  While existing mathematical descriptions can accurately account for phenomena
at microscopic scales (e.g. molecular dynamics), these are often
high-dimensional, stochastic and their applicability over macroscopic time
scales of physical interest is computationally infeasible or impractical. In
complex systems, with limited physical insight on the coherent behavior of
their constituents, the only available information is data obtained from
simulations of the trajectories of huge numbers of degrees of freedom over
microscopic time scales. This paper discusses a Bayesian approach to deriving
probabilistic coarse-grained models that simultaneously address the problems of
identifying appropriate reduced coordinates and the effective dynamics in this
lower-dimensional representation. At the core of the models proposed lie
simple, low-dimensional dynamical systems which serve as the building blocks of
the global model. These approximate the latent, generating sources and
parameterize the reduced-order dynamics. We discuss parallelizable, online
inference and learning algorithms that employ Sequential Monte Carlo samplers
and scale linearly with the dimensionality of the observed dynamics. We propose
a Bayesian adaptive time-integration scheme that utilizes probabilistic
predictive estimates and enables rigorous concurrent s imulation over
macroscopic time scales. The data-driven perspective advocated assimilates
computational and experimental data and thus can materialize data-model fusion.
It can deal with applications that lack a mathematical description and where
only observational data is available. Furthermore, it makes non-intrusive use
of existing computational models.
