# Application of ANN in Predicting the Cantilever Wall Deflection in Undrained Clay
## 3 Results and Discussions  
### Performance of the Proposed ANN Model  
Figure 8 shows the performance of the proposed ANN model. As can be seen in the figure, the MSE for training, testing, and validation decreases rapidly as the number of epochs increases. The minimum MSE is reached at the ninth epoch with a value of 4.6964 \(\times\) 10\({}^{-4}\), indicating that the performance of the proposed ANN model is good.  
Figure 9a compares the cantilever wall deflection derived from the proposed ANN model with the actual numerical data in the training process. In general, there is a fairly good agreement between the two datasets. Figure 9b shows the evolution of the error with the change in the number of samples. The histogram of the error values with the mean and standard deviation (StD) during the training is also shown in Figure 9c. It can be seen that most of the errors during the training process are close to 0. A similar phenomenon was also observed when comparing the results of the ANN model and the corresponding numerical data for the testing and validation, as well as for all data (see Figures 10-12). The above results confirm the effectiveness of the developed ANN model and show that it is reliable in predicting the cantilever wall deflection in undrained clay.  
Furthermore, the actual values of the cantilever wall deflection \(\delta\) were compared with the values predicted by the ANN model using the regression analysis, as shown in Figure 13. It can be observed that the values of \(R^{2}\) for the training, testing, validation, and all data are 0.9823, 0.9135, 0.9915, and 0.9811, respectively. They are all quite close to 1, which again indicates the acceptable performance of the developed ANN model.  
Figure 8: Performance of the developed ANN model.  
Figure 9: The performance of training data: (**a**) Predicted versus actual data; (**b**) error; (**c**) the histogram of error values.  
Figure 10: The performance of testing data: (**a**) Predicted versus actual data; (**b**) error; (**c**) the histogram of error values.  
Figure 11: The performance of validation data: (**a**) Predicted versus actual data; (**b**) error; (**c**) the histogram of error values.  
Figure 12: The performance of all data: (**a**) Predicted versus actual data; (**b**) error; (**c**) the histogram of error values.  
Two commonly used performance indicators, root mean square error (RMSE) and mean absolute error (MAE), were selected to evaluate the network performance. The expression for each indicator is given in Equations (5) and (6). The value of the MAE represents the absolute differences between the predicted and actual values, while the RMSE represents the existing deviation between the predicted and actual values. The results of two indicators for training, testing, validation, and all data are shown in Figure 14. This again demonstrates that the predicted cantilever wall deflection using the developed ANN model were quite similar to the actual values. These results confirm that the developed ANN model performs well and could be successfully used to coherently predict the cantilever wall deflection in undrained clay. Equations (5) and (6) are calculated as follows:  
\[\text{RMSE}= \sqrt{\frac{1}{N}\sum_{i=1}^{N}(e_{i})^{2}} \tag{5}\] \[\text{MAE}= \frac{\sum_{i=1}^{N}\lvert e_{i}\mid}{N} \tag{6}\]  
Figure 13: Regression of the developed ANN model: (**a**) training data; (**b**) testing data; (**c**) validation data; (**d**) and all data.  
where \(N\) represents the input datasets number; \(e_{i}\) is the difference between the predicted and actual results.