  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 Artificial Neural Network (ANN) 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,
and engineering controls. Numerical results demonstrate that the TgNN model
achieves much better predictability, reliability, and generalizability than ANN
models due to the physical/engineering constraints in the former.
