## 4 Breakdown Pressure Prediction with a Multi-Layer Neural Network
### Dataset Preparation and Model Verification
The loss and accuracy results for the training and testing sets are shown in Figures 8 and 9. As illustrated, the loss function was initially high, but upon increasing the number of iterations, the error between the actual and predicted values decreased for both the training and testing sets, eventually stabilizing. After stabilization, both training and testing accuracy exceeded 90%, indicating that the multi-layer neural network model effectively learned and predicted breakdown pressure. Figure 8: Performance of the neural network model on the training set: (**a**) Training loss curve, illustrating the model’s reduction in prediction error across epochs, which demonstrates the model’s learning process and its convergence toward minimizing error on known data; (**b**) training accuracy curve, showing the model’s predictive accuracy on the training set over epochs, indicating the model’s capacity to accurately capture the relationships between variables affecting breakdown pressure. Figure 9: Performance of the neural network model on the testing set: (**a**) Testing loss curve, showing the reduction in prediction error across epochs, which indicates the model’ ability to generalize and converge toward minimizing prediction error on unseen data; (**b**) testing accuracy curve, representing the model’s predictive accuracy on the testing set over epochs, reflecting its capability to generalize and correctly predict breakdown pressure under various experimental conditions. Metrics commonly used to evaluate prediction performance include the coefficient of determination (\(R\)), root mean square error (\(RMSE\)), and mean absolute error (\(MAE\)). A higher \(R\) value, closer to 1, indicates a stronger correlation between predicted and actual values, while lower \(RMSE\) and \(MAE\) values reflect higher prediction accuracy. The formulae for these evaluation metrics are provided in Table 3, where \(n\) represents the number of data points, \(y_{i}^{*}\) and \(y_{i}\) are the actual and predicted values, respectively, and \(\overline{y^{*}}\) and \(\overline{y}\) represent the mean of actual and predicted values, respectively. To further validate the prediction performance of the trained model, we compared the predicted values from the training set with the actual values and performed a regression analysis of the predicted versus actual values for the testing set. The coefficients \(R\), \(RMSE\), and \(MAE\) were calculated, as shown in Table 3. Figure 10 illustrates that the data points for both the training and testing samples are closely aligned with the ideal fit line, where the actual value equals the predicted value. The \(R\), \(RMSE\), and \(MAE\) values for the training set (testing set) were 0.9482 (0.9123), 3.424 (4.421), and 2.283 (3.188), respectively. Compared to the PSO-BPNN model in the literature [19], which achieved an \(R\) value of 0.6971 for the test set, the multi-layer neural network improved the \(R\) value by 23.6%. These results demonstrate the neural network's effectiveness in learning and predicting breakdown pressure, indicating the strong predictive capability of the trained model. Notably, Figure 11 shows the model's predicted breakdown pressure for the fracturing experiments, with an average relative error of approximately 16% between the predicted and experimental values. This highlights the model's accuracy in predicting breakdown pressure and its potential as an effective tool for optimizing fracturing operation parameters. \begin{table}
\begin{tabular}{c c} \hline \hline
**Index** & **Calculation** \\ \hline \(R\) & \(R=\frac{\sum_{i}\left(y_{i}^{*}-\overline{y}^{*}\right)\left(y_{i}-\overline{y }\right)}{\sqrt{\sum_{i=1}^{n}\left(y_{i}-\overline{y}\right)^{2}}\sqrt{\sum_{i =1}^{n}\left(y_{i}-\overline{y}\right)^{2}}}\) \\ \(RMSE=\sqrt{\frac{1}{n}\sum\limits_{i=1}^{n}\left(y_{i}-y_{i}^{*}\right)^{2}}\) \\ \(MAE=\frac{1}{n}\sum\limits_{i=1}^{n}\left|y_{i}^{*}-y_{i}\right|\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Equations for calculating \(R\), \(RMSE\), and \(MAE\).