## 4 Results and Discussion  
The topological structure of the BPANN obtained in this study is presented in Figure 1, where the number of input layer neurons is three, representing \(G\), \(P\) and \(FS\), the number of output layer neurons is one, representing the \(MRR\), and the optimum number of neurons in the single hidden layer is 10, which was obtained according to a trial and error process. The learning process of the BPANN is presented in Table 1 of Appendix A. The error training curve is presented in Figure 1 of Appendix B.  
A linear least-squares fit [18] was adopted to compare the prediction precisions of the previously proposed regression models and the presently proposed BPANN model. The order of the linear fit was determined by inspecting the residual errors of the approximation via the MSE [27; 28].  
According to our past study [5], the conventional linear regression equation is given as follows:  
\[MRR=5.037-0.003\times G-0.023\times P+0.214\times FS \tag{3}\]  
The improved nonlinear regression equation is given as follows:  
\[MRR=0.904+5.041\times\sin\frac{68\pi}{G}-0.023\times P+0.214\times FS \tag{4}\]  
The best linear fits of the \(MRR\) predictions obtained for the conventional linear regression model, improved nonlinear regression model and the testing sets of the BPANN model with respect to the experimental values are presented in Figure 2a-c, respectively.  
Figure 1: Structure of the BPANN obtained (\(3\times 10\times 1\)).  
The modelling performances, i.e., the slope \(M\), intercept \(B\), \(R\) and \(R^{2}\), of the three methods are presented in Table 2, where the ideal values are \(M=1\), \(B=0\), \(R=1\) and \(R^{2}=1\). Here, the prediction precision increases as the obtained values approach the ideal values.  
From Figure 2 and Table 2, we note that the BPANN model has the highest prediction precision of all models considered. This indicates that the BPANN model provides better prediction results than either of the regression models.  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Three Modelling Methods** & \(M\) & \(B\) & \(R\) & \(R^{2}\) \\ \hline (a) Conventional linear regression model & 0.149 & 3.57 & 0.387 & 0.150 \\ (b) Improved nonlinear regression model & 0.859 & 0.603 & 0.927 & 0.859 \\ (c) Back-propagation artificial neural network model & 0.884 & 0.371 & 0.948 & 0.899 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Prediction precision indicators of the three modelling methods, where the ideal values are \(M=1\), \(B=0\), \(R=1\) and \(R^{2}=1\).  
Figure 2: Correlation between the predicted values and the experimental values along with the best linear fit to the data: (**a**) the conventional linear regression model; (**b**) the improved nonlinear regression model; and (**c**) the BPANN model.  
To intuitively demonstrate the prediction precision obtained by the three models, the experimentally measured values and the corresponding prediction values for the sixteen experimental datasets are presented in Figure 3. We observe that the BPANN model and the improved nonlinear regression model are useful for predicting the MRR in the USM process; however, the results in Figure 2 and Table 2 more clearly demonstrate that the BPANN model provides the best results. Meanwhile, the conventional linear regression model is useless for predicting the MRR in the USM process.