# Bond Strength Assessment of Concrete-Corroded
## 3 Application of Artificial Neural Network (ANN) Model  
### ANN Model  
Among soft computing techniques, ANN is usually the best option to study regression-type problems related to various parameters [40]. It is considered as a regression technique that captures higher nonlinearity between independent and dependent variables. In this study, an ANN structure which consists of a feed-forward multilayer perceptron, an error back-propagation technique of gradient descent and momentum and scaled data were utilized. The schematic of the feed-forward multilayer perception model is shown in Figure 2. In the model, the input and hidden layer may have a bias node and there can be one or more hidden layer(s). The calculation procedure for the link weight value can be seen in our previous studies [43; 44]. In this study, the input layer contains independent variables (corrosion level, crack width, etc.) and one bias node (used as the intercept of each perceptron). The number of hidden layers is one. The node number of hidden layers is equal to the node number of input layer (including one bias node). The node value in the hidden layer can be calculated as follows  
\[v_{j}=\left(1+\exp\left(-1\times\sum_{i=1}^{I}x_{i}w_{ij}\right)\right)^{-1},j =1,\ldots,J \tag{1}\]  
where \(v_{j}\) is node \(j\) in the hidden layer; \(x_{i}\) is node \(i\) in the input layer; and \(w_{ij}\) is the weight between nodes in the input and hidden layer. For the output layer, the dependent variable is bond strength. The node value in the output layer can be calculated as below  
\[y=\left(1+\exp\left(-1\times\sum_{j=1}^{J}v_{j}w_{j1}\right)\right)^{-1} \tag{2}\]  
where \(y\) is the node in the output layer. Error \(E\) between the experimental and calculated data can be considered as below  
\[E=0.5(d-y)^{2} \tag{3}\]  
where \(d\) is the experimental data. Error signals from the output layer and the hidden layer can be calculated by Equations (4) and (5), respectively:  
\[\delta_{y}=(d-y)y(1-y) \tag{4}\]  
\[\delta_{p}=v_{j}\left(1-v_{j}\right)\delta_{y}w_{j1},\ \ j=1,\ldots,J \tag{5}\]  
Figure 2: Schematic of feed-forward multilayer perception model.  
Weights between hidden and output layers and input and hidden layer can be adjusted using the following amounts, respectively:  
\[\Delta w_{j1}=\alpha\delta_{y}v_{j},\;\;j=1,\ldots,J \tag{6}\]  
\[\Delta w_{ij}=\alpha\delta_{p}x_{i},\;\;i=1,\ldots,I;\;j=1,\ldots,J \tag{7}\]  
where \(\alpha\) is learning rate, which was 1.0 in this study. Furthermore, to quicken the convergence to the optimal solutions, momentum can be considered as follows:  
\[\Delta w_{j1}^{n}=\alpha\delta_{y}v_{j}+\beta\Delta w_{j1}^{n-1},\;\;j=1, \ldots,J \tag{8}\]  
\[\Delta w_{ij}^{n}=\alpha\delta_{p}x_{i}+\beta\Delta w_{ij}^{n-1},\;\;i=1, \ldots,I;\;j=1,\ldots,J \tag{9}\]  
where \(n\) is iteration number of error back-propagation and \(\beta\) is momentum constant (the value is 0.8 in this study). This momentum technique accelerates the training process (weight adjustment) in flat regions of the error surface and prevents fluctuations in the weights. The values of learning rate and momentum constant were chosen based on typically used ones [50].