## 4 Methodology
All four models use the Adam optimizer with a learning rate of 0.0002, a beta of 0.001, and a batch size of 64, and are trained for 2000 epochs. They all incorporate model collapse monitoring, but the WGAN stands out with its multiple critic updates per generator update. These architectural differences make each variant suitable for different types of data and training scenarios, with the DCGAN and WGAN often performing well on complex, high-dimensional data, while the GAN and TSAGAN can be effective for simpler or specific time-series datasets. Figure 4: (**a**) Workflow of the hybrid network for AE source localization (**b**) Architecture of the Inception network for regression. The GAN architecture for data augmentation, detailed in Figure 5, involves a generator and a discriminator, both of which are sequential models. The generator \(G\) transforms a noise vector z into synthetic AE data \(G(z)\) that mimics the real data distribution. The discriminator \(D\) evaluates both \(x_{\text{real}}\) and generated data \(G(z)\), outputting a probability of the input being real. The training process alternates between updating the discriminator and the generator. The key equations governing this process are presented as follows. The first step is to compute the discriminator real loss:  
\[\mathcal{L}_{\mathcal{D}_{\text{fa}}}=-\text{log}(D(x_{\text{real}})) \tag{1}\]  
Then, we derive the discriminator fake loss as:  
\[\mathcal{L}_{\mathcal{D}_{\text{fa}}}=-\text{log}(1-D(G(z))) \tag{2}\] \begin{table}
\begin{tabular}{c|c|c|c|c} \hline \hline
**Feature** & **GAN** & **DCGAN** & **TSAGAN** & **WGAN** \\ \hline \multirow{2}{*}{Generator Architecture} & Fully connected layers & 1D transposed convolutions, Dense layers (128, 512, 1024) & Fully connected layers (128, 512, 1024) & Fully connected layers (128, 512, 1024) \\ \hline \multicolumn{4}{c}{Noise vector} \\ \hline Input & \multicolumn{4}{c}{Noise vector} \\ \hline Output layer & Dense layer with \multicolumn{4}{c|}{Dense layer with tanh} & Dense layer with \multicolumn{4}{c|}{Dense layer with \multicolumn{4}{c}{LeakyReLU}} \\ \hline Discriminator & Fully connected layers (1024, 512, 64) & 1D convolutions, Dense layers & Fully connected layers (1024, 512, 64) & 1D convolutions, Dense layers \\ \hline Loss Function & Binary cross-entropy & Binary cross-entropy & Binary cross-entropy & Wasserstein loss \\ \hline Optimizer & \multicolumn{4}{c}{Adam (learning rate = 0.0002, beta = 0.001)} \\ \hline Batch Size & 64 & 64 & 64 & 64 \\ \hline Epochs & 2000 & 2000 & 2000 & 2000 \\ \hline \multirow{2}{*}{Special Features} & Model collapse & Model collapse & Model collapse & Multiple critic updates, Model collapse \\  & monitoring & monitoring & monitoring & monitoring \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Specifications for the GAN Variants. Figure 5: Architecture of the generator and discriminator networks in the GAN for AE signal augmentation. By combining the two terms, the overall discriminator loss is:  
\[\mathcal{L}_{\mathcal{D}}=0.5\times\left(\mathcal{L}_{\mathcal{D}_{\mathrm{ra}}}+ \mathcal{L}_{\mathcal{D}_{\mathrm{Rx}}}\right) \tag{3}\]  
And the overall generator loss is derived from the input noise vector:  
\[\mathcal{L}_{\mathcal{G}}=-\mathrm{log}(D(G(z))) \tag{4}\]  
The discriminator aims to maximize the probability of correctly classifying real and fake samples, while the generator aims to minimize the probability of the discriminator correctly classifying generated samples as fake. During training, we updated the discriminator by maximizing:  
\[E[\mathrm{log}(D(x_{\mathrm{real}}))]+E[\mathrm{log}(1-D(G(z)))] \tag{5}\]  
and then updated the generator by minimizing:  
\[E[\mathrm{log}(1-D(G(z)))] \tag{6}\]  
To prevent mode collapse, we implemented a model edge collapse threshold \(\tau\). If \(|\mathcal{L}_{\mathcal{D}}-\mathcal{L}_{\mathcal{G}}|>\tau\) for a certain number of consecutive iterations, we reinitialized the models.