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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 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 iterative local updating ensemble smoother in solving the inverse problem shows that it can achieve accurate inversion results and predictive uncertainty estimates.

Plain Language Summary High-dimensional inverse problems are often computationally expensive since a large number of forward model evaluations are usually required. A computationally efficient alternative is to replace in the inversion process the forward model with an accurate but fast-to-evaluate surrogate model. However, most existing surrogate methods suffer from the curse of dimensionality. In this paper, we develop a deep autoregressive neural network-based surrogate method to efficiently solve such high-dimensional inverse problems. The autoregressive neural network can efficiently obtain accurate approximations for the time-dependent outputs of forward models with time-varying inputs. In addition, a dense convolutional network architecture is employed to transform the surrogate modeling task to an image-to-image regression problem by leveraging the robust capability of convolutional networks in image-like data processing. The curse of dimensionality was tackled though a series of inherent nonlinear projections of the input into low-dimensional latent spaces. It is shown that the proposed network provides an accurate surrogate of a transport system with high-dimensional uncertain inputs using relatively limited training data. Reliable inversion results are then obtained with a rather minor computational cost by using the surrogate to replace the forward model in the inversion process.