## 2 Thermal Error Modeling
### Introduction of CNN-LSTMNN
#### 2.2.2 Long Short-Term Memory Neural Network  
The long short-term memory neural network (LSTMNN) can effectively extract continuous-time features. In contrast to RNN, LSTMNN exhibits distinctive features and advantages, such as capturing long-term dependence, alleviating the disappearance and explosion of gradients, processing variable-length sequences, learning gating mechanisms, and carrying out parallel computing. LSTMNN can automatically learn the important information and ignore the unimportant parts in the input data through the gating mechanism, which can control the flow of information inside the memory unit. This makes LSTMNN more resistant to noise and redundant information and improves the robustness of the model. Moreover, in traditional RNN, the backpropagated gradient is prone to the  
Figure 2: Calculation process diagram of CNN s core steps. problem of gradient vanishing or gradient explosion on long sequences, which leads to difficulties with the training and learning of deep networks. However, LSTMNN effectively alleviates this problem through the gating mechanism, which enables the gradient to be effectively transmitted over longer time steps, further improving the training efficiency and performance of the network. The underlying structure of the LSTMNN s hidden layer is characterized by the addition of three thresholds and two states, namely, the forgetting threshold \(f_{t}\), the input threshold \(i_{t}\) and the output threshold \(o_{t}\), as well as the cellular state \(C_{t}\) and the candidate state \(\widetilde{C}_{t}\). They are computed as follows:  
\[h_{t}=tanh(C_{t})*o_{t} \tag{12}\]  
\[C_{t}=\widetilde{C}_{t}*i_{t}+C_{t-1}*f_{t} \tag{13}\]  
\[\widetilde{C}_{t}=\tanh(h_{t-1}*W_{hc}+x_{t}*W_{xc}+b_{c}) \tag{14}\]  
\[i_{t}=\sigma(h_{t-1}*W_{hi}+x_{t}*W_{xi}+b_{i}) \tag{15}\]  
\[f_{t}=\sigma\Big{(}h_{t-1}*\ W_{hf}+x_{t}*W_{xf}+b_{f}\Big{)} \tag{16}\]  
\[o_{t}=\sigma(h_{t-1}*W_{ho}+x_{t}*W_{xo}+b_{o}) \tag{17}\]  
where \(h_{t-1}\) denotes the immediate memory at the preceding time step \(t-1\) : \(h_{t}\) represents the immediate memory at the ongoing time step \(t\): \(x_{t}\) is the input characteristic at the current time; \(b_{i}\), \(b_{f}\), and \(b_{o}\) are the to-be-trained bias items; \(W_{xi}\), \(W_{xf}\), and \(W_{xo}\) are the to-be-trained parameter matrices of \(x_{t}\); \(W_{hi}\), \(W_{hf}\), and \(W_{ho}\) are the to-be-trained parameter matrices of \(h_{t}\); cellular state \(C_{t}\) represents sustained memory, \(\widetilde{C}_{t}\) denotes temporary memory; and \(\sigma\) stands for using sigmoid to activate the function. The diagram illustrating the structural units of LSTMNN is illustrated in Figure 3.  
The design of the CNN-LSTM neural network (CNN-LSTMNN) is depicted in Figure 4. CNN uses a pair of convolutional layers and a pair of pooling layers, whereas the LSTMNN part uses two LSTM layers and two dropout layers. The dropout layer can discard a certain proportion of the neurons in the last hidden layer, which alleviates the over-fitting problem. Table 1 showcases the parameter configurations of CNN-LSTMNN. Figure 3: Structural unit diagram of LSTMNN. _Appl. Sci._**2024**, _14_, 381 positions of the N pelicans can be represented by an N \(\times\) M matrix, denoted as matrix P, as follows:  
\[P=\left[ \begin{array}{c}P_{1}\\ \vdots\\ P_{i}\\ \vdots\\ P_{N}\end{array} \right]_{N\times M}=\left[ \begin{array}{ccccc}p_{1,1}&\cdots&p _{1,j}&\cdots&p_{1,M}\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ p_{i,1}&\cdots&p_{i,j}&\cdots&p_{i,M}\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ p_{N,1}&\cdots&p_{N,j}&\cdots&p_{N,M}\end{array} \right]_{N\times M},\ i=1,2,\ldots,N;j=1,2,\ldots,M; \tag{18}\]  
where \(p_{i,j}\) denotes the \(i\)-th pelican s site in the \(j\)-th dimension. At the beginning, the site of the pelican is randomly distributed, so \(p_{i,j}\) can be initialized in a random manner within the specified lower and upper boundaries of the prey hunting range for the pelican at index \(i\) in the \(j\)-th dimension.