# NewApproach to Predict the Motion Characteristics of Single Bubbles in Still Water
## 3 Results
### Prediction Model for the Drag Coefficient of Single Bubbles  
For bubble drag coefficient prediction model development, the data obtained from experiments were distributed into three portions. Among the datasets, 692 datasets were used for training, 148 datasets were used for validation, while another 148 datasets were kept for testing purpose.  
The FBNN model was built with one input layer, three hidden layers, and one output layer, as shown in Figure 10. There were three nodes in the input layer, which are Re number, Eo number, and We number, and only one node in the output, which was bubble drag coefficient. In each hidden layer, there were 10 hidden neurons, and tansig function was selected as the transfer function. By setting the learning rate to 0.001, the number of training to 1000, and the momentum factor to 0.85, the model exhibited good performance during the training process.  
From Figures 11 and 12, the R\({}^{2}\)-value for both the validation and testing plot is obviously high, reaching 0.96 and 0.92, respectively. The value is approximate to 1, which indicates that the developed model has achieved a good performance in terms of the coefficient of determination.  
Figure 10: Architecture of the drag coefficient prediction model.  
Figure 9: Relationship between R\({}^{2}\) and root mean squared error (RMSE) in both training and testing sets with respect to different number of samples.  
In addition, in order to have further verification on the suitability of the model, the RMSE value is calculated. It records at 0.0946 and 0.1534, respectively, for validation set and testing set. Such a low RMSE value indicates that the model is well-performed from the perspective of RMSE.  
In order to further show that the developed AI model has the ability to serve as an alternative or replace the need for mathematical calculation, 150 datasets were randomly picked. The predicted values obtained from the model were compared with values calculated through the empirical formulas.  
As shown in Figure 13, the drag coefficient (\(\mathrm{C_{D}}\)) values calculated through the formula as proposed by [PERSON] et al. [35] show a larger deviation from the actual \(\mathrm{C_{D}}\) values. Meanwhile, the predicted \(\mathrm{C_{D}}\) values from the developed AI model and calculated \(\mathrm{C_{D}}\) values from the formula introduced by [PERSON] et al. [36] are in good agreement with the actual values.  
From the perspective of the coefficient of determination, the FBNN model exhibits \(\mathrm{R^{2}}\) value of 0.9593, while the formulas proposed by [PERSON] [35] and [PERSON] [36] show \(\mathrm{R^{2}}\) values of 0.9192 and 0.9362, respectively. Also, the RMSE value is calculated for each case, recorded at 0.1291 for the developed FBNN model, 0.2412 for [PERSON]'s formula [35], and 0.1686 for [PERSON]'s formula [36].  
Based on the above statistical analyses, the predicted value from the developed FBNN model has the highest accuracy if compared with the empirical formulas suggested by the previous researchers.  
Figure 11: Plot of the predicted drag coefficient against the actual drag coefficient for the validation set.  
Figure 12: Plot of the predicted drag coefficient against the actual drag coefficient for the testing set.