# Supervised parallel-in-time algorithm for long-time Lagrangian simulations of stochastic dynamics: Application to hydrodynamics

## TL;DR

This paper introduces a supervised parallel-in-time algorithm that accelerates long-time stochastic Lagrangian simulations by leveraging a macroscopic Navier-Stokes model as a predictor, effectively bridging microscopic and macroscopic dynamics.

## Contribution

The novel SPASD method uses a low-fidelity macroscopic model to supervise and correct high-dimensional microscopic simulations, enabling efficient long-time stochastic dynamics simulations.

## Key findings

- Accelerates long-time Lagrangian simulations of stochastic dynamics.
- Uses a macroscopic Navier-Stokes model as a predictor for microscopic simulations.
- Effectively recovers microscopic details and fluctuations through correction.

## Abstract

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.