  We introduce a generalized machine learning framework to probabilistically
parameterize upper-scale models in the form of nonlinear PDEs consistent with a
continuum theory, based on coarse-grained atomistic simulation data of
mechanical deformation and flow processes. The proposed framework utilizes a
hypothesized coarse-graining methodology with manifold learning and
surrogate-based optimization techniques. Coarse-grained high-dimensional data
describing quantities of interest of the multiscale models are projected onto a
nonlinear manifold whose geometric and topological structure is exploited for
measuring behavioral discrepancies in the form of manifold distances. A
surrogate model is constructed using Gaussian process regression to identify a
mapping between stochastic parameters and distances. Derivative-free
optimization is employed to adaptively identify a unique set of parameters of
the upper-scale model capable of rapidly reproducing the system's behavior
while maintaining consistency with coarse-grained atomic-level simulations. The
proposed method is applied to learn the parameters of the shear transformation
zone (STZ) theory of plasticity that describes plastic deformation in amorphous
solids as well as coarse-graining parameters needed to translate between
atomistic and continuum representations. We show that the methodology is able
to successfully link coarse-grained microscale simulations to macroscale
observables and achieve a high-level of parity between the models across
scales.
