## 4 Prediction Method  
### Preprocessing of Sample Data  
According to the analysis in Section 2.3, this paper selects three main factors, which are jet pressure, jet angle and impact time, as the input parameters of the prediction model, and erosion depth as the predicted output. The sample data are obtained by the orthogonal experimental design of L\({}_{25}\)(5\({}^{3}\)). The orthogonal experimental design has the characteristic of \"uniformity and decentralization, orderliness and comparable\". The 125 data obtained by this hybrid orthogonal experimental design method are sufficient and reliable as the whole input data set for neural network, and satisfactory prediction results can be obtained [30; 40; 46; 47]. Factors and levels of the tests are shown in Table 7. Table 8 records the results of the submerged jet erosion test on refractory bricks by an orthogonal table of L\({}_{25}\)(5\({}^{3}\)). The standoff distance is fixed at 10 mm.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \(P_{C}\) & \(P_{m}\) & \(P_{C1}\) & \(P_{C2}\) & \(P_{m1}\) & \(P_{m2}\) & \
\begin{tabular}{c} **Mind-Evolution** \\ **Iteration** \\ \end{tabular} &
\begin{tabular}{c} **Number of** \\ **Sub-Populations** \\ \end{tabular} \\ \hline GA & 0.5 & 0.1 & — & — & — & — & — & — \\ MAGA & — & — & 0.8 & 0.4 & 0.2 & 0.02 & 50 & 10 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Setting parameters of different algorithms.  
BP neural network requires a large amount of effective sample data for training, but due to the constraints of the experimental conditions, it is difficult to obtain a sufficient number of experimental samples. Therefore, the virtual sample method [46] can be used to increase the number of samples data. In the experiment, due to the inevitable errors caused by the instrumentation, operation and measurement, there is a slight deviation between the input variable value and the experimental value, and this deviation is normal. Existing research [47] considers that all the input values within this deviation range correspond to the same output value, and usually the fluctuation range is \(\pm\Delta\) = 0.2%. Then, each experimental sample can generate \(2^{3}\) virtual samples, although not all these virtual samples are needed for training. With the orthogonal experimental design of L\({}_{4}\)(\(2^{3}\)), four virtual samples can be used to represent all these virtual samples corresponding to each experimental sample. As shown in Table 9, four virtual samples are generated based on the first experimental sample of test No.1. In this way, a total of 100 virtual samples can be generated.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Levels**} & \multicolumn{3}{c}{**Factors**} \\ \cline{2-4}  & \(p\) **(MPa)** & _\(\alpha\)_ **(\({}^{\circ}\))** & \(t\) **(s)** \\ \hline
1 & 4 & 30 & 20 \\
2 & 5 & 45 & 30 \\
3 & 6 & 60 & 40 \\
4 & 7 & 75 & 50 \\
5 & 8 & 90 & 60 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Factors and levels of orthogonal experimental design.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Test No.**} & \multicolumn{3}{c}{**Factors**} & \multicolumn{3}{c}{**Evaluation Indexes**} \\ \cline{2-5}  & \(p\) **(MPa)** & _\(\alpha\)_ **(\({}^{\circ}\))** & \(t\) **(s)** & \(H\) **(cm)** \\ \hline
1 & 4 & 30 & 20 & 0.332 \\
2 &  
In order to avoid the difference in the magnitude of the data affecting the recognition accuracy of the network, the sample is normalized by the proportional compression method, as the Equation (10), so that the data mapping range is from \(-1\) to \(1\).  
\[x^{\prime}=x^{\prime}_{\text{min}}+\frac{x^{\prime}_{\text{max}}-x^{\prime}_{ \text{min}}}{x_{\text{max}}-x_{\text{min}}}(x-x_{\text{min}}), \tag{10}\]  
where \(x\) is the original input data, \(x_{\text{max}}\) and \(x_{\text{min}}\) are the maximum and minimum values of the original data, respectively, and \(x^{\prime}\) is the normalized input data.