## 2 Network Models  
### Generative Adversarial Network Model  
The basic model of the original GAN [9] consists of two parts: a generator and a discriminator. Drawing inspiration from the game theory, the generator and discriminator confront each other and constantly play binary minimax games. Throughout this process, the generator gradually learns to capture the distribution of actual samples. Generally speaking, the training is considered to be complete when the confrontation between the generator and the discriminator satisfies the Nash equilibrium. Figure 1 shows the basic model of the GAN. The generator (\(G\)) input is a random variable \(x\), and the output is called the generated sample \(\hat{y}\). The discriminator (\(D\)) input is the generated sample \(\hat{y}\) and the actual sample \(y\), and the output is the determination result  
of the input sample. The training goal of \(G\) is to make the generated sample \(\hat{y}\) conform to the distribution of the actual sample \(y\) as much as possible so that it can hide from \(D\). The training goal of \(D\) is to distinguish whether the input sample is a generated sample or an actual sample as much as possible. In the straight game between the two, the samples generated by \(G\) gradually tend to the distribution of actual samples, and the ability of \(D\) to distinguish true and false samples gradually improves. Until it reaches equilibrium, the entire network structure can generate new samples that fulfill the utilization requirements. Thus, the training objective of the whole network can be expressed as minimizing the distribution distance between \(y\) and \(\hat{y}\) and maximizing the discrimination ability of D to the input samples. The objective function expression can be obtained as follows:  
\[\min_{G}\max_{D}F(G,D)=E_{y\sim P_{data(y)}}[\log D(y)]+E_{x\sim P_{x(x)}}[ \log(1-D(G(x))] \tag{1}\]  
The generator and discriminator constitute a fundamental neural network architecture consisting of an input layer, a hidden layer, and an output layer. This architecture incorporates linear and nonlinear transformations, as shown in Figure 2 below. The information flow only passes in one direction, with random noise being propagated from the input node to the hidden layer and subsequently to the output layer for generating the required motor samples. The network, as the simplest ANN model [20], can map complex relationships between backpropagation learning noise and motor samples without rigorous mathematical formulas. For the generator, \(x\) is the input random noise matrix and \(y\) is the generated motor sample. As for the discriminator, \(x\) is the actual motor sample or the generated motor sample, and \(y\) is the score (0-1) of the input sample, which represents the possibility of the discriminator to determine that the sample is an actual sample. \(z_{i}^{l}\) and \(a_{i}^{l}\) are the inputs and  
Figure 1: Basic model diagram of GAN. Figure 2: Structure diagram of BP neural network. outputs of the \(i\)-th neuron in the \(l\)-th layer, and \(z^{l}\) and \(a^{l}\) are the inputs and outputs of the \(l\)-th neuron, \(z^{l}_{i}\in z^{l},a^{l}_{i}\in a^{l},i=1,2,3,\ldots,n\). On the right side of Figure 2 is the structure of each neuron. It can be seen that each element in the input matrix is linearly calculated with its corresponding weights and biases. Then, the nonlinear operation of the activation function is performed. Without the activation function, the number of hidden layers of the network will be meaningless. No matter how many layers are added, the linear calculation will be a simple accumulation. The matrix operations of each layer are as follows:  
\[z^{l}=\omega^{l}\cdot a^{(l-1)}+b^{l} \tag{2}\]  
\[a^{l}=f^{l}(z^{l}) \tag{3}\]  
where \(\omega^{l}\) is the weight matrix of the layer, \(b^{l}\) is the bias matrix of the layer, and \(a^{l}\) is the activation function of the layer.