## 3 Prediction of Manned Aircraft Trajectory in Terminal Areas
### Model Construction  
The gated recurrent unit (GRU) structure is utilized to train the neural network for manned aircraft trajectory prediction. The loss function is expressed as:  
\[Loss=\frac{1}{n}\sum_{i=1}^{n}\left(y(x)_{i}-y_{i}\right)^{2} \tag{2}\]  
where \(y(x)_{i}\) is the predicted position of the aircraft at the time \(i\); and \(y_{i}\) is the actual position of the aircraft at the time \(i\).  
In the process of neural network training, to minimize the loss function, the stochastic gradient descent method is used to solve the problem. The minimized loss function and corresponding model parameters are obtained. Figure 1 depicts the GRU architecture for aircraft trajectory prediction.  
As mentioned above, the small drone will only collide with the manned aircraft during the take-off and landing phases, and for each trajectory, data in the take-off and landing phases are filtered. With the longitude, latitude, altitude, ground speed, and heading angle of the manned aircraft, the input to the network is a three-dimensional dataset with the shape of the batch number of samples * number of features * timestep. The data samples are divided into the training sets, validation sets, and test sets according to the ratio of 80%, 10%, and 10%, respectively.  
The structural parameters and internal parameters of the neural network are initialized, including the number of hidden layers, the number of neurons in a single layer, the number of batch training, the number of iterations, and the learning rate. The neural network parameters are iteratively updated by using the manned aircraft trajectory training dataset. In each epoch process, a certain number of samples are randomly selected from the trajectory data training set, and the loss function value is calculated according to the data labels corresponding to the actual value of the aircraft trajectory. The test set is substituted into the trained network structure for performance testing.  
The long short-term memory structure  
The long short-term memory (LSTM) network was proposed by [PERSON] and [PERSON] for addressing the vanishing gradients problem in RNNs [30]. It has been proven to perform well on sequence-based tasks with long-term dependencies. Compared with the traditional artificial neural network, the LSTM network realizes the combination of long-term and short-term memory by setting special structures such as the forgetting gate, input gate and output gate. Based on the LSTM network, the historical data of the drone trajectory, including the longitude, latitude, altitude, ground speed, and heading angle of the manned aircraft, are input into the model. The output is the trajectory prediction result in the next moment. Based on the standard LSTM network, the learning rate is automatically adjusted to avoid accuracy degradation and local convergence caused by rapid convergence.  
The support vector regression structure  
Support vector regression (SVR) is a supervised learning algorithm that can be used to predict time series data. SVR trains using a symmetrical loss function, which equally penalizes high and low misestimates. A flexible tube of minimal radius is formed symmetrically around the estimated function. In this way, the absolute values of errors less than a certain threshold \(\varepsilon\) are ignored both above and below the estimate. The points outside the tube are penalized, but those within the tube, either above or below the function, receive no penalty. One of the main advantages of SVR is that its computational complexity does not depend on the dimensionality of the input space. Additionally, it has excellent generalization capability, with high prediction accuracy [31]. The inputs and outputs of the model are similar to those in the LSTM and GRU models.