidrule(l){2-5} 
                     & WTFTP            & 0.23                                                     & 60.00                                               & 0.0290\\ \bottomrule 
    \end{tabular}
    }
}
\end{table}

As shown in Table~\ref{table:modelcomplexity}, our proposed FlightPatchNet achieves the greatest efficiency and has relatively fewer parameters compared to other models. For multi-step prediction, the DMS-based models demonstrate significant improvements in computational performance compared to the IMS-based models. In addition, FlightPatchNet is lightweight compared to FlightBERT++ and FlightBERT, which indicates our model can provide a promising solution for real-time air transportation management. 
