  Inverse modeling for computing a high-dimensional spatially-varying property
field from indirect sparse and noisy observations is a challenging problem.
This is due to the complex physical system of interest often expressed in the
form of multiscale PDEs, the high-dimensionality of the spatial property of
interest, and the incomplete and noisy nature of observations. To address these
challenges, we develop a model that maps the given observations to the unknown
input field in the form of a surrogate model. This inverse surrogate model will
then allow us to estimate the unknown input field for any given sparse and
noisy output observations. Here, the inverse mapping is limited to a broad
prior distribution of the input field with which the surrogate model is
trained. In this work, we construct a two- and three-dimensional inverse
surrogate models consisting of an invertible and a conditional neural network
trained in an end-to-end fashion with limited training data. The invertible
network is developed using a flow-based generative model. The developed inverse
surrogate model is then applied for an inversion task of a multiphase flow
problem where given the pressure and saturation observations the aim is to
recover a high-dimensional non-Gaussian permeability field where the two facies
consist of heterogeneous permeability and varying length-scales. For both the
two- and three-dimensional surrogate models, the predicted sample realizations
of the non-Gaussian permeability field are diverse with the predictive mean
being close to the ground truth even when the model is trained with limited
data.
