## 1 Introduction
In recent years, there has been an increased focus on incorporating data-driven and machine learning-based approaches to forecast hydrocarbon production [17; 18]. These approaches view such production prediction as a time-series forecasting problem [19]. By utilizing time series production as training data, these methods can effectively predict well production by capturing the trends and characteristics present in the hydrocarbon production data. Deep neural networks have been extensively studied for the time series forecasting of hydrocarbon production due to their ability to handle nonlinear data. Previous studies on hydrocarbon production prediction using deep learning models can be categorized based on the count of input and forecasted features, as illustrated in Table 1. These prediction categories are classified into univariate, covariate, and multivariate scenarios [19], as visually depicted in Figure 1.  
In Figure 1a, the univariate prediction entails the projection of the future values of a single variable based on its historical data. In certain instances, when predicting the values of a single variable, it is necessary to consider the relationships among multiple variables and to model their associations to forecast the forthcoming value of the single variable. This form of prediction is termed covariate forecasting. Within the context of covariate forecasting, both one or multiple time series data and additional physical factors governing production are considered. While the number of features predicted in the covariate forecasting remains single, akin to the univariate prediction, it incorporates multiple input features, as depicted in Figure 1b. Beyond the aforementioned predictive paradigms, there are multivariate predictions that focus on capturing the associations among multiple variables to forecast the values of the variables. These prediction scenarios are depicted in Figure 1c. These investigations have demonstrated the impressive effectiveness and accuracy of deep neural networks in the prediction of hydrocarbon production, as evidenced by the results presented in Table 1.  
Recent scholarly pursuits have delved into the exploration of the potential of physics-constrained machine learning (PCML) methodologies for projecting hydrocarbon production. These methods excel at capturing the inherent physical dependencies between variables through the design of neural network architectures. In the realm of PCML, physical insights are integrated into the neural network framework, wherein designated neurons or modules are endowed with explicit physical interpretations [25]. Notably, [PERSON] et al. [30] put forth an integrated neural network comprising LSTM and MLP architectures for the prediction of geothermal productivity in multilateral wells, incorporating the physical constraints into the model to enhance its predictive capability. A hybrid network was devised, wherein the integration of physical constraints and LSTM outputs serves as input for the MLP. Through this integration, the intricate physical dependencies and geothermal production were systematically explored. Similarly, [PERSON] et al. [25] introduced a composite neural network encompassing Bidirectional Gated Recurrent Unit (BiGRU) and Deep Hybrid Neural Network (DHNN) components. This approach leverages the complementary strengths of physics knowledge and machine learning, effectively capturing the dynamics of time-series production. By considering the static and dynamic, temporal and spatial aspects of the fractured wells as the physical constraints shaping long-term production predictions, this PCML technique facilitated the generation of oil production forecasts that were more precise and dependable. Motivated by the recent advancements in the application of PCML to the realm of time-series production forecasting, this study introduces a hybrid neural network named GRU-MLP, which seamlessly integrates the architectures of GRU and MLP.