  Identification of a groundwater contaminant source simultaneously with the
hydraulic conductivity in highly-heterogeneous media often results in a
high-dimensional inverse problem. In this study, a deep autoregressive neural
network-based surrogate method is developed for the forward model to allow us
to solve efficiently such high-dimensional inverse problems. The surrogate is
trained using limited evaluations of the forward model. Since the relationship
between the time-varying inputs and outputs of the forward transport model is
complex, we propose an autoregressive strategy, which treats the output at the
previous time step as input to the network for predicting the output at the
current time step. We employ a dense convolutional encoder-decoder network
architecture in which the high-dimensional input and output fields of the model
are treated as images to leverage the robust capability of convolutional
networks in image-like data processing. An iterative local updating ensemble
smoother (ILUES) algorithm is used as the inversion framework. The proposed
method is evaluated using a synthetic contaminant source identification problem
with 686 uncertain input parameters. Results indicate that, with relatively
limited training data, the deep autoregressive neural network consisting of 27
convolutional layers is capable of providing an accurate approximation for the
high-dimensional model input-output relationship. The autoregressive strategy
substantially improves the network's accuracy and computational efficiency. The
application of the surrogate-based ILUES in solving the inverse problem shows
that it can achieve accurate inversion results and predictive uncertainty
estimates.
