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.

翻译：由数据驱动的轨迹减序模型(ROMs)往往对培训数据敏感,因而缺乏可靠性。我们提议从随机初始条件下由多轨组成的数据中建立一个由随机初始条件下的多轨构成的稳妥的ROM关闭(S-ROM)。S-ROM是一个从数据中推断出的适当正方位分解(POD)模式系数的低维时间序列模型。因此,S-ROM实现了空间和时间的减少,导致模拟数量序列的速度比全序模型快。我们表明,在轨迹增加时,S-ROD估计模式和参数会汇合。因此,S-ROM在培训数据大小增加时是稳健的。我们用1D Burgers 方程式展示S-ROM, 方位值为 $\ = 0.002美元, 初始条件随机。数字结果可以核实趋同。此外,S-ROM还从新的初始条件和远超出培训范围的预测时,得出了准确的轨迹谱性预测结果。