## 3 Methodology and Comparative Experimental Setup
### Methodology
#### 3.1.2 Standard Artificial Intelligence for Modeling  
* BP Neural Network;  
The BP neural network model, proposed by [PERSON] and [PERSON], is capable of simultaneously learning and storing a large number of \"input-output\" mapping relationships, and is one of the most widely applied neural network models today [33]. Each node represents a function, and each connection between two nodes represents a weighted value of the signal they are connected to, called the weight, as shown in Figure 3. To reach the minimum, the specific form of back-propagation is as follows:  
(1) Error function definition  
The error function is the layer-by-layer transfer of the function between the nodes in the hidden layer, after the training sample passes through the input layer, and the output layer passes through the output layer to obtain the sampling error after the corresponding error square function, \(E_{P}\)[34]:  
\[E_{P}=\frac{1}{2}\sum_{j=1}^{m}\left(t_{j}^{p}-y_{j}^{p}\right)^{2} \tag{7}\]  
where \(E_{P}\) is the sampling error, \(t_{j}^{p}\) is the desired output, and \(y_{j}^{p}\) is the actual output produced by the neural network on the input.  
For \(P\) samples, the global error is  
\[E=\frac{1}{2}\sum_{P=1}^{P}\sum_{j=1}^{m}\left(t_{j}^{p}-y_{j}^{p}\right)= \sum_{j=1}^{m}E_{P} \tag{8}\]  
(2) Output layer weight change  
Using the cumulative error BP algorithm to adjust \(w_{jk}\), lowering the global error \(E\), i.e.,  
\[\Delta w_{jk}=-\eta\frac{\partial E}{\partial w_{jk}}=-\eta\frac{\partial}{ \partial w_{jk}}\left(\sum_{P=1}^{P}E_{p}\right)=\sum_{P=1}^{P}\left(-\eta \frac{\partial E_{P}}{\partial w_{jk}}\right) \tag{9}\]  
where \(\eta\) is the learning rate (%), and \(w_{jk}\) is the output layer weight.  
(3) Implied weight change  
\[\Delta v_{ki}=-\eta\frac{\partial E}{\partial v_{ki}}=-\eta\frac{\partial}{ \partial v_{ki}}\left(\sum_{P=1}^{P}E_{p}\right)=\sum_{P=1}^{P}\left(-\eta \frac{\partial E_{P}}{\partial v_{ki}}\right) \tag{10}\]  
where \(v_{ki}\) is the implicit layer weight.  
Define the error signal as  
\[\delta_{zk}=-\frac{\partial  
* Elman Neural Network;  
The Elman neural network model is a class of local regression networks with extreme computational power, connecting local memory units with local feedback. Compared to the general static neural network, Elman has an additional association layer, which is used as the input of the implicit layer and as the input of the implicit layer at the previous moment, and has the characteristics of fast approximation and good dynamic characteristics [35]. The specific structure is shown in Figure 4[36]. The mathematical model of the Elman neural network is  
\[x_{c}(k)=\alpha\cdot x_{c}(k-1)+h(k-1) \tag{13}\]  
where \(x_{c}(k)\) is the take-up output at moment \(k\) and \(\alpha\) is the feedback gain of the self-connection.  
\[h(k)=f\Big{(}w^{1}u(k-1)+w^{3}x_{c}(k)+b_{h}\Big{)} \tag{14}\]  
where \(h(k)\) is the implied layer output value; and \(w^{1}\), \(w^{3}\) are the connection weights from the input layer to the middle layer and the takeover layer to the middle layer, respectively.  
\[y(k)=g\Big{(}w^{2}h(k)+b_{y}\Big{)} \tag{15}\]  
where \(g(\cdot)\) is the transfer function of the output neuron, and \(w^{2}\) is the connection weight of the middle layer to the output layer.  
Figure 4: Schematic diagram of Elman model structure.  
Figure 3: Structure of back-propagation (BP) neural networks.