Deep learning of subsurface flow via theory-guided neural network

Active researches are currently being performed to incorporate the wealth of scientific knowledge into data-driven approaches (e.g., neural networks) in order to improve the latter's effectiveness. In this study, the Theory-guided Neural Network (TgNN) is proposed for deep learning of subsurface flow. In the TgNN, as supervised learning, the neural network is trained with available observations or simulation data while being simultaneously guided by theory (e.g., governing equations, other physical constraints, engineering controls, and expert knowledge) of the underlying problem. The TgNN can achieve higher accuracy than the ordinary Deep Neural Network (DNN) because the former provides physically feasible predictions and can be more readily generalized beyond the regimes covered with the training data. Furthermore, the TgNN model is proposed for subsurface flow with heterogeneous model parameters. Several numerical cases of two-dimensional transient saturated flow are introduced to test the performance of the TgNN. In the learning process, the loss function contains data mismatch, as well as PDE constraint, engineering control, and expert knowledge. After obtaining the parameters of the neural network by minimizing the loss function, a TgNN model is built that not only fits the data, but also adheres to physical/engineering constraints. Predicting the future response can be easily realized by the TgNN model. In addition, the TgNN model is tested in more complicated scenarios, such as prediction with changed boundary conditions, learning from noisy data or outliers, transfer learning, learning from sparse data, and engineering controls. Numerical results demonstrate that the TgNN model achieves much better predictability, reliability, and generalizability than DNN models due to the physical/engineering constraints in the former.