## 5 Conclusions  
In this paper, a hybrid GRU-MLP deep learning model was developed to facilitate the prediction of shale gas production. This model seamlessly integrates both the historical production data and the inherent physical constraints into its framework. The GRU network was harnessed to capture the nonstationary patterns of the production data, exploiting its long-term memory capability. Conversely, the MLP component was enlisted to unveil the intricate and multifaceted nonlinear relationships existing between the production data and the prevailing physical constraints.  
Through a comprehensive analysis conducted on the field data, the proposed physics-constrained GRU-MLP model demonstrated its ability to effectively capture the intricate and dynamic patterns that characterize production sequences. Furthermore, the model successfully considered the nonlinear dependencies between geological properties, fracture geometry, fracturing treatment parameters, and production outcomes. In comparison to the original LSTM and GRU models that did not incorporate such constraints, the GRU-MLP model showcased better performances in both short-term and long-term forecasting tasks.  
Extending the findings of the research to other shale reservoirs holds significant potential for advancing production forecasting. Despite originating from a specific dataset, the inherent adaptability and generalizability of the methodology make it suitable for such expansion. Exploring diverse shale reservoirs in a similar manner aims to enhance predictive precision and foster a broader comprehension of reservoir dynamics. This research endeavor aligns with the dedication to provide a versatile and transferable tool to optimize production across diverse shale reservoirs.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Predicted Well** & **Model** & **Input Variable** & _RMSE_ & _MAE_ & _MAPE_ \\ \hline Well1 & GRU & Well1 & 72.39 & 38.82 & 0.019 \\ Well2 & Well2 & 156.97 & 99.26 & 0.048 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Evaluation metrics using different well production data.  
Figure 12: Comparisons of 150-day predictions using well 1 and well 2 historical data (GRU).  
**Author Contributions:** Conceptualization, X.M. and J.Z.; methodology, X.M.; validation, M.H. and R.Z.; formal analysis, X.M.; investigation, X.M. and J.Z.; data curation, X.M.; writing--original draft preparation, X.M.; writing--review and editing, X.M. and [PERSON]; visualization, [PERSON] and [PERSON]; funding acquisition, [PERSON] and [PERSON]. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by the National Natural Science Foundation of China (Grant Nos. 51974253, 51934005, 52004219), the Natural Science Basic Research Program of Shaanxi (Grant Nos. 2017)M5109 and 2020Q-781), the Scientific Research Program Funded by Education Department of Shaanxi Province (Grant Nos. 18 JS085 and 20 JS117), and the Graduate Student Innovation and Practical Ability Training Program of Xi'an Shiyou University (Grant No. YCS21211021).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data that support the findings of this study are available from the corresponding author, [PERSON], upon reasonable request.  
**Conflicts of Interest:** The authors declare no conflict of interest.