## 3 Modeling Method
### Artificial Neural Networks
#### 3.1.2 Construction of Neural Network  
In this study, the collected experimental data of the moment redistribution coefficient were randomly divided into a training set and a testing set. Since the purposes in different experimental studies are diverse and the test programs are time-consuming and costly, datasets with complete variable information collected from the published literature are relatively few. To improve the applicability and generalization of the training model, 85% of the data were selected as training data, which were used to fit the parameters of the neural network, such as the weights and the bias in each layer. The remaining 15% of the data were used as testing data, which were independent of the training process and used to test the accuracy and applicability of the trained model. The performance of the neural network models can be evaluated by two indexes: mean squared error (MSE) and coefficient of correlation (\(R^{2}\)), as given by Equations (3) and (4):  
\[\text{MSE}=\frac{1}{n}\sum_{i=1}^{n}\left(t_{i}-c_{i}\right)^{2} \tag{3}\]  
\[R^{2}=1-\frac{\sum\limits_{i=1}^{n}\left(t_{i}-c_{i}\right)^{2}}{\sum\limits_{ i=1}^{n}\left(t_{i}-\overline{t}\right)^{2}} \tag{4}\]  
where \(n\) is the total number of the data, \(c_{i}\) and \(t_{i}\) are the predicted value and the target value of the \(i\)th data, and \(\overline{t}\) is the average of the target values. A prediction model is considered to be better fitting when MSE and \(R^{2}\) are closer to 0 and 1, respectively. Eight nodes were used in the input layer because there were eight influential parameters for the moment redistribution studied. Before training the input data, normalizations for the input and target data should be performed since the sigmoid activation function is sensitive to the variations between 0 and 1. In this way, the stability and convergence rate of the training process can be improved. A linear relationship was used to scale the data from 0.1 to 0.9, given by Equation (5):  
\[x_{i,\text{scaled}}=\left(0.9-0.1\right)\times\frac{x_{i}-x_{\text{min}}}{x_{ \text{max}}-x_{\text{min}}}+0.1 \tag{5}\]  
where \(x_{i,\text{scaled}}\) is the normalized value, \(x_{i}\) is the value of the variable, and \(x_{\text{min}}\) and \(x_{\text{max}}\) are the minimum and maximum values of the variable, respectively. The function of the hidden layer is to determine the inherent relationship between the input and output layers based on the training of the data. One hidden layer was used in the present neural network. The number of the nodes in the hidden layer is of great importance to the design of the neural network. An excessive number of the nodes results in an extension of the training time and the over-fitting of the data, which decreases the computational efficiency. However, if the number of the nodes is not enough, insufficient information leads to a decrease in the computational accuracy. Currently, there are no general rules to exactly determine the number of nodes. It is often based on experience, and the optimum result is obtained through trial and error. In this study, the number of the nodes in the hidden layer was selected from 6 to 15. Figures 8 and 9 show the regression values and MSE of the neural network with different numbers of nodes. It can be seen that the model is optimal when 13 nodes were used in the hidden layer. Hence, the final network configuration used in present study is illustrated in Figure 10 and the values of the parameters for the neural network were determined as follows:  
Number of input layer units = 8 Number of hidden layer units = 13 Number of output layer units = 1 Momentum rate = 0.9 Learning rate = 0.4  
Error after learning = 8 \(\times\) 10\({}^{-8}\)  
Learning cycle = 20,000  
The weight matrix from input layer to hidden layer:  
\[\left[