  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. The second category is direct-deep-learning-inversion methods,
in which TgNN constrained with geostatistical information, named TgNN-geo, is
proposed for direct inverse modeling. 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. Moreover, both the advantages and disadvantages are
further analyzed for the proposed two categories of deep-learning based
inversion methods.
