Lu Fei
Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218, USA.
Entropy (Basel). 2020 Nov 30;22(12):1360. doi: 10.3390/e22121360.
We present a class of efficient parametric closure models for 1D stochastic Burgers equations. Casting it as statistical learning of the flow map, we derive the parametric form by representing the unresolved high wavenumber Fourier modes as functionals of the resolved variable's trajectory. The reduced models are nonlinear autoregression (NAR) time series models, with coefficients estimated from data by least squares. The NAR models can accurately reproduce the energy spectrum, the invariant densities, and the autocorrelations. Taking advantage of the simplicity of the NAR models, we investigate maximal space-time reduction. Reduction in space dimension is unlimited, and NAR models with two Fourier modes can perform well. The NAR model's stability limits time reduction, with a maximal time step smaller than that of the K-mode Galerkin system. We report a potential criterion for optimal space-time reduction: the NAR models achieve minimal relative error in the energy spectrum at the time step, where the K-mode Galerkin system's mean Courant-Friedrichs-Lewy (CFL) number agrees with that of the full model.
我们提出了一类用于一维随机伯格斯方程的高效参数封闭模型。将其视为流映射的统计学习，我们通过将未解析的高波数傅里叶模式表示为已解析变量轨迹的泛函来推导参数形式。简化模型是非线性自回归（NAR）时间序列模型，其系数通过最小二乘法从数据中估计。NAR模型可以准确地再现能量谱、不变密度和自相关。利用NAR模型的简单性，我们研究了最大时空约简。空间维度的约简是无限制的，具有两个傅里叶模式的NAR模型可以表现良好。NAR模型的稳定性限制了时间约简，其最大时间步长小于K模式伽辽金系统的时间步长。我们报告了一个最优时空约简的潜在标准：在时间步长上，当K模式伽辽金系统的平均库朗 - 弗里德里希斯 - 勒维（CFL）数与全模型的CFL数一致时，NAR模型在能量谱中实现最小相对误差。
