## 4 Results
### Artificial Neural Network (ANN)  
In this study, all artificial neural network (ANN) modeling was performed using MATLAB (R2020a: The Math Works Inc., Natick, MA, USA). Table 6 displays the results of the Bayesian regularization (BR) and Levenberg-Marquardt (LM) algorithms for all networks. The table contains the \(R^{2}\) and mean absolute error (MAE) values for both test and training datasets. The results are based on the best-performing network among all 45 neurons and three training repetitions, as determined by the highest \(R^{2}\) and lowest MAE values. The findings indicate, that the network with two hidden layers achieved the highest performance for both the BR and LM algorithms. Moreover, the average accuracy of the BR algorithm was superior to that of the LM algorithm. Figure 8 depicts the comparison between the predicted values of the artificial neural network (ANN) model and the actual values of laboratory experiments. The results indicate that the ANN model exhibits a very high level of precision in predicting the California bearing ratio (CBR) values for both the training and testing databases. Table 7 displays the best outcomes of utilizing the ANN modeling method to predict the California bearing ratio (CBR) for the mixtures of album sludge and soil. The performance indicators of the ANN model are exhibited for both the training and testing datasets. The ANN model yielded a mean absolute error (MAE) of 0.392 and 0.303, for the training and testing datasets, respectively; the mean squared error (MSE) values for the training and testing datasets were 0.200 and 0.116, respectively; the root mean squared error (RMSE) values obtained for the training and testing datasets were 0.447 and 0.341, respectively; the mean squared log error (MSLE) values were 0.005 and 0.002, for the training and testing datasets, respectively; while the root mean squared log error (RMSLE) values were 0.074 and 0.046, for the training and testing datasets, respectively. The ANN model yielded  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Number of Hidden Layers**} & **R-Test** & **R-Train** & **MAE-Test** & **MAE-Train** \\ \hline \multirow{6}{*}{
\begin{tabular}{c} **Neural** \\a high coefficient of determination (\(R^{2}\)), of 0.989 and 0.980, for the training and testing datasets, respectively, implying that the model is highly precise in predicting the CBR values for mixtures of alum sludge and soil. Figure 8: The results of the best ANN model predicting the CBR values for the (**a**) training and (**b**) testing databases. One of the key determinants of the precision and complexity of an artificial intelligence (AI) model, is the number of neurons present in each hidden layer. To find the optimal number of neurons, a group of analyses is conducted. Figure 9a shows the accuracy of the ANN model, based on the number of neurons present in each hidden layer. According to the results, the accuracy of the ANN model does not improve significantly beyond a certain number of neurons, which in this case is 10 neurons. Therefore, it is essential to determine the optimal number of neurons, to achieve the best accuracy while minimizing the complexity of the model. Figure 9b displays the model's error rate for each neuron. The results show that the error rate remains almost constant after the 10 th neuron. Thus, the optimal number of neurons for the ANN model was determined to be 10. This information is crucial for researchers and developers of AI models, as it allows them to minimize the complexity while achieving the best accuracy. By optimizing the number of neurons, the performance of the AI model can be significantly enhanced. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Performance Metrics** & **Training Database** & **Testing Database** \\ \hline MAE & 0.392 & 0.302 \\ MSE & 0.200 & 0.116 \\ RMSE & 0.447 & 0.341 \\ MSLE & 0.005 & 0.002 \\ RMSLE & 0.074 & 0.046 \\ R\({}^{2}\) & 0.989 & 0.980 \\ \hline \hline \end{tabular}
\end{table}