  The data-based discovery of effective, coarse-grained (CG) models of
high-dimensional dynamical systems presents a unique challenge in computational
physics and particularly in the context of multiscale problems. The present
paper offers a probabilistic perspective that simultaneously identifies
predictive, lower-dimensional coarse-grained (CG) variables as well as their
dynamics. We make use of the expressive ability of deep neural networks in
order to represent the right-hand side of the CG evolution law. Furthermore, we
demonstrate how domain knowledge that is very often available in the form of
physical constraints (e.g. conservation laws) can be incorporated with the
novel concept of virtual observables. Such constraints, apart from leading to
physically realistic predictions, can significantly reduce the requisite amount
of training data which enables reducing the amount of required, computationally
expensive multiscale simulations (Small Data regime). The proposed state-space
model is trained using probabilistic inference tools and, in contrast to
several other techniques, does not require the prescription of a fine-to-coarse
(restriction) projection nor time-derivatives of the state variables. The
formulation adopted is capable of quantifying the predictive uncertainty as
well as of reconstructing the evolution of the full, fine-scale system which
allows to select the quantities of interest a posteriori. We demonstrate the
efficacy of the proposed framework in a high-dimensional system of moving
particles.
