  A Theory-guided Auto-Encoder (TgAE) framework is proposed for surrogate
construction and is further used for uncertainty quantification and inverse
modeling tasks. The framework is built based on the Auto-Encoder (or
Encoder-Decoder) architecture of convolutional neural network (CNN) via a
theory-guided training process. In order to achieve the theory-guided training,
the governing equations of the studied problems can be discretized and the
finite difference scheme of the equations can be embedded into the training of
CNN. The residual of the discretized governing equations as well as the data
mismatch constitute the loss function of the TgAE. The trained TgAE can be used
to construct a surrogate that approximates the relationship between the model
parameters and responses with limited labeled data. In order to test the
performance of the TgAE, several subsurface flow cases are introduced. The
results show the satisfactory accuracy of the TgAE surrogate and efficiency of
uncertainty quantification tasks can be improved with the TgAE surrogate. The
TgAE also shows good extrapolation ability for cases with different correlation
lengths and variances. Furthermore, the parameter inversion task has been
implemented with the TgAE surrogate and satisfactory results can be obtained.
