# Phased Antenna-Array Synthesis Using Taylor-Series Expansion and Neural Networks
## 5 Neural Networks for Synthesis and Optimization of Antenna Arrays
### Taylor-Neural Network Architectures  
Taylor-Neural Network architectures combine the advantages of the Taylor method with the learning capabilities of neural networks to optimize antenna array designs. This hybrid approach allows for a more efficient synthesis of radiation patterns by leveraging the strengths of both techniques. Below is a detailed exploration of their interaction. Figure 28 illustrates the architecture of the Taylor-neural network. This figure presents a schematic representation of how the Taylor series expansion is integrated with the neural network model. It likely shows the layers, nodes, and activation functions used within the network, as well as how the Taylor series is applied to enhance the network's ability to model complex patterns or solve specific problems. This architecture may be used for applications such as pattern recognition, signal processing, or optimization tasks, where the combination of Taylor series and neural networks provides an efficient method for approximation and learning. The Taylor method is a widely used technique for antenna array synthesis, particularly in phased arrays, where it aims to control the radiation pattern  
Figure 27: Flowchart of the proposed hybrid synthesis process combining Taylor series initialization with neural network refinement. by optimizing the phase distribution across the antenna elements. The method involves using Taylor series expansion to define the phase distribution \(\Phi(\theta)\), which dictates the desired radiation pattern in a specific direction. The phase distribution for a uniformly spaced linear array with \(N\) elements and a desired radiation pattern can be expressed as follows:  
\[\Phi(\theta)=\sum_{k=0}^{M}a_{k}(\theta-\theta_{0})^{k} \tag{15}\]  
where  
* \(\theta\) is the angle of observation;
* \(a_{k}\) are the Taylor-series coefficients;
* \(M\) is the order of the Taylor expansion;
* \(\theta_{0}\) is the central angle where the main lobe is located. The Taylor method allows for controlling the main lobe width and sidelobe levels by adjusting the values of the coefficients \(a_{k}\). For instance, the coefficients can be adjusted to minimize sidelobes or create a specific shape for the main lobe, depending on the application. Although the Taylor method is effective in many cases, it involves limitations, especially when dealing with complex antenna array designs that involve multiple constraints such as non-uniform spacing, irregular element distributions, or specific performance goals. These constraints make it difficult to achieve optimal results using the Taylor method alone. Furthermore, the Taylor method does not account for nonlinearities in real-world systems or the interactions between different antenna elements, which could lead to suboptimal performance in practical applications. Neural networks can address these limitations by learning complex, nonlinear relationships between the input parameters (such as element positions, excitation amplitudes, etc.) and the desired output (such as radiation patterns). By using a training dataset of antenna configurations and their corresponding radiation patterns, the neural network can learn the optimal configuration for the antenna array. Figure 28: Taylor-neural network architecture. A feed-forward neural network for antenna array optimization can be formulated as follows:  
\[y=f(\mathbf{X};\mathbf{W},b) \tag{16}\]  
where  
* \(y\) is the output (the optimized radiation pattern or phase distribution);
* \(\mathbf{X}\) is the input vector containing parameters such as element positions, excitation amplitudes, and other design factors;
* \(\mathbf{W}\) represents the weights of the network, and
* \(b\) is the bias term. The network is trained to minimize a loss function that measures the difference between the desired radiation pattern and the output produced via the network. A feedforward, fully connected neural network was used, consisting of three hidden layers with 64, 32, and 16 neurons, respectively.