Stochastic Data-Driven Variational Multiscale Reduced Order Models

TL;DR

This paper introduces a stochastic reduced order model (S-ROM) that is robust, convergent with more data, and capable of fast, accurate trajectory predictions with uncertainty quantification for complex dynamical systems.

Contribution

The paper develops a novel stochastic ROM framework that improves robustness, convergence, and predictive accuracy over traditional data-driven ROMs.

Findings

S-ROM converges as the number of trajectories increases.

S-ROM provides accurate long-term trajectory predictions.

S-ROM quantifies uncertainty due to unresolved scales.

Abstract

Trajectory-wise data-driven reduced order models (ROMs) tend to be sensitive to training data, and thus lack robustness. We propose to construct a robust stochastic ROM closure (S-ROM) from data consisting of multiple trajectories from random initial conditions. The S-ROM is a low-dimensional time series model for the coefficients of the dominating proper orthogonal decomposition (POD) modes inferred from data. Thus, it achieves reduction both space and time, leading to simulations orders of magnitude faster than the full order model. We show that both the estimated POD modes and parameters in the S-ROM converge when the number of trajectories increases. Thus, the S-ROM is robust when the training data size increases. We demonstrate the S-ROM on a 1D Burgers equation with a viscosity $\nu= 0.002$ and with random initial conditions. The numerical results verify the convergence. Furthermore, the S-ROM makes accurate trajectory-wise predictions from new initial conditions and with a prediction time far beyond the training range, and it quantifies the spread of uncertainties due to the unresolved scales.