  Data assimilation for parameter and state estimation in subsurface transport
problems remains a significant challenge due to the sparsity of measurements,
the heterogeneity of porous media, and the high computational cost of forward
numerical models. We present a physics-informed deep neural networks (DNNs)
machine learning method for estimating space-dependent hydraulic conductivity,
hydraulic head, and concentration fields from sparse measurements. In this
approach, we employ individual DNNs to approximate the unknown parameters
(e.g., hydraulic conductivity) and states (e.g., hydraulic head and
concentration) of a physical system, and jointly train these DNNs by minimizing
the loss function that consists of the governing equations residuals in
addition to the error with respect to measurement data. We apply this approach
to assimilate conductivity, hydraulic head, and concentration measurements for
joint inversion of the conductivity, hydraulic head, and concentration fields
in a steady-state advection--dispersion problem. We study the accuracy of the
physics-informed DNN approach with respect to data size, number of variables
(conductivity and head versus conductivity, head, and concentration), DNNs
size, and DNN initialization during training. We demonstrate that the
physics-informed DNNs are significantly more accurate than standard data-driven
DNNs when the training set consists of sparse data. We also show that the
accuracy of parameter estimation increases as additional variables are inverted
jointly.
