# Artificial Neural Network-Based Model for Prediction of Frost Heave Behavior of Silty Soil Specimen
## 4 Prediction of Frost Heave Ratio Using the Artificial Neural Network Model  
### Establishment of an Artificial Neural Network  
In this study, an ANN for the estimation of frost heave ratio was designed with three layers: an input layer, a hidden layer, and an output layer. The input layer stores and provides data to the ANN network, whereas the hidden layer, which is constructed with general neurons, connects the input layer to the output layer. As shown in Figure 7, the developed ANN model has a 2-5-1 structure: with two neurons in the input layer, five neurons in the hidden layer, and one neuron in the output layer. Each neuron has an input parameter that is the weighted sum of the output from every neuron in the previous layer. This sum is passed through a transfer function to provide an outgoing signal to the next layer. Finally, the output layer stores the value predicted by the network. For each neuron, the total input value can be obtained as follows. \[z=Wx+b \tag{22}\]  
where \(W\) is the weight matrix, which stores the weights of every connection between the current and the preceding layer. A vector \(x\) contains all output signal values from the previous layer, whereas the vector \(b\) comprises the bias value at the current layer. The input value is transformed within neurons via a transfer function. Thus, Equation (22) can rewrite as follows. \[y=f(Wx+b) \tag{23}\]  
where \(f\) is the transfer function, which usually adds non-linearity to the network to try to fit the network. Therefore, the network is able to produce an output that fits within \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & \(\lambda_{s}\) & \(k_{0}\) & \(\zeta\) \\ \hline \multirow{2}{*}{\(\lambda_{s}\)} & Correlation coefficient & 1 & 0.114 & \(-\)0.330 ** \\  & \(p\)-value (two sided) & & \(>\)0.1 & 0.000 \\ \hline \multirow{2}{*}{\(k_{0}\)} & Correlation coefficient & 0.114 & 1 & 0.805 ** \\  & \(p\)-value (two sided) & \(>\)0.1 & & 0.000 \\ \hline \multirow{2}{*}{\(\zeta\)} & Correlation coefficient & \(-\)0.330 ** & 0.805 ** & 1 \\  & \(p\)-value (two sided) & 0.000 & 0.000 & \\ \hline \hline \end{tabular} \({}^{*}\)Correlation coefficient is significant at two sides (\(p<0.05\)). ** Correlation coefficient is significant at two sides (\(p<0.01\)).  
\end{table} 
Table 2: Pearson correlation analysis between the three variables. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **B** & **Standard Error** & **t** & \(p\)**-Value** & **VIF** \\ \hline Constant & 6.958 & 0.104 & 66.655 & & \\ Hydraulic conductivity & 7.68 \(\times\) 10\({}^{9}\) & 2.41 \(\times\) 10\({}^{8}\) & 31.963 & \(<\)0.01 & 1.013 \\ Thermal conductivity & \(-\)0.372 & 0.023 & \(-\)16.011 & \(<\)0.01 & 1.013 \\ R\({}^{2}\) & 0.829 & & & & \\ adjR\({}^{2}\) & 0.827 & & & & \\ \hline \hline \end{tabular} \({}^{*}\)In non-standardized coefficient, t B/standard error, VIF: variance inflation.  
\end{table} 
Table 3: Regression analysis between three variables. the proper value. Without transfer functions, the network could only be able to provide a linear output when compared with its input signal. To guarantee the performance of the network, the Nguyen-Widrow method [50] was adopted to produce the initial weight and bias. In this study, the back-propagation technique was applied to the training procedure. According to this method, the procedure includes two phases. First, the feed-forward phase involves the passing of all data from the input layer to the output layer according to the weighted sum of the output from every connected neuron in the preceding layer. A transfer function is applied to estimate the output value within neurons. Thus, the predicted output is estimated at the output layer. The difference between the predicted value and the expected value is obtained by a cost function. In this study, the quadratic cost function was applied as follows. \[C=\sum(y_{i}-\exp_{i})^{2} \tag{24}\]  
where \(y_{i}\) is the predicted value obtained by ANN while \(\exp_{i}\) is the expected value from the dataset.