  The automated construction of coarse-grained models represents a pivotal
component in computer simulation of physical systems and is a key enabler in
various analysis and design tasks related to uncertainty quantification.
Pertinent methods are severely inhibited by the high-dimension of the
parametric input and the limited number of training input/output pairs that can
be generated when computationally demanding forward models are considered. Such
cases are frequently encountered in the modeling of random heterogeneous media
where the scale of the microstructure necessitates the use of high-dimensional
random vectors and very fine discretizations of the governing equations. The
present paper proposes a probabilistic Machine Learning framework that is
capable of operating in the presence of Small Data by exploiting aspects of the
physical structure of the problem as well as contextual knowledge. As a result,
it can perform comparably well under extrapolative conditions. It unifies the
tasks of dimensionality and model-order reduction through an encoder-decoder
scheme that simultaneously identifies a sparse set of salient lower-dimensional
microstructural features and calibrates an inexpensive, coarse-grained model
which is predictive of the output. Information loss is accounted for and
quantified in the form of probabilistic predictive estimates. The learning
engine is based on Stochastic Variational Inference. We demonstrate how the
variational objectives can be used not only to train the coarse-grained model,
but also to suggest refinements that lead to improved predictions.
