## 3 Bit Footage and ROP Prediction Model
According to Equation (3), it is preliminarily determined that the number of neurons in the hidden layer is 5-13, so it is necessary to further analyze the determination coefficient \(R^{2}\) (also called coefficient of determination, goodness of fit) of linear regression under different numbers of neurons in the hidden layer by combining the training results of the BP neural network. \(R^{2}\) is calculated by the total sum of squares \(TSS\) and the residuals sum of squares \(RSS\) of the sample. The specific calculation formula is shown in Formula (5). \[R_{2}=1-\frac{RSS}{TSS}=1-\frac{\sum\left(Y_{actual}-Y_{predict}\right)^{2}}{ \sum\left(Y_{actual}-Y_{mean}\right)^{2}} \tag{5}\]  
The determination coefficient has a range of [0, 1], and the closer it is to 1, the better the model's performance. On the other side, the closer it is to 0, the lower the model's performance. As illustrated in Figure 5, the optimal number of hidden layer units in the Shaximiao Formation-Ziliujing Formation ROP prediction model is ten, and the optimal number of hidden layer units in the Xujiahe Formation ROP prediction model is eleven. The goodness of fit is greater than 85%, and under the selected number of hidden layers, the predicted demand can be satisfied. The model's accuracy is outstanding. Mean squared error (MSE) is often used in regression tasks as a loss function. It calculates the average of the squared differences between the predicted and actual values. The calculation formula of MSE is shown in Formula (6). \[MSE=\frac{1}{n}\sum_{i=1}^{n}\left(y_{i}-\hat{y}_{i}\right)^{2} \tag{6}\]  
\(n\) is the number of samples; \(y_{i}\) are the truthful data; and \(\hat{y}_{i}\) are the predicted data. The MSE results for calculating the prediction model of the ROP are shown in Table 3. It is found that in Shaximiao Formation to Ziliujing Formation, when the number of hidden layer neurons is 10, the minimum MSE is 1.562. When the number of hidden layer neurons is 11, the minimum MSE is 0.583 in Xujiahe Formation. The results correspond to the determination coefficient, which verifies the good training performance of the neural network. Figure 6 depicts the prediction result of the BP neural network. It can provide a theoretical basis for the optimization of structural characteristics and the prediction of the application effect of shale gas bits in specific formations in Luzhou block, Sichuan province. Figure 5: Determination coefficient of prediction model under different number of neurons in hidden layer. Figure 6: \(Cont.\) \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Shaximiao Formation to Ziliujing Formation** & \multicolumn{2}{c}{**Xujiahe Formation**} \\ \hline
**Number of Hidden Layer Neurons** & **Mean Squared Error (MSE)** & **Number of Hidden Layer Neurons** & **Mean Squared Error (MSE)** \\ \hline
5 & 3.537 & 5 & 1.545 \\
6 & 3.824 & 6 & 1.279 \\
7 & 2.412 & 7 & 1.185 \\
8 & 2.285 & 8 & 0.962 \\
9 & 1.856 & 9 & 0.804 \\
10 & 1.562 & 10 & 0.787 \\
11 & 1.884 & 11 & 0.583 \\
12 & 2.050 & 12 & 0.671 \\
13 & 1.968 & 13 & 0.727 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The MSE results for calculating the prediction model of ROP (m/h).