## 3 Results and Discussion  
### MLP Network  
The MLP network is trained on the training dataset to determine the structure parameters. The optimal number of neural network layers is 3 and the number of neurons in each layer is 128. The batch size and epochs are 5000 and 100, respectively. The ReLU activation function and Adam optimizer are used in the training process. The network is implemented based on the TensorFlow framework [25]. The MAE between prediction values and true values is 50.12 m\({}^{2}\)/d. To validate the shale gas productive prediction model based on MLP, we select 100 groups of different geological and fracturing reservoir parameters and compare the prediction results of the model with the real values. The comparison of the true values and values predicted by MLP networks of daily shale gas production and the relative error between the true values and predictions are shown in Figure 1. The black broken line is the true value of daily production, and the red circular scatter is the predictions of the model. The red scattered points are all located on the black broken line, which shows that the model can accurately and stably forecast the gas production. Figure 1 also shows the relative errors between the predicted values and true data. The average relative error is 2.85% and the maximum relative error is 12.9%. It was found that there are only three data points with relative errors greater than 8%, and the corresponding daily productions are small. At this time, the absolute errors of the model prediction results are small. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Value** \\ \hline Fracture cluster & 3, 5, 7, 9 \\ Half-length of fracture (m) & 60, 80, 100, 120 \\ Fracture conductivity (mD\(\cdot\)m) & 100, 200, 300, 400 \\ Permeability (nD) & 100, 200, 300, 400 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters for training data. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Cluster**} & **Half-Length of** & **Fracture (m)** & **Fracture** (mD\(\cdot\)m)** & **Permeability** & **Production** & **Daily Production** \\  & **Fracture (m)** & **Conductivity (mD\(\cdot\)m)** & **(nD)** & **Time (d)** & **(m\({}^{3}\)/d)** \\ \hline
1 & 9 & 120 & 400 & 400 & 1 & 106,485.00 \\
2 & 9 & 120 & 400 & 400 & 2 & 89,001.90 \\
3 & 9 & 120 & 400 & 400 & 3 & 79,231.30 \\
4 & 9 & 120 & 400 & 400 & 4 & 72,545.10 \\
5 & 9 & 120 & 400 & 400 & 5 & 67,481.10 \\ \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) \\
255,996 & 3 & 60 & 100 & 100 & 996 & 289.90 \\
255,997 & 3 & 60 & 100 & 100 & 997 & 289.75 \\
255,998 & 3 & 60 & 100 & 100 & 998 & 289.60 \\
255,999 & 3 & 60 & 100 & 100 & 999 & 289.43 \\
256,000 & 3 & 60 & 100 & 100 & 1000 & 289.28 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Dataset of shale gas production. Furthermore, we analyze the prediction performance of the MLP network for shale gas production. Figure 2 depicts the prediction performance for the daily shale gas production under three groups of geological and fracturing reservoir parameters. The geological and fracturing reservoir parameters of the three cases are as follows: in case 1, the number of perforating clusters is 9, the half-length of hydraulic fracture is 120 m, the hydraulic fracture conductivity is 400 MD\(\cdot\)m, and the matrix permeability is 400 nd; in case 2, the number of perforation clusters is 5, the half-length of hydraulic fracture is 100 m, the hydraulic fracture conductivity is 300 MD\(\cdot\)m, and the matrix permeability is 300 nd; in case 3, the number of perforation clusters is 3, the half-length of hydraulic fracture is 60 m, the hydraulic fracture conductivity is 100 MD\(\cdot\)m, and the matrix permeability is 100 nd. The parameters in case 1 are the maximum values in the dataset, and the parameters in case 3 are the minimum values in the dataset. The predictions of the three groups of examples are in good agreement with the actual daily production, which further validated the prediction model.