  We develop a physics-informed machine learning approach for large-scale data
assimilation and parameter estimation and apply it for estimating
transmissivity and hydraulic head in the two-dimensional steady-state
subsurface flow model of the Hanford Site given synthetic measurements of said
variables. In our approach, we extend the physics-informed conditional
Karhunen-Lo\'{e}ve expansion (PICKLE) method for modeling subsurface flow with
unknown flux (Neumann) and varying head (Dirichlet) boundary conditions. We
demonstrate that the PICKLE method is comparable in accuracy with the standard
maximum a posteriori (MAP) method, but is significantly faster than MAP for
large-scale problems. Both methods use a mesh to discretize the computational
domain. In MAP, the parameters and states are discretized on the mesh;
therefore, the size of the MAP parameter estimation problem directly depends on
the mesh size. In PICKLE, the mesh is used to evaluate the residuals of the
governing equation, while the parameters and states are approximated by the
truncated conditional Karhunen-Lo\'{e}ve expansions with the number of
parameters controlled by the smoothness of the parameter and state fields, and
not by the mesh size. For a considered example, we demonstrate that the
computational cost of PICKLE increases near linearly (as $N_{FV}^{1.15}$) with
the number of grid points $N_{FV}$, while that of MAP increases much faster as
$N_{FV}^{3.28}$. We demonstrated that once trained for one set of Dirichlet
boundary conditions (i.e., one river stage), the PICKLE method provides
accurate estimates of the hydraulic head for any value of the Dirichlet
boundary conditions (i.e., for any river stage).
