## 4 Results and Discussion  
Table 5 and Figure 4 demonstrate that the average absolute relative error (AARE) in the train and test sets is 8.48% and 8.64%, respectively. To evaluate the impact of the settling area ratio, we compared the prediction accuracy of the ANN model with and without the input of the settling area ratio. Our findings show that the prediction accuracy could be enhanced by including the settling area ratio as an input, which validates that settling orientation is a critical factor affecting particle settling velocity. The weights and biases used in the ANN model are provided in Appendix A. Table 6 and Figure 5 illustrate the accuracy of the proposed ANN model in comparison to other empirical correlations [16; 29; 42; 43]. Although the model proposed by [PERSON] et al. [29] provides precise predictions in Newtonian fluids, it is unsuitable for power-law fluids. In contrast, the ANN model can accurately predict the settling velocity of both spherical and non-spherical particles in both Newtonian and power-law fluids. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{2}{c}{**AARE of Train Set**} & \multicolumn{2}{c}{**AARE of Test Set**} \\ \cline{2-5}  & **Newtonian** & **Power-Law** & **Newtonian** & **Power-Law** \\ \hline [PERSON] and [PERSON] [42] & 15.47\% & / & 15.52\% & / \\ [PERSON] and [PERSON] [43] & / & 89.14\% & / & 87.53\% \\ [PERSON] et al. [29] & 3.96\% & / & 3.89\% & / \\ [PERSON] et al. [16] & / & 14.03\% & / & 13.85\% \\ ANN model in this study & 9.69\% & 7.63\% & 8.41\% & 8.79\% \\ \hline \hline \end{tabular} Note: ‘/’ in Table 6 represents that this model is unsuitable for this fluid type.  
\end{table} 
Table 6: Comparison of ANN model with empirical correlations. Figure 4: Comparison of predicted settling velocity by ANN model and measured settling velocity in train and test sets ((a) ANN model with input \(S\); (b) ANN model without input \(S\)). The inclusion of Newtonian fluid and spherical particle data slightly reduced the ANN model's precision compared to high-accuracy models specific to these conditions. However, it significantly enhanced the model's accuracy for non-Newtonian fluids. For instance, the prediction error for non-Newtonian fluids was markedly lower than existing models. In conclusion, while the dataset imbalance slightly affected the model's performance in predicting Newtonian fluid and spherical particle settling velocities, it considerably improved its applicability and accuracy for non-spherical particles in non-Newtonian fluids. As for the sensitivity of this imbalance, it could be found that an imbalance in the dataset can influence a model's performance and generalizability. These analyses involved testing the model's performance with varied proportions of spherical and non-spherical particles, as well as Newtonian and non-Newtonian fluids. The results indicated that while there is some sensitivity to these variations, the overall predictive accuracy of the model remained robust across different scenarios. This suggests that the proposed model can generalize well despite the uneven distribution of data points. Figure 5: Comparison of settling velocity predictions by different models with measured values (**a**): [PERSON] (2019), (**b**): [PERSON] et al. (2019), (**c**): [PERSON] et al. (2016), (**d**): ANN model. To compare with other advanced algorithms, K-fold cross-validation (K = 10) is used to determine the most suitable model for this study. The evaluation focused on the coefficient of determination as a measure of model accuracy, defined as Equation (4):  
\[R^{2}=\frac{SS_{\text{regression}}}{SS_{\text{total}}} \tag{4}\]  
where \(SS_{\text{regression}}\) is the regression sum of squares, representing the fit error of the regression model, and \(SS_{\text{total}}\) is the total sum of squares, indicating the dispersion of the total data.