Qi Di, Liu Jian-Guo
Department of Mathematics, Purdue University, 150 North University Street, West Lafayette, Indiana 47907, USA.
Department of Mathematics and Department of Physics, Duke University, Durham, North Carolina 27708, USA.
Chaos. 2023 Oct 1;33(10). doi: 10.1063/5.0160057.
We propose a high-order stochastic-statistical moment closure model for efficient ensemble prediction of leading-order statistical moments and probability density functions in multiscale complex turbulent systems. The statistical moment equations are closed by a precise calibration of the high-order feedbacks using ensemble solutions of the consistent stochastic equations, suitable for modeling complex phenomena including non-Gaussian statistics and extreme events. To address challenges associated with closely coupled spatiotemporal scales in turbulent states and expensive large ensemble simulation for high-dimensional systems, we introduce efficient computational strategies using the random batch method (RBM). This approach significantly reduces the required ensemble size while accurately capturing essential high-order structures. Only a small batch of small-scale fluctuation modes is used for each time update of the samples, and exact convergence to the full model statistics is ensured through frequent resampling of the batches during time evolution. Furthermore, we develop a reduced-order model to handle systems with really high dimensions by linking the large number of small-scale fluctuation modes to ensemble samples of dominant leading modes. The effectiveness of the proposed models is validated by numerical experiments on the one-layer and two-layer Lorenz '96 systems, which exhibit representative chaotic features and various statistical regimes. The full and reduced-order RBM models demonstrate uniformly high skill in capturing the time evolution of crucial leading-order statistics, non-Gaussian probability distributions, while achieving significantly lower computational cost compared to direct Monte-Carlo approaches. The models provide effective tools for a wide range of real-world applications in prediction, uncertainty quantification, and data assimilation.
我们提出了一种高阶随机统计矩封闭模型，用于在多尺度复杂湍流系统中对主导阶统计矩和概率密度函数进行高效的集合预测。通过使用一致随机方程的集合解对高阶反馈进行精确校准，统计矩方程得以封闭，适用于对包括非高斯统计和极端事件在内的复杂现象进行建模。为应对与湍流状态下紧密耦合的时空尺度以及高维系统昂贵的大集合模拟相关的挑战，我们引入了使用随机批次方法（RBM）的高效计算策略。这种方法显著减少了所需的集合规模，同时准确捕捉到基本的高阶结构。每次样本更新仅使用一小批小尺度波动模式，并通过在时间演化过程中对批次进行频繁重采样确保精确收敛到完整模型统计量。此外，我们通过将大量小尺度波动模式与主导主模式的集合样本相联系，开发了一种降阶模型来处理真正高维的系统。所提出模型的有效性通过对单层和两层洛伦兹96系统的数值实验得到验证，这些系统展现出具有代表性的混沌特征和各种统计状态。完整和降阶的RBM模型在捕捉关键主导阶统计量的时间演化、非高斯概率分布方面均表现出一致的高技能，同时与直接蒙特卡罗方法相比，计算成本显著降低。这些模型为预测、不确定性量化和数据同化等广泛的实际应用提供了有效的工具。
