Corresponding author: Email address: changhaibin@pku.edu.cn Key Points: Two categories of innovative deep-learning based inverse modeling methods are proposed and compared. The deep-learning surrogate-based inversion methods can accelerate the inversion process significantly. Abstract Deep-learning has achieved good performance and shown great potential for solving forward and inverse problems. In this work, two categories of innovative deep-learning based inverse modeling methods are proposed and compared. The first category is deep-learning surrogate-based inversion methods, in which the Theory-guided Neural Network (TgNN) is constructed as a deep-learning surrogate for problems with uncertain model parameters. By incorporating physical laws and other constraints, the TgNN surrogate can be constructed with limited simulation runs and accelerate the inversion process significantly. Three TgNN surrogate-based inversion methods are proposed, including the gradient method, the iterative ensemble smoother (IES), and the training method. In TgNN-geo, two neural networks are introduced to approximate the respective random model parameters and the solution. Since the prior geostatistical information can be incorporated, the direct-inversion method based on TgNN-geo works well, even in cases with sparse spatial measurements or imprecise prior statistics. Although the proposed deep-learning based inverse modeling methods are general in nature, and thus applicable to a wide variety of problems, they are tested with several subsurface flow problems. It is found that satisfactory results are obtained with a high efficiency.

Subsurface flow problems usually involve some degree of uncertainty. Consequently, uncertainty quantification is commonly necessary for subsurface flow prediction. In this work, we propose a methodology for efficient uncertainty quantification for dynamic subsurface flow with a surrogate constructed by the Theory-guided Neural Network (TgNN). The TgNN here is specially designed for problems with stochastic parameters. In the TgNN, stochastic parameters, time and location comprise the input of the neural network, while the quantity of interest is the output. The neural network is trained with available simulation data, while being simultaneously guided by theory (e.g., the governing equation, boundary conditions, initial conditions, etc.) of the underlying problem. The trained neural network can predict solutions of subsurface flow problems with new stochastic parameters. With the TgNN surrogate, the Monte Carlo (MC) method can be efficiently implemented for uncertainty quantification. The proposed methodology is evaluated with two-dimensional dynamic saturated flow problems in porous medium. Numerical results show that the TgNN based surrogate can significantly improve the efficiency of uncertainty quantification tasks compared with simulation based implementation. Further investigations regarding stochastic fields with smaller correlation length, larger variance, changing boundary values and out-of-distribution variances are performed, and satisfactory results are obtained.