## 3 Solution Procedure
### Artificial Neural Network (ANN) Method
To calculate the loss function, we use the following mean square error (MSE):  
\[f(\mathbf{W})=\frac{1}{|X|}{\sum_{i=1}^{|X|}}{(\overline{\omega}_{i}-\theta(X _{i};\mathbf{W}))}^{2} \tag{33}\]  
where \(X\) is the training dataset, \(\overline{\omega}_{i}\) is the ground truth value for the input \(X_{i}\), and \(\theta(X_{i};\mathbf{W})\) denotes the predicted output of the ANN model \(\theta(\cdot)\), given weights \(W\) and the input \(X_{i}\). The optimal solution of the problem is the following:  
\[\theta(\mathbf{W}^{*})=\text{arg}_{\mathbf{W}}\text{min}f(\mathbf{W}) \tag{34}\]  
To reach this optimal solution, we optimize our model at each iteration using the following stochastic gradient descent with momentum (SGDM):  
\[\theta(\mathbf{W})=\mathbf{W}-\alpha v_{t+1} \tag{35}\]  
\[v_{t+1}=\beta v_{t}+(1-\beta)\
abla\mathbf{w}f(\mathbf{W}) \tag{36}\]  
where \(v_{0}=0\), \(\alpha\) is the learning rate, and \(\beta\) is the momentum weight. However, the performance of an ANN model is significantly influenced by the selection of the activation function, the number of hidden layers, their respective layer sizes, the learning rate, and the momentum weight. In order to optimize these architectural aspects, we use a grid search-based approach, systematically exploring various combinations of these hyperparameters. Our results indicate that using ReLU as the activation function, with a network depth of 3 and 35 neurons for each hidden layer, \(\alpha\) as 0.01, and \(\beta\) as 0.9, yields the best performance on our dataset. For a visual representation of the final architecture of our ANN model, please refer to Figure 3.  
We implemented the ANN model using PyTorch [24]. Moreover, we designed an interface for the purpose of input provision and prediction retrieval, as illustrated in Figure 4. Then, the model was trained by the selected hyperparameters for a maximum of 300 epochs on the dataset collected from [5]. The resulting training and test losses can be seen in Figure 5. After 300 epochs, the model achieved an MSE of \"0.00047\" in the training dataset. Furthermore, Table 1 shows the actual and predicted frequencies for the given inputs. Through the results, the ANN model demonstrated that it predicts accurately. Figure 2:0 ANN model architecture. The set off variables from \(x_{1}\) to \(x_{22}\) represent the input vector. Figure 2:1 Interface for frequency prediction of ANN model. Figure 2:2 Training and evaluation losses through training epochs. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **Inputs** & & & **Ref. [5]** & **Present** & & & **Error** \\ \hline
**Beam Type** & **Beam Theory** & _L/h_ & \(p\) & & & & **ANN Prediction** & **(\%)** \\ \hline \multirow{8}{*}{S-S} & \multirow{8}{*}{HSDT} & & 0 & 5.1527 & 5.1527 & 5.1404 & 0.24 \\  & & & 0.5 & 4.4107 & 4.4107 & 4.3784 & 0.73 \\  & & & 1 & 3.9904 & 3.9903 & 3.9163 & 1.86 \\  & & & 2 & 3.6265 & 3.6264 & 3.6419 & 0.42 \\  & & & 5 & 3.4014 & 3.4014 & 3.4209 & 0.57 \\  & & & 10 & 3.2817 & 3.2817 & 3.2072 & 2.27 \\  & & & 0 & 5.4603 & 5.4603 & 0.18 \\  & & & 0.5 & 4.6511 & 4.6511 & 4.5956 & 1.19 \\  & & & 1 & 4.2051 & 4.2049 & 4.1425 & 1.49 \\  & & & 2 & 3.8361 & 3.8361 & 3.7991 & 0.96 \\  & & & 5 & 3.6485 & 3.6485 & 3.6715 & 0.63 \\  & & & 10 & 3.539 & 3.5390 & 3.4599 & 2.24 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dimensionless frequencies (\(\varpi\)) of FG beams for the first mode.