# Phased Antenna-Array Synthesis Using Taylor-Series Expansion and Neural Networks
## 5 Neural Networks for Synthesis and Optimization of Antenna Arrays
### Benefits of the Taylor-Neural Network Architecture
The BFGS Quasi-Newton (BFG) algorithm also offers fast convergence, while resilient backpropagation (RP) is robust to noise, performing well even with inconsistent data. Bayesian regularization (BR) is particularly effective for small datasets, preventing overfitting. The Scaled Conjugate  
Gradient (SCG) algorithm is memory-efficient, making it suitable for large-scale problems. Conjugate Gradient with Powell/Beale Restarts (CGB) offers balanced performance, providing a trade-off between speed and accuracy, while Fletcher-Powell Conjugate Gradient (CGF) ensures stable convergence. Polak-Ribiere Conjugate Gradient (CGP) excels with sparse data, handling many zero values efficiently. The One-Step Secant (OSS) algorithm is known for its fast convergence, while Variable Learning Rate Backpropagation (GDX) adapts its learning rate during training for improved performance. Basic gradient descent (GD) is simple and easy to implement but may not converge as efficiently, whereas gradient descent with momentum (GDM) accelerates convergence by incorporating momentum, helping avoid local minima. Each algorithm offers specific advantages, and the choice depends on factors like data size, speed, and memory constraints. Figure 29 shows the progress of the [PERSON] neural network training at epoch 74. At this stage, the performance goal has been achieved, meaning the model has converged to an optimal solution according to the specified performance criteria. This suggests that the training process has effectively adjusted the model's parameters to meet the desired objectives, with a significant reduction in error over the iterations. This figure could also represent the convergence curve, showing the decrease in the cost function or error during the training iterations. Figure 30 illustrates the evolution of the mean squared error (MSE) for the [PERSON] neural network. MSE is a key indicator of model quality, measuring the difference between predicted values and actual values. A low MSE indicates good performance, as it shows that the model's predictions are close to the expected results. This figure likely shows the progressive reduction in the MSE as the neural network learns and adjusts its parameters. A steadily decreasing MSE curve would suggest that the network is effectively converging to a solution with minimal prediction error. Figure 31 presents the correlation coefficient (\(R\)) achieved with the Taylor neural network, which is 0.99999. Such a high correlation coefficient indicates an extremely strong relationship between the predicted values and the actual values. In other words, the model's predictions are almost identical to the expected results. This level of performance is exceptional and suggests that the model is very well fitted to the training data. A correlation of 0.99999 indicates that the model has learned with great accuracy, which is a strong indicator of the quality of the training and the model's ability to generalize to new data. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Abbreviation** & **Algorithm** & **Performance** \\ \hline LM & Levenberg-Marquardt & Fast convergence \\ BFG & BFGS Quasi-Newton & Quick convergence \\ RP & Resilient Backpropagation & Noise resistant \\ BR & Bayesian Regularization & Small data effective \\ SCG & Scaled Conjugate Gradient & Memory efficient \\ CGB & Powell/Beale Restarts & Balanced performance \\ CGF & Fletcher-Powell Conjugate & Reliable convergence \\ CGP & Polak-Ribiere Conjugate Gradient & Sparse data-efficient \\ OSS & One-Step Secant & Fast convergence \\
**GDX** & **Variable Learning Rate** & **Adaptive rate** \\ GD & Gradient Descent & Simple implementation \\ GDM & Momentum Gradient Descent & Faster convergence \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of neural network-training algorithms. Figure 32 illustrates the SOM (self-organizing map) neighbor weight distances. The SOM is a type of unsupervised neural network used for clustering and dimensionality reduction.