## 4 Result and Discussion
### Algorithm's Performance Evaluation  
The performance of the proposed algorithms is summarized in Table 4 based on the metrics, such as RMSE, MAE, and R\({}^{2}\) for the training and test dataset. RMSE is plotted in Figure 8 for different algorithms. It is observed that RF and BNN performance on test and train data is very similar and lowest. Both methods surpass the RMSE score for ANN followed by SVM. MAE score is plotted in Figure 9 with different algorithms. It is found that the performance of the BNN and RF is similar and lowest in comparison to ANN and SVM, while RF maintains a marginal accuracy than BNN. The R\({}^{2}\) plot is also analyzed (shown in Figure 10) with different algorithms and it is found that the test and train datasets are very concise and clear. The accuracy of RF and BNN are close to 1, whereas, it is near 0.9 for both ANN and SVM. Figure 6: BNN predicted pressure plotted with observed pressure loss: (**a**) training dataset and (**b**) test dataset. Figure 7: Random Forest predicted pressure plotted with observed pressure loss: (**a**) training dataset and (**b**) test dataset. by BNN at 3.275% and for ANN and SVM it is estimated to be 11.52% and 18.13%, respectively. The average R\({}^{2}\) for RF and BNN is 0.988 and 0.989, while ANN and SVM exhibits 0.9 and 0.875, respectively. The performance of RF and BNN are optimum and can accurately determine the pressure drop of Herschel-Bulkley fluids in eccentric and concentric annulus with higher accuracy compared to ANN and SVM. In Figures 11 and 12, the error distribution for train and test datasets are plotted for Bayesian neural network and random forest. The black outline in Figures 11 and 12 denotes the theoretical normal distribution while the red color represents the distribution of the residual error of the test and train dataset respectively. The error distribution for Bayesian neural network is uniformly distributed around mean of \(-1.25\) with standard deviation of 6.7 for train dataset, while same for test dataset is \(-0.9\) with standard deviation of  
the experimental results. A comparison is performed in Figures 17 and 18 exhibiting pressure drop predictions from this study and [PERSON] [19] for a selected range of annulus diameter (0.020828 m) and fluid XCD5 (Appendix B). The BNN shows a slight resemblance to the derived results of [PERSON] [19], while the RF predicted data points lies around the vicinity of the theoretical results, hence, concluding the better performance of the later method. RF is an ensemble model, where multiple decision trees are created based on randomly sampled datasets to approximate and lower the error. The model also quantifies the contribution of each input parameter towards predicting output. Subsequently, this property can be used to determine the importance of each input parameter to predict the desired output. The percentage-wise bar plot is shown in Figure 19 for each input parameter that have effect on the predicted output values. The diameter ratio (Di/Do) is the highest dominating parameter with contribution of approximately 58% out of the six-input followed by flow rate m\({}^{3}\)/s i.e., 37%. The diameter ratio and flow rate account for 95% dominance in predicting the pressure loss, while the Herschel-Bulkley fluid parameters has low accountability of remaining 5% where the consistency Index (K) has a share of 3.5% while flow index and shear stress have less than 1% contribution and eccentricity has negligible effect of the pressure loss. Figure 14: Observed and predicted pressure loss is plotted with flow rate for RF algorithm for low pressure regime. Figure 13: Observed and predicted pressure loss is plotted with flow rate for RF algorithm for high pressure regime. Figure 16: Observed and predicted pressure loss is plotted with flow rate for BNN algorithm for low pressure regime. Figure 17: Comparative analysis of the theoretical solution by [PERSON] et al. (2019) and the BNN.