## 4 Prediction Method
### Prediction Model of Optimized BP Neural Network  
BP neural network [48] is a multi-layer feedforward network with a strong prediction ability. The topology of the single hidden layer is shown in Figure 8a. Figure 8b embodies the mathematical model of the \(j\) artificial neuron in each layer.  
Variables \(x_{1}\), \(x_{2}\),... \(x_{\text{n}}\) are the inputs from each neuron in the previous layer. \(\omega j_{1}\), \(\omega j_{2}\),... \(\omega j_{n}\) are the connection weights from each neuron in the previous layer to the \(j\) neuron. \(Sj\) is the input of the \(j\) neuron. \(\theta_{j}\) is the threshold. \(y_{i}\) is the output of the \(j\) neuron. Activation function is Equation (11). The activation function of the hidden layer uses the tan-sigmoid function with Equation (12). The Levenberg-Marquardt algorithm is used to train the network. According to the principle of minimum root mean square error, the neuron number in a single hidden layer was set to \(5\) by the empirical formula as Equation (13) [40].  
\[\left\{\begin{array}{l}y_{i}=f(S_{j})\\ S_{j}=\sum\limits_{i=1}^{n}\omega_{j}x_{i}+\theta_{j}\end{array}\right.. \tag{11}\]  
\[\tan\text{sig}(S_{j})=\frac{2}{1+e^{-2S_{j}}}-1. \tag{12}\]  
\[n_{h}=\sqrt{n_{i}+n_{0}}+a, \tag{13}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & \(p\) **(MPa)** & \(\alpha\) **(\({}^{\circ}\))** & \(t\) **(s)** \\ \hline
1 & 3.992 & 29.94 & 19.96 \\
2 & 3.992 & 30.06 & 20.04 \\
3 & 4.008 & 29.94 & 20.04 \\
4 & 4.008 & 30.06 & 19.96 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Virtual samples based on the first experimental sample.  
Figure 8: Prediction model of a back-propagating (BP) neural network: (**a**) The topology of the single hidden layer, (**b**) Model of the \(j\) artificial neuron.  
\(\text{APE}_{\text{max}}\) is the maximum absolute percentage error, which reflects the deviation of the worst point by Equation (15). APE reflects the deviation of each predicted point from the experimental value by Equation (16). \[\text{APE}_{\text{max}}=\text{max}\{\text{APE}\}\] (15) \[\text{APE}_{i}=\frac{\left|\hat{y}_{i}-y_{i}\right|}{y_{i}}i=1,2,3\ldots n.\] (16)
3. MAPE is the mean absolute percentage error, which reflects the accuracy of the prediction results by Equation (17). \[\text{MAPE}=\frac{1}{n}\sum\limits_{i=1}^{n}\frac{\left|\hat{y}_{i}-y_{i} \right|}{y_{i}}.\] (17)
4. RMSE is the root mean square error and is the most important indicator by Equation (18). This indicator reflects the accuracy of the prediction model. The smaller the value, the closer the predicted result is to the experimental value. It means that the prediction accuracy of this model is higher. \[\text{RMSE}=\sqrt{\frac{\sum\limits_{i=1}^{n}\left(\hat{y}_{i}-y_{i}\right)^{ 2}}{n}}.\] (18)
5. \(R^{2}\) is the coefficient of determination, which reflects the reliability of the prediction model, and is an important indicator of the goodness of fit between the predicted and experimental values. It is calculated by Equation (19). When \(0.8\leq R^{2}\leq 1\), it means a very strong correlation; When \(0.6\leq R^{2}\leq 0.8\), it means a strong correlation; \(0.2\leq R^{2}\leq 0.4\) means a weak correlation; \(0\leq R^{2}\leq 0.2\) means no correlation. The closer \(R^{2}\) is to 1, the higher the prediction accuracy and goodness of fit. \[R^{2}=\frac{(n\sum\limits_{i=1}^{n}\hat{y}_{i}y_{i}-\sum\limits_{i=1}^{n}\hat {y}_{i}\sum\limits_{i=1}^{n}y_{i})^{2}}{(n\sum\limits_{i=1}^{n}\hat{y}_{i}^{2 }-(\sum\limits_{i=1}^{n}\hat{y}_{i})^{2})(n\sum\limits_{i=1}^{n}y_{i}^{2}-( \sum\limits_{i=1}^{n}y_{i})^{2})}\] (19)