  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.
