## 2 Materials and Methods
### Generalization Ability Verification  
After completing the preliminary analysis of the dataset, this study then used the remaining 20% of the dataset for the implementation of production forecasting. In order to comprehensively evaluate the performance of the model, appropriate evaluation indicators were selected for detailed analysis. Especially when dealing with regression tasks such \begin{table}
\begin{tabular}{c c c c} \hline
**Pipeline** & **Method** & **Hyperparameter** & **Range of Values** \\ \hline \multirow{5}{*}{PR-NN} & PCA & n\_components & 2/4/6 \\ \cline{2-4}  & \multirow{4}{*}{Neural network} & hidden\_layer\_sizes & (50), (100), (200), (100, 50), (200, 100), \\  & & hidden\_layer\_sizes & (300, 200, 100), (400, 300, 200, 100) \\  & & activation function & “tanh”, “relu”, “logistic” \\  & & alpha & 0.0001/0.001/0.01/0.1 \\  & & max\_iter & 3000/5000/7000 \\ \hline \multirow{5}{*}{PFR-XGB} & PolynomialFeatures & poly\_degree & 2/3/4 \\ \cline{2-4}  & \multirow{4}{*}{XGBoost} & learning\_rate & 0.01/0.1/0.5 \\  & & n\_estimators & 50/100/200 \\ \cline{1-1}  & & max\_depth & 1/2/3 \\ \cline{1-1}  & & min\_child\_weight & 1/3/5 \\ \cline{1-1}  & & booster & “gbtree”/”gblinear” \\ \hline \multirow{5}{*}{PFR-RF} & PolynomialFeatures & poly\_degree & 2/3/4 \\ \cline{2-4}  & \multirow{4}{*}{Random Forest} & n\_estimators & 400/500/600 \\ \cline{1-1}  & & max\_depth & 2/6/10 \\ \cline{1-1}  & & min\_samples\_split & 10/15/20 \\ \cline{1-1}  & & min\_samples\_leaf & 4/6/8 \\ \cline{1-1}  & & max\_features & “auto”/”sqrt” \\ \cline{1-1}  & & bootstrap & True/False \\ \hline \multirow{5}{*}{PFR-NN} & PolynomialFeatures & poly\_degree & 2/3/4 \\ \cline{1-1}  & & hidden\_layer\_sizes & (50), (100), (200), (100, 50), (200, 100), \\ \cline{1-1}  & & hidden\_layer\_sizes & (300, 200, 100), (400, 300, 200, 100) \\ \cline{1-1}  & & activation function & “tanh”, “relu”, “logistic” \\ \cline{1-1}  & & alpha & 0.0001/0.001/0.01/0.1 \\ \cline{1-1}  & & max\_iter & 3000/5000/7000 \\ \hline \end{tabular}
\end{table} 
Table 2: _Cont._as production forecasting, this study uses several common evaluation criteria, including mean absolute error (MAE), mean square error (MSE), and decision score coefficient (R\({}^{2}\)). \[MAE=\frac{1}{n}\sum\limits_{k=1}^{n}{\left|{{y_{k}}-{\hat{y}_{k}}}\right|} \tag{7}\]  
\[MSE=\frac{1}{n}\sum\limits_{k=1}^{n}{\left({{y_{k}}-{\hat{y}_{k}}}\right)^{2}} \tag{8}\]  
\[R^{2}=1-\frac{\sum\limits_{k=1}^{n}{\left({{y_{k}}-{\hat{y}_{k}}}\right)^{2}}} {\sum\limits_{k=1}^{n}{\left({{y_{k}}-{\overline{y}}}\right)^{2}}} \tag{9}\]  
where \(y_{k}\) is the kth real production (m\({}^{3}\)/d), \({\hat{y}_{k}}\) is the kth predicted production (m\({}^{3}\)/d), \({\overline{y}}\) is the mean real production of all wells (m\({}^{3}\)/d), \({\overline{y}}\) is the mean predicted production of all wells (m\({}^{3}\)/d), and \(n\) is the total count of specimens. [PERSON] introduced the percentage of accuracy-precision (PAP) metric as a novel approach for model selection in energy science and engineering. This approach emphasizes the PAP's utility in objectively comparing experimental and modeling data, and as a function for optimization, advocating its effectiveness in identifying models that best balance accuracy and precision [55]. Meanwhile, for oil and gas field production predictions, Taylor diagrams offer a valuable graphical method for comparing predicted and observed datasets [56], such as predicted production, pressure drops, and fluid flow patterns. In this study, Taylor diagrams are utilized to examine the accuracy of the model in relation to alternative machine learning models. These diagrams can incorporate various measures of the predicted and actual values. Specifically, they can display the standard deviation (SD), Pearson correlation coefficient (PCC), and root-mean-square error (RMSE) between the predicted and observed data, providing a comprehensive overview of the model's performance.