# Pore Water Pressure Prediction Based on Machine Learning Methods--Application to an Earth Dam Case
## 4 Results and Discussion  
### Model Performance and Comparison  
The results of ML models for C2 are presented in Figure 7. They show the powerful prediction capability of the four methods. This may be due to a number of factors. Firstly, the model input variables (hydrostatic, season, time, etc.) explain a large portion of the variations in PWP, for example, higher PWP for a full reservoir (and lower for a low reservoir level), higher PWP in winter and lower PWP in summer. In addition, several values of the hyper-parameters were tested before arriving at a satisfactory model. These ML models can be useful in detecting pathology or abnormal behavior of the dam, such as an increase in PWP, that is not explained by a reversible variation in the level of the reservoir or by the season. It should be remembered that the case study relates to the prediction of time series. The dataset has been divided randomly into training data sets (352), validation datasets (44), and test datasets (44). Table 2 shows the optimized hyperparameters applied in the RF-SA model. Table 3 presents the prediction performance and efficiency of the ML models on the training and test datasets. Table 4 shows the performance of the prediction models on the test datasets. For the static PWP prediction, there are two types of models: one is established by the MLP method; the other is obtained by random forest combined with simulated annealing (RF-SA). Hyperparameters are set before the learning process begins. For the MLP model, the optimized hyperparameters (learning rate = 0.01, decay = 1 \(\times\) 10\({}^{-6}\), and momentum = 0.9) used a test size equal to 30% of the dataset. Both the testing and training R\({}^{2}\) values reached 0.98 (as seen in Figure 6(a)), signifying the robust prediction capability of the MLP model. For the RF-SA model, the R\({}^{2}\) reaches 0.98, as shown in Figure 7b, which is similar to the performance of the MLP model. The single RF training time is so rapid that it only takes several seconds, and commonly, single training results do not seem to be very accurate (R\({}^{2}\) = 0.92). After optimizing the hyperparameters, the model performance improved. All of the implications and optimal values of the hyperparameters by applying SA in the RF models are shown in Table 2. The optimized model is a distribution of 14 decision trees, and every tree would re-split less than 18 times. The SA processing increases the accuracy (R\({}^{2}\) increases from 0.92 to 0.98), while on average, the efficiency of RF-SA is a little lower than MLP by contrast to the CPU training time shown in Table 3. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Hyperparameters** & **Meanings** & **Value** \\ \hline n\_estimators & The number of trees in the forest. & 14 \\ max\_depth & The maximum depth of the tree. & 18 \\ min\_samples\_leaf & The minimum number of samples & 1 \\  & required to be at a leaf node. & \\  & The number of random seeds to ensure & \\ random\_state & the same results when inputting the & 11 \\  & same number. & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Optimized hyperparameters applied in RF-SA model of C2 piezometer after optimization. Figure 7: The performance of PWP (C2) model (both training and testing sets) and the R\({}^{2}\) of each model: (**a**) MLP (R\({}^{2}\) = 0.9935); (**b**) RF-SA (R\({}^{2}\) = 0.9878); (**c**) GRU (R\({}^{2}\) = 0.9639); (**d**) RNNs (R\({}^{2}\) = 0.9622). The PWP prediction models are also built by the classic RNNs and GRU methods. The training results show that the accuracy of RNNs and GRU models could achieve 0.96 and 0.97, respectively. Comparing all ML models, the MLP model can predict PWP most accurately. The reason why deep learning methods perform inadequately is that the database consists of 440 groups, which is not enough for training a deep learning model. It is evident that the larger the number of databases, the more accurate the deep learning model, which has powerful robustness for both interpolation and exploitation.