## 4 Methodology
### Generative Adversarial Network (GAN) Module
Eventually, a dynamic equilibrium is reached, in which the generator produces samples that closely match the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Module** & **Content** & **Parameter Setting** & \begin{tabular}{c} **Output Size** \\ **(B = Batch Size)** \\ \end{tabular} \\ \hline \multirow{3}{*}{1} & \begin{tabular}{c} Input (Latent Noise) \\ Repeat \\ Permute \\ \end{tabular} & latent dim = 64 & \begin{tabular}{c} (B, latent\_dim) \\ (B, 256, latent\_dim) \\ (B, latent\_dim, 256) \\ \end{tabular} \\ \hline \multirow{3}{*}{2} & \multirow{3}{*}{Conv1D} & in channels = 8 & \multirow{3}{*}{(B, 64, 256)} \\  & & kernel = 3 & \\ \cline{1-1}  & & padding = 2 & \\ \cline{1-1}  & & out channels = 64 & \\ \cline{1-1}  & \
\begin{tabular}{c} BatchNorm +ReLU \\ Permute \\ \end{tabular} & & &
\begin{tabular}{c} (B, 64, 256) \\ (B, 256, 64) \\ \end{tabular} \\ \hline \multirow{3}{*}{4} & \multirow{3}{*}{LSTM} & input size = 64 & \multirow{3}{*}{(B, 256, 256)} \\  & & hidden = 128 & \\ \hline \multirow{3}{*}{5} & Linear + & input size = 256 & \multirow{3}{*}{(B, 256, 8)} \\  & Tanh & output size = 8 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Generator network architecture and settings. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Module** & **Content** & **Parameter Setting** & \begin{tabular}{c} **Output Size** \\ **(B = Batch Size)** \\ \end{tabular} \\ \hline \multirow{3}{*}{1} & \begin{tabular}{c} Input (Time-series data) \\ Permute \\ Conv1D \\ \end{tabular} & \begin{tabular}{c} in channels = 8 \\ kernel = 5 \\ padding = 2 \\ out channels = 64 \\ \end{tabular} & \begin{tabular}{c} (B, 256, 8) \\ (B, 8, 256) \\ (B, 64, 256) \\ \end{tabular} \\ \hline \multirow{3}{*}{2} & BatchNorm + ReLU & \multirow{3}{*}{(B, 64, 256)} \\  & Dropout & \(p\) = 0.3 & \begin{tabular}{c} (B, 64, 256) \\ (B, 256, 64) \\ (B, 256, 256) \\ \end{tabular} \\ \cline{1-1}  & Permute & input size = 64 & \begin{tabular}{c} (B, 256, 64) \\ (B, 256, 256) \\ \end{tabular} \\ \cline{1-1}  & LSTM & hidden size = 128 & \begin{tabular}{c} (B, 256) \\ (B, 256, 256) \\ \end{tabular} \\ \hline \multirow{3}{*}{5} & \begin{tabular}{c} TimeStep Extract \\ Linear \\ Sigmoid \\ \end{tabular} & \
\begin{tabular}{c} input size = 256 \\ output size = 1 \\ \end{tabular} &
\begin{tabular}{c} (B, 256) \\ (B, 1) \\ (B, 1) \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Discriminator network architecture and settings. distribution of real data, making it increasingly difficult for the discriminator to accurately identify their origin. In this process, the generator aims to maximize the probability that the discriminator classifies generated (fake) samples as real, whereas the discriminator seeks to correctly predict the authenticity of input samples, i.e., maximize the probability of identifying real samples correctly and minimize the probability of misclassifying fake samples as real. This adversarial objective can be formulated through corresponding loss functions, as expressed in the following Equations (1) and (2):  
\[L\_G=-E\_\{z\sim p\_z(z)\}[log(D(G(z)))] \tag{1}\]  
\[L\_D=-(E\_\{x\sim p\_data\}(x))\{log(D(x))]+E\_\{z\sim p\_z(z)\}[log(1-D(G(z)))]\} \tag{2}\]  
Here, \(L\_G\) denotes the loss function of the generator and \(L\_D\) denotes the loss function of the discriminator. The notation \(z\sim p\_z(z)\) represents a random noise vector sampled from a prior distribution, while \(x\sim p\_\{data\}\) denotes a real sample drawn from the distribution of fault data. \(G(z)\) refers to the synthetic sample generated by the generator from the input noise \(z\), and \(D(x)\) represents the probability assigned by the discriminator that input \(x\) is real.