随机系统降阶建模的有效Mori-Zwanzig方程
Effective Mori-Zwanzig equation for the reduced-order modeling of stochastic systems
摘要：
通过对随机动力系统观测量的有效Mori-Zwanzig（EMZ）方程的亚椭圆分析，证明了所得到的EMZ方程的半群估计可用于驱动系统在平衡态和非平衡态下观测统计量的先验估计.此外，我们引入第一原理和数据驱动的方法来近似EMZ记忆核，并证明了数据驱动的参数化方案的收敛性使用的记忆核的正则性估计。通过对Fermi-Pasta-Ulam链模型的朗之万动力学的蒙特卡罗模拟，对分析结果进行了数值验证.通过同一个例子，我们也证明了所提出的记忆核近似方法的有效性。
英文摘要：
Built upon the hypoelliptic analysis of the effective Mori-Zwanzig (EMZ) equation for observables of stochastic dynamical systems, we show that the obtained semigroup estimates for the EMZ equation can be used to drive prior estimates of the observable statistics for system in the equilibrium and non-equilibrium state. In addition, we introduce both first-principle and data-driven methods to approximate the EMZ memory kernel, and prove the convergence of the data-driven parametrization schemes using the regularity estimate of the memory kernel. The analysis results are validated numerically via the Monte-Carlo simulation of the Langevin dynamics for a Fermi-Pasta-Ulam chain model. With the same example, we also show the effectiveness of the proposed memory kernel approximation methods.
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