## 4 Methodology  
This section outlines the overall framework of the innovative hybrid method. Our study employs a novel approach that combines advanced data augmentation techniques with an adapted Inception architecture to enhance the accuracy and robustness of AE source localization in complex structures. This methodology integrates the custom-designed GAN for data augmentation with an Inception network specifically adapted for regression tasks. The overall process flow is shown in Figure 4a. It commences with the collection of AE signals, each labeled with coordinates in the form of \((d_{i},\theta_{i})\) for 35 distinct positions. These signals undergo a training/test split. The training data are then augmented using four different GAN architectures (GAN, DCGAN, WGAN, and TSAGAN) to address dataset imbalance and scarcity issues. This multi-GAN augmentation approach is crucial as it generates synthetic AE signals that closely emulate the characteristics of real data, effectively expanding the dataset and improving the model's ability to generalize across various AE source locations. Each GAN variant offers unique strengths in data synthesis, allowing for a comprehensive augmentation strategy. The augmented training data from each GAN feed into an Inception network specifically adapted for regression tasks, forming the core of our hybrid approach. This adaptation of the Inception network, originally designed for image classification, enables effective processing of AE signals across multiple scales, capturing both local and global features crucial for accurate localization. This trained network is then utilized to predict AE source locations from the test data. Figure 4b specifies the architecture of the Inception network used for regression. It initiates with an input layer, followed by an Inception Module that processes data through multiple parallel pathways. The Inception Module is particularly adept at AE signal processing as it can simultaneously extract features at different scales, which is essential given the complex nature of AE waveforms. The outputs are concatenated and passed through batch normalization. A ReLU activation function is then applied, followed by global average pooling to reduce spatial dimensions. global average pooling is employed instead of traditional fully connected layers to minimize the number of parameters, mitigate overfitting, and maintain spatial information. Finally, a dense layer produces the output, predicting the AE source location as continuous coordinates. The four GAN architectures--GAN, DCGAN, TSAGAN, and WGAN--share a common foundation but differ in key aspects of their design and training approach, which is shown in Table 1. The original GAN uses fully connected layers in both the generator and discriminator, with a structure of (128, 512, 1024) neurons for the generator and (1024, 512, 64) for the discriminator. It employs LeakyReLU activations throughout, with batch normalization (momentum 0.8) in both networks. The DCGAN introduces convolutional layers, specifically using 1D transposed convolutions in the generator and 1D convolutions in the discriminator, which are particularly effective for capturing spatial or temporal patterns in the data. It typically uses ReLU activations in the generator and LeakyReLU in the discriminator, with a _tanh_ activation in the final generator layer. The TSAGAN, tailored for time series data, reverts to a fully connected architecture similar to the original GAN but is optimized for sequential data. The key innovation of the WGAN lies not in its architecture, which is similar to the DCGAN with convolutional layers, but in its use of the Wasserstein loss function and weight clipping in the discriminator (now called a critic). This change allows for more stable training and potentially better-quality results. The WGAN also typically includes dropout in the discriminator, a feature that is not present in the other architectures.