## 3 Artificial Neural Network (ANN) Model for Structural Characterization
### Mathematical Approach for Artificial Neural Network (ANN) Model
The addition of the inputs and their corresponding weights results to a summation function, as shown in Equation (1):  
\[Q_{j}=\sum_{i=1}^{n}(x_{i})(w_{ij})+(b_{j}), \tag{1}\]  
Equation (1): ANN architecture/model with inputs, weights and bias,  
where,  
\(x_{i}\) (\(i=1,\,2,\,3,\,\ldots\,\ldots\,\ldots\,\), \(n\)) = input neurons;  
\(w_{ij}\) (\(i=1,\,2,\,3,\,\ldots\,\ldots\,\ldots\,\), \(n\); \(j=1,\,2,\,3,\,\ldots\,\ldots\,\ldots\,\), \(m\)) = respective weights for \"\(n\)\" input and \"\(m\)\" hidden neurons;  
\(Q_{j}\) = the net input = addition of inputs and their corresponding weights = the summation of the weighted input for the \(j\)th neuron;  
\(w_{ij}\) = weight from the \(i\)th input neutron to the \(j\)th neuron in the hidden layer;  
\(b_{j}\) = threshold (also called as bias) associated with node \(j\). The net input (\(Q_{j}\)) is then passed through an activation function \"\(f(\,)\)\", and the output \"\((Y)\)\" of the node is computed as:  
\[Y=w_{p0}+\sum_{j=1}^{m}\big{(}w_{pj}\big{)}\Big{(}f\Big{(}\sum_{i=1}^{n}(x_{i} )(w_{ij})+(b_{j})\Big{)}\Big{)}, \tag{2}\]  
Equation (2): Outputs of the ANN model,  
where,  
\(p\) = the number of outputs (\(p\) = 0 if it is one);  
\(f(Q_{j})\) = the activation function. The typically used activation functions in the ANN model are the Rectified Linear Units (ReLU), sigmoid, Hyperbolic tangent (tanh), etc. The ReLU function, as shown below in Equation (3), is the most common activation function. 'ReLU' is the abbreviation for the Rectified Linear Unit activation function [42]. It is frequently implemented in neural networks to accelerate the training and mitigate the disappearing gradient problem [42]. The value is returned by the \"ReLU\" activation function if the input is positive; otherwise, it is set to zero [42]. A sigmoid function is defined as a mathematical function with an \"S\" shaped curve, and it represents the logistic function, as shown below. The Sigmoid function, unlike the stepwise function, includes non-linearity in the ANN model. This implies that the outputs received from the neurons do not represent a linear combination of inputs and weights. This will support the nonlinear activation function for classifying and assisting the nonlinear decision boundaries in the data. \[f\big{(}Q_{j}\big{)}=\frac{1}{\Big{(}1+e^{(-Q_{j})}\big{)}}, \tag{3}\]  
Equation (3): ANN model with nonlinear weights. Equation (3) shows that the complexity of the ANN model is dependent on the number of hidden layers and the learning accuracy of the ANN model. The accuracy is accomplished by increasing the number of hidden layers in ANN that causes a proportionate number of needed training repetitions. Figure 8 shows the typical process of ANN model training. In the typical process of ANN model training, the pass/fail criterion is defined as the threshold, and then the ANN is initialized. After that step is completed, the forward propagation is executed to calculate the error. The error calculation in this context is also known as a backward propagation. After that step in the ANN training process, ANN weights are updated to evaluate the ANN accuracy threshold. Once the ANN accuracy threshold is achieved, then the ANN model training process is completed. If the ANN accuracy threshold is not obtained, the forward propagation step is executed, and the whole step is repeated until the ANN model training is completed.