# Enhancing Beamforming Efficiency Utilizing Taguchi Optimization and Neural Network Acceleration
## 4 Neural Networks for Synthesis and Optimization of Antenna Arrays
### Taguchi-Neural Network Architectures  
When integrating neural networks with the Taguchi method for antenna array optimization, a comprehensive understanding of the underlying mathematical framework is crucial [65]. The Taguchi method typically employs a loss function, often derived from the signal-to-noise ratio (SNR), aimed at optimizing antenna array performance. One common formulation is the smaller-the-better loss function, shown below. \[L=-10\log_{10}\left(\frac{1}{n}\sum_{i=1}^{n}\left(\frac{1}{\phi_{i}}\right)^{ 2}\right) \tag{34}\]  
Here, \(\phi_{i}\) represents the observed performance from individual experiments. This loss function encapsulates the overall quality of the antenna array, with lower values indicating better performance. Neural networks are then utilized to predict the output performance \(\hat{\phi}\) based on th einput parameters \(X\) and network parameters \(\theta\). Mathematically, this relationship is expressed as \(\hat{\phi}=f(X;\theta)\), where the function \(f\) encapsulates the mapping learned by the neural network. The optimization objective is to minimize the loss function \(L(X)\) subject to any parameter constraints. During the training phase of the neural network, optimization algorithms such as gradient descent are employed to iteratively update the network parameters \(\theta\). Specifically, the parameters are updated according to  
\[\theta:=\theta-\alpha\
abla J(\theta), \tag{35}\]  
where \(\alpha\) represents the learning rate and \(\
abla J(\theta)\) denotes the gradient of the loss function with respect to the network parameters. These equations form the backbone of the mathematical framework when combining neural networks with the Taguchi method for antenna array optimization. By leveraging neural networks, engineers can effectively design and optimize antenna arrays, leading to enhanced performance and efficiency in various applications [66, 67, 68] (Table 8). We employed the backpropagation algorithm, a widely-used method in neural network training, to train our model. The neural network architecture consisted of three layers: an input layer with 15 neurons, a hidden layer with 50 neurons, and an output layer with 10 neurons (see Figure 8). To measure the performance of the network during training, we utilized the Mean Squared Error (MSE) as the performance criterion. Our goal was to minimize the MSE to below 0.0001, indicating a high level of accuracy in the model's predictions (see Figure 9). During the training process (see Figure 10), we set parameters to monitor the progress and control the learning dynamics. We configured the network to display progress updates every 50 training epochs, allowing us to track the improvement of the model over time. Additionally, we set the learning rate to 0.03 to regulate the size of weight updates during training. To ensure effective convergence of the model, we specified the maximum number of training epochs as 30,000. This parameter defines the number of iterations the training algorithm undergoes to optimize the network parameters. Furthermore, we employed a momentum coefficient of 0.9, which helps accelerate convergence by considering past weight updates during parameter optimization [70]. We chose the backpropagation algorithm for several reasons. First, backpropagation is widely recognized for its effectiveness in training artificial neural networks, with proven successes in domains such as computer vision and natural language processing. Additionally, its implementation is relatively straightforward, and is supported by numerous online resources and well-established code libraries. The flexibility of backpropagation allows it to adapt to a wide variety of neural network structures and to work with different data types and architectures(see Figure 11). Furthermore, it provides efficient optimization of network weights through techniques such as gradient descent, and is widely supported by popular libraries such as TensorFlow and PyTorch.