## 1 Introduction
The GRU-MLP model is designed to forecast well productivity within shale gas reservoirs while simultaneously accounting for the physical constraints associated with shale gas production. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Categories** & **Author (Year)** & **Well Production** & **Neural Network** & **Description** \\ \hline \multirow{5}{*}{Univariate} & [PERSON] et al. (2019) [20] & \begin{tabular}{c} Monthly shale gas \\ production \\ \end{tabular} & LSTM & \begin{tabular}{c} LSTM outperforms \\ DCA methods. \\ \end{tabular} \\ \cline{2-5}  & [PERSON] et al. (2019) [21] & Daily oil production & Deep LSTM (DLSTM) & \begin{tabular}{c} DLSTM outperformed \\ RNN and GRU. \\ \end{tabular} \\ \cline{2-5}  & [PERSON] et al. (2022) [22] & Oil production rate & LSTM & \begin{tabular}{c} LSTM outperformed \\ traditional prediction \\ methods \\ \end{tabular} \\ \cline{2-5}  & [PERSON] et al. (2022) [23] & Daily shale gas & LSTM & \begin{tabular}{c} LSTM outperforms \\ ARIMA and DCA \\ \end{tabular} \\ \hline \multirow{5}{*}{Covariate} & [PERSON] et al. (2021) [24] & Oil production rate & \begin{tabular}{c} Physics-guided \\ model \\ \end{tabular} & \begin{tabular}{c} Physics-guided \\ model outperforms LSTM \\ \end{tabular} \\ \cline{2-5}  & [PERSON] et al. (2022) [25] & Daily oil rate & \begin{tabular}{c} Bidirectional GRU and \\ Hybrid network \\ \end{tabular} & \begin{tabular}{c} BiGRU-DHNN \\ outperforms \\ RNN, GRU, BiGRU and \\ LSTM \\ \end{tabular} \\ \hline \multirow{5}{*}{Multivariate} & [PERSON] et al. (2019) [26] & \begin{tabular}{c} Water saturation, \\ formation pressure and \\ oil production \\ \end{tabular} & LSTM, GRU, RNN & \begin{tabular}{c} LSTM is better than \\ GRU and standard RNN \\ \end{tabular} \\ \cline{2-5}  & [PERSON] et al. (2021) [27] & Oil and water rates & LSTM & \
\begin{tabular}{c} RNN-based proxy model \\ for well-control \\ optimization \\ \end{tabular} \\ \cline{2-5}  & [PERSON] et al. (2022) [28] & Daily oil, water, gas & LSTM &
\begin{tabular}{c} LSTM outperforms \\ numerical reservoir \\ simulation \\ \end{tabular} \\ \cline{2-5}  & [PERSON] et al. (2022) [29] & Gas, water flow & GRU-MLP combined & GRU-MLP is superior to \\ \cline{2-5}  & rates & neural network & RNN, GRU and LSTM \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of recent studies using deep neural networks in hydrocarbon production forecasting. Embedded within the GRU-MLP model, the MLP component is trained to apprehend the intricate nonlinear correlation existing between the production data and the accompanying physical constraints, including formation properties and engineering parameters. On the other hand, the GRU component is harnessed to capture the sequential relationships innate to the production data. This paper offers three key contributions: (i) the creation of a novel hybrid GRU-MLP architecture designed for well production prediction within shale gas reservoirs; (ii) a comprehensive comparative evaluation of the proposed GRU-MLP model with established LSTM and GRU models, demonstrating its improved predictive performance; and (iii) the integration of geological attributes, fracture geometry, and fracturing treatment parameters as inputs, enabling a holistic assessment of intricate interactions impacting production outcomes. Collectively, these contributions introduce innovative methodologies and pragmatic advancements to the realm of production prediction in shale gas reservoirs. By incorporating the physical constraints, the model not only enhances the predictive accuracy but also aligns the prognostications with the physical principles governing shale gas reservoir dynamics.