# NewApproach to Predict the Motion Characteristics of Single Bubbles in Still Water
## 3 Results  
### Prediction Model for the Final Velocity of Single Bubbles  
There is a nonlinear yet complex relationship between the final velocity of single bubbles and the dimensionless parameters (Re number, Eo number, and We number). Therefore, a feed forward back propagation neural network (FBNN) was developed to predict the final velocity.  
1000 experiment datasets were collected and separated according to the ratio of 70:15:15. 700 datasets were used to train the model, 150 datasets were used for calibration and validation, and the remaining150 datasets were reserved for testing purposes.  
While designing the architecture of the model, tansig activation function was selected as the transfer function, while gradient descent was chosen as the optimization method. As shown in Figure 6, the input layer consisted of three nodes (Re number, Eo number, and We number) while the output layer had only one node (final velocity). There were three hidden layers where each of the layers consisted of 10 hidden neurons. In addition, the learning rate was set to 0.01, the number of training was fixed at 1000, and the momentum factor was 0.9. With such a combination of the hyper parameters, the model exhibited good convergence performance during the training process.  
Figure 5: Process for machine learning implementation.  
Figure 6: Architecture of the final velocity prediction model.  
Figures 7 and 8 show the model performance in terms of coefficient of determination, R\({}^{2}\) for the model validation process, and model testing process respectively, recorded at 0.93 and 0.83. A high R\({}^{2}\)-value indicates a strong correlation between the actual value and predicted value, where the fitting degree is considerably high.  
In order to further verify the appropriateness of the FBNN model, root mean squared error (RMSE) was chosen. A smaller RMSE value indicates a higher level of model accuracy. The equation of RMSE is shown as in Equation (15):  
\[\text{RMSE}=\sqrt{\frac{1}{2n}\sum_{x-1}^{n}(z(x)-y(x))^{2}} \tag{15}\]  
where \(n\) is the number of samples, \(z(x)\) is the predicted value, and \(y(x)\) is the actual value. The calculated RMSE is 0.0741 for the validation set and 0.0518 for the testing set. Such a low RMSE value indicates that the developed model has achieved a relatively good performance. It has the ability to reflect the relationship between the dimensionless number (Re number, Eo number, and We number) and bubble final velocity with a considerably high level of accuracy.  
On the other hand, Figure 9 depicted the relationship between R\({}^{2}\) and RMSE in both training and testing sets under varying sample size. With the increases of the sample size, the R\({}^{2}\) value shows a  
Figure 8: Plot of the predicted final velocity against the actual final velocity for the testing set.  
Figure 7: Plot of the predicted final velocity against the actual final velocity for the validation set.  
decreasing trend in training set, but an increasing trend in testing set. Meanwhile, from the perspective of RMSE, such a value decreases gradually in training set but increases gently in testing set when the number of sample increases. Only with the sample size of around 900, the RMSE value for both training and testing set shows an almost similar value, meaning that the model can provide a considerably high level of accuracy with that particular sample size. In other words, it is appropriate to choose a slightly higher sample size of 1000 in this study.