## 3 Solution Procedure
### Artificial Neural Network (ANN) Method  
An artificial neural network (ANN) is a computational model that is inspired by the structure and function of biological neural networks in the human brain. It comprises interconnected nodes, known as neurons, organized systematically into layers. These neurons work in harmony to process and transform input data into meaningful output. The configuration of an ANN, in terms of the number of neurons within each layer (i.e., height) and the number of layers (i.e., depth), is a critical factor that influences the network's capacity to perform a given task. In an ANN, data are typically fed into the input layer, and information flows through the network's hidden layers to the output layer. Each connection between neurons is associated with a weight \(w\), which influences the strength of the signal transmitted between neurons. These weights are adjusted during the training process where the network learns to map input data to the desired output. In the present research, it is a supervised learning approach, since we have access to the corresponding outputs for the inputs. Specifically, all neurons in an ANN receive one or more input values from the previous layer, which are each associated with weights, denoted as \(\{x_{1}\), \(x_{2}\), \(\ldots\), \(x_{n}\}\) and \(\{w_{1}\), \(w_{2}\), \(\ldots\), \(w_{n}\}\), respectively. Then, a neuron performs \(g(\sum_{i=1}^{n}w_{i}x_{i}+b)\) with bias \(b\) and activation function \(g(\cdot)\). Here, the \(g(\cdot)\) function introduces nonlinearity into the model, for example, widely used activation functions are sigmoid, tanh, ReLU, and leaky ReLU. In this study, we aimed to predict \(\overline{\omega}\) (i.e., dimensionless frequency) from 4 inputs, which are the beam theory, beam type, \(L/h\), and \(p\), as shown in Figure 2. Since the first three variables represent categorical values, for instance, \(L/h\) is drawn from a finite set of [5; 10; 20; 30; 100], we are encouraged to use a categorization technique. One commonly used approach is one-hot encoding, which transforms a categorical input denoted as \(x_{cat}\), with \(d\) distinct categories, into a one-hot encoded vector \(x_{cat}^{obs}\in R^{\text{a.t.}}\). In this encoding, the corresponding category is assigned the value 1, while all other categories are assigned the value 0 in \(x_{cat}^{obs}\). Moreover, the final variable, denoted as \(p\), represents a numerical value of \(x_{num}\). Consequently, we use a normalization procedure with the objective of centering its mean at 0 and standardizing its deviation to 1, shown as \(x_{num}^{norm}\). This normalization step's purpose is to enhance training stability and accelerate the convergence of neural networks towards optimal solutions. Prior to training, the dataset is partitioned into two distinct subsets: the training dataset and the test dataset. Given the inherent susceptibility of datasets in this domain to overfit ANN models, we use dropout [23] with a rate of \(\{0.1\}\). The dropout technique is commonly used for its efficacy in addressing overfitting by randomly dropping out a fraction of neurons during each training iteration. Figure 2: Input pre-processing for the ANN model. During the training process, the model is trained on the training dataset by optimizing the model weights \(\mathbf{W}\), using a loss function \(f(\mathbf{W})\) that is the expected loss over the training dataset:  
\[f(\mathbf{W})=\mathbb{E}_{\xi}\Big{[}\widehat{f}(\mathbf{W};\xi)\Big{]} \tag{32}\]  
where \(\xi\) is a random data sample from the training dataset and \(\widehat{f}(\mathbf{W};\xi)\) is the loss function for this sample.