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Deep‐learning has achieved good performance and demonstrated 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 method, and the training method. The second category is direct‐deep‐learning‐inversion methods, in which TgNN constrained with geostatistical information, named TgNN‐geo, is proposed as the deep‐learning framework for direct inverse modeling. In TgNN‐geo, two neural networks are introduced to approximate the random model parameters and the solution, respectively. In order to honor prior geostatistical information of the random model parameters, the neural network for approximating the random model parameters is first trained by using observed or generated realizations. Then, by minimizing the loss function of TgNN‐geo, the estimation of model parameters and the approximation of the model solution can be simultaneously obtained. 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 high efficiency. Moreover, both the advantages and disadvantages are further analyzed for the proposed two categories of deep‐learning‐based inversion methods.

Deep‐Learning‐Based Inverse Modeling Approaches: A Subsurface Flow Example