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.

中文翻译：

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基于深度学习的逆建模方法：地下流示例

深度学习取得了良好的性能，并显示出解决正反问题的巨大潜力。在这项工作中，提出并比较了两类创新的基于深度学习的逆建模方法。第一类是基于深度学习替代的反演方法，其中将理论导向神经网络（TgNN）构造为用于模型参数不确定的问题的深度学习替代。通过合并物理定律和其他约束条件，可以使用有限的模拟运行来构造TgNN替代项，并显着加速反演过程。提出了三种基于TgNN代理的反演方法，包括梯度法，迭代集成平滑器法和训练法。第二类是直接深度学习反转方法，其中，TgNN受地统计信息的约束，被称为TgNN-geo，被建议作为直接逆建模的深度学习框架。在TgNN-geo中，引入了两个神经网络分别逼近随机模型参数和解。为了遵守随机模型参数的先验地统计信息，首先通过使用观察或生成的实现来训练用于逼近随机模型参数的神经网络。然后，通过最小化TgNN-geo的损失函数，可以同时获得模型参数的估计和模型解的近似值。由于可以合并先前的地统计信息，因此即使在空间测量稀疏或先前统计不精确的情况下，基于TgNN-geo的直接反演方法也能很好地工作。尽管所提出的基于深度学习的逆建模方法本质上是通用的，因此可以适用于各种各样的问题，但是它们还是针对一些地下流动问题进行了测试。发现以高效率获得令人满意的结果。此外，针对提出的两类基于深度学习的反演方法，进一步分析了优缺点。