  Lagrangian particle methods based on detailed atomic and molecular models are
powerful computational tools for studying the dynamics of microscale and
nanoscale systems. However, the maximum time step is limited by the smallest
oscillation period of the fastest atomic motion, rendering long-time
simulations very expensive. To resolve this bottleneck, we propose a supervised
parallel-in-time algorithm for stochastic dynamics (SPASD) to accelerate
long-time Lagrangian particle simulations. Our method is inspired by bottom-up
coarse-graining projections that yield mean-field hydrodynamic behavior in the
continuum limit. Here as an example, we use the dissipative particle dynamics
(DPD) as the Lagrangian particle simulator that is supervised by its
macroscopic counterpart, i.e., the Navier-Stokes simulator. The low-dimensional
macroscopic system (here, the Navier-Stokes solver) serves as a predictor to
supervise the high-dimensional Lagrangian simulator, in a predictor-corrector
type algorithm. The results of the Lagrangian simulation then correct the
mean-field prediction and provide the proper microscopic details (e.g.,
consistent fluctuations, correlations, etc.). The unique feature that sets
SPASD apart from other multiscale methods is the use of a low-fidelity
macroscopic model as a predictor. The macro-model can be approximate and even
inconsistent with the microscale description, but SPASD anticipates the
deviation and corrects it internally to recover the true dynamics.
