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.
Inverse modeling aims to infer uncertain parameters of a system with noisy observations of the system response, which has been widely utilized in various scientific and engineering practices, such as seismic inversion (Bunks et al., 1995), petroleum reservoir history matching (Oliver et al., 2008), aquifer parameter estimation (Carrera & Neuman, 1986a, 1986b, 1986c), and medical imaging (Arridge, 1999). Under certain conditions, inverse problems can be viewed as optimization problems, in which the model parameters are modified, such that the predictions from forward models can match the measurements. In addition, prior knowledge can be used to regularize the objective function and to obtain the maximum a posteriori estimate of the model parameters. The gradient-based method is a straightforward and common way to perform inverse modeling tasks. Anterion et al. (1989) presented a rigorous analytical method to calculate the gradients of observations with respect to reservoir characteristics, which can then be used to assist the process of parameter adjustment for history matching. For calculating the required gradients in inverse modeling, the adjoint method is a frequently utilized technique (
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