# Meander Structure Analysis Techniques Using Artificial Neural Networks
## 2 Materials and Methods
### Structure of the Convolutional Neural Network  
In this subsection, the convolutional neural network (CNN) model is investigated for the analysis and synthesis of meander structures. The presented CNN model was used to predict the electromagnetic (EM) responses of the meander structure and speed up the design process. The layering architecture enables CNNs to learn the underlying nonlinear relationships effectively, making these models particularly adept at handling the complexities in microwave device data for more accurate predictions. A prediction was made by mapping the relationship between the geometric dimensions of the meander structure and its EM responses. In total, three separate CNN models were trained for each EM response. All three models shared the same structural architecture and training settings. This consistency in structure and settings provided a uniform framework which allowed for comparative analysis and validation. The designed CNN consisted of a total of 18 layers, including four convolutional layers, each followed by a batch normalization layer and an activation layer of rectified linear units (ReLUs). The first two ReLU layers were further linked to the average pooling layers with a size of 2 and a stride of 2, reducing the spatial dimensions of the feature maps while improving the network's ability to extract and generalize nonlinear EM responses as input data. The final ReLU layer led to a dropout layer with a probability of 10%, which helped prevent overfitting by randomly omitting some network connections during training. This architecture also included a fully connected layer which integrated the learned features, culminating in a regression output layer designed to predict 397 response values. The network had a total of 1.6 million learning parameters, allowing it to effectively capture and predict the characteristics of nonlinear microwave devices. The architecture of the investigated model is shown in Figure 5.  
The model designed for this application began with an input layer which accepts images \(64\times 64\) pixels in size. The first convolutional layer had 4 filters, the second had 8 filters, and the third and fourth layers each had 16 filters. These filters were \(3\times 3\) in size, ideal for capturing the fine details in an image without losing spatial resolution. To maintain the output size, padding was applied in each convolutional layer, ensuring that the dimension of the output feature map remained unchanged. The stride, which defines the step size of the filter as it moves through the image, was set to 1. The dilation factor was also set to \(1\times 1\), indicating that there was no dilation in the convolution process. The size of the mini-batch was set to 128. The network was trained over 200 epochs. A key aspect of training was the initial learning rate, established to be 0.0001. A piecewise learning rate schedule was implemented. Furthermore, the learning rate drop factor was set to 0.1, and the learning rate drop period was in all 20 epochs, helping to gradually  
Figure 4: The general structure of the time delay neural network (TDNN), with \(x=[L_{\mathrm{c}}\ N\ 1\ 2a\ c\ f]\), \(Y=[t_{\mathrm{d}}]\), and \(N_{\mathrm{d}}\ =\ 5\). accelerate the convergence of the network to optimal performance. The MSE was employed as the loss function. In this study, the number of filters used in the investigated CNN architecture was notably lower compared with typical CNN structures. This reduction was primarily attributed to the nature of the input data, consisting of grayscale images, which inherently possess less complexity than their colored counterparts as they contain only intensity information, rather than the additional color channel data present in RGB images. Consequently, a lower number of filters was sufficient to capture the essential features and patterns in the grayscale images, as there was a reduced requirement to process and distinguish between multiple color channels [47].