## 4 Neural Network Analysis
### BP Neural Network  
The BP neural network is composed of an input layer, hidden layer, and output layer. It is a multilayer neural network with error back propagation [4; 5]. BP neural network can learn the mapping relationship between input and output because it uses the back propagation algorithm to train the network. In the training process, the input data are first transmitted to the input layer of the network, and then the output value is obtained by feedforward calculation. The network output is compared with the expected output, and the error is calculated. Then, the error is back-propagating from the output layer to the input layer, and the connection weights in the network are adjusted to minimize the error. This process of adjusting weights uses a gradient descent algorithm to update the weights by calculating the partial derivatives of each layer of neurons. After the training of the BP neural network is completed, it can be used to predict or classify new input data. It can be applied to fields such as pattern recognition, data mining, artificial intelligence, and so on. It should be noted that the BP neural network may be plagued by over-fitting problems, and appropriate network structure design and regularization methods are needed to avoid over-fitting and achieve better generalization performance. The specific process is as follows: The data information is transmitted to the neural network, and a processing method is selected from the input layer and the result is transmitted to the hidden layer. The number of hidden layers can be one or more. If the input problem is different, the hidden layer is also different. The information processed by the input layer is then transmitted to the hidden layer. By processing the information of the hidden layer, the obtained information is transmitted to the output layer, and finally the output layer processes and outputs the results. If there is a deviation between the output layer, result and the expected output, the error is propagated backward from the output layer, and the network connection weights and thresholds of the output layer, the hidden layer, and the input layer are updated. The forward propagation is repeated to calculate the output and adjust the reverse error until the output value meets the desired output [21; 22]. In Figure 2, \(J_{1},J_{2},J_{3}\ldots J_{a}\) are the input values of the BP neural network, \(K_{1},K_{2},K_{3}\ldots K_{b}\) are the predicted values of the BP neural network, \(S_{cd}\) and \(S_{de}\) are the weights of the BP neural network. The BP neural network can be regarded as a nonlinear function. The input value is the independent variable of the function, and the output value is the dependent variable of the function. When the number of input nodes is \(a\) and the number of output nodes is \(b\), the BP neural network expresses the function mapping relationship from \(a\) variables to \(b\) variables. The BP neural network can make the network have associative memory and predictive ability only by training the network first. The training steps of BP neural network are as follows:  
1. Network initialization. According to the input sequence \((J,K)\), the number of network input layer nodes \(a\), the number of hidden layer nodes \(l\), and the number of output layer nodes \(b\) are determined. The connection weights \(S_{cd}\) and \(S_{de}\) between the input layer, the hidden layer, and the output layer neurons are initialized. The hidden layer threshold \(x\) and the output layer threshold \(y\) are initialized, and the learning rate and neuron activation function are given. 2. Hidden layer output calculation. According to the input vector \(J\), the connection weight \(S_{cd}\) between the input layer, the hidden layer, and the hidden layer threshold \(x\), the hidden layer output \(H\) is calculated. \[H_{d}=f(\sum_{c=1}^{a}S_{cd}J_{c}-a_{d}),d=1,2,\ldots,l\] (1) In Equation (1), \(l\) is the number of hidden layer nodes, and \(f\) is the hidden layer activation function. This function has a variety of expressions.