# Bond Strength Assessment of Concrete-Corroded
## 3 Application of Artificial Neural Network (ANN) Model
### Comparison between ANN and other Statistical Methods  
The commonly used regression-type methods are multiple linear regression (MLR), multiple non-linear regression (MNR) and ANN [44; 45]. For fair comparison, the proposed ANN model utilizes identical data for training and validation without testing as other statistical techniques do in the engineering field with relatively few data.  
To prove that ANN is the most appropriate method among them for the assessment of bond strength, three methods were used for calculation using the same database that has 242 series of data (\(n=242\)). The details of the database are shown in Appendix A. For all methods, the data of corrosion level and cover-to-diameter ratio were used as the inputs, which are generally considered the most important parameters affecting the bond strength degradation, while the relative bond strength is the output.  
For bond strength assessment, the MLR model considering the corrosion level and cover-to-diameter ratio is given as  
\[y=c_{1}x_{1}+c_{2}x_{2}+c_{3}, \tag{10}\]  
where \(c_{1}\), \(c_{2}\) and \(c_{3}\) are the regression coefficients; \(x_{1}\) is the corrosion level; and \(x_{2}\) is the cover-to-diameter ratio. In this study, based on the fitting, \(c_{1}=-0.05299\), \(c_{2}=0.05061\) and \(c_{3}=0.8371\).  
For the MNR model, scale factors are needed, and the best fitting equation is given as  
\[y=y_{\max}\bigg{(}c_{1}\bigg{(}\frac{x_{1}}{x_{1\max}}\bigg{)}^{c_{1}}+c_{2} \bigg{(}\frac{x_{2}}{x_{2\max}}\bigg{)}^{c_{2}}+c_{3}\bigg{)}, \tag{11}\]  
where \(c_{1}\), \(c_{2}\), \(c_{3}\), \(e_{1}\) and \(e_{2}\) are the regression coefficients; \(x_{1}\) is the corrosion level; \(x_{2}\) is the cover-to-diameter ratio; and \(x_{1\max}\), \(x_{2\max}\) and \(y_{\max}\) are the scale factors, which are the maximum values for the corrosion level, the cover-to-diameter ratio and bond strength, respectively. From the fitting, the regression coefficients in this study were \(c_{1}=-0.6714\), \(c_{2}=0.1956\), \(c_{3}=0.565\), \(e_{1}=0.5465\) and \(e_{2}=1.6756\).  
As shown in Figure 3, the model-predicted results were compared with experimental results, and only unconfined cases (without stirrups) were considered for calculation. Based on the results, the coefficients of determination (\(R^{2}\)) and root mean square error (RMSE) were calculated for comparison because they are two commonly used statistical parameters that show the accuracy of the model. On comparing \(R^{2}\), the results from the ANN method have the best fitting with the experimental results. While \(R^{2}\) is 0.455 for MLR and 0.535 for MNR, it is 0.667 for ANN, which is considerably higher.  
Further, in terms of RMSE, the results from the ANN method are the smallest, which indicates it has less error. Thus, the ANN method is worthy of further application to bond strength assessment.  
Since the ANN method achieves the best evaluation result and epochs are important for accuracy, as identified in previous studies, different epochs were considered for the calculation. Here, epochs in artificial neural network means the number of operations of the entire training dataset. To obtain best training results, we have to properly set the maximum number of epochs (not too many but also not too few). In this study, epochs in the range 100-100,000 were selected. As shown in Figure 4, the calculated results show that the agreement between the predicted and experimental results could be improved by increasing epochs; however, the difference is insignificant. With fewer epochs, the calculated results can be determined quickly, and it is very promising to adopt this approach in the future. Therefore, the default number of epochs was 1000.  
Figure 3: Comparison of the predicted results using: (**a**) multiple linear regression (MLR); (**b**) multiple non-linear regression (MNR); and (**c**) artificial neutral network (ANN).