Predictive modeling of physical processes in heterogeneous media requires innovations in mathematical and computational thinking. While multiscale approaches have been successful in modeling the effects of fine scales to macroscopic response, a significant grant challenge remains in understanding the effects of topological uncertainties in characterization of properties. We will briefly address major limitations in physical modeling in heterogeneous media including data-driven models of stochastic input, the curse of stochastic dimensionality, stochastic coarse graining and development of inexpensive surrogate stochastic models. A number of examples will be discussed including an information theoretic approach to coarse graining in materials science and deformation of random polycrystalline materials.