Empirical evaluated SDE modelling for dimensionality reduction systems and its predictability estimates
Empirical evaluated SDE modelling for dimensionality reduction systems and its predictability estimates
复制标题
降维系统的实证评估 SDE 建模及其可预测性估计
DOI：
10.1007/s13160-017-0296-2
复制
发表时间：
2018
影响因子：
0.9
通讯作者：
S. Kusuoka and Y. Saiki
中科院分区：
文献类型：
--
作者：
N. Nakano;M. Inatsu;S. Kusuoka and Y. Saiki
本文开发并验证了一种基于随机微分方程（SDE）框架的非线性动力系统降维系统的经验建模方法。根据概率论反问题的数学定理，我们推导了漂移向量和扩散矩阵的经验评估公式。着眼于洛伦兹系统的低维动力系统，我们凭经验重建了一个近似投影二维平面上原始时间序列的 SDE。数值 SDE 生成的解的集合方差的分布与预测时间序列的轨迹的分布非常吻合，这表明 SDE 建模具有表示局部可预测性的能力。此外，我们还将我们的 SDE 构造方法与传统的 Mori-Zwanzig 投影算子方法进行比较，该方法用于推导降维系统的广义 Langevin 方程，以评估通过该方法推导的 SDE 模型的适用性。
This paper develops and validates a method of empirical modelling for a dimensionality-reduced system of a nonlinear dynamical system based on the framework of the stochastic differential equation (SDE). Following the mathematical theorem corresponding to some inverse problem of the probability theory, we derive the empirically evaluating formulae for the drift vector and diffusion matrix. Focusing on a low-dimensional dynamical system of the Lorenz system, we empirically reconstruct an SDE that approximates the original time-series on the projected 2-dimensional plane. The distribution of the ensemble variance of solutions generated by the numerical SDE well agrees with that of the trajectories of the projected time-series, which indicates the ability of the SDE modelling to represent local predictability. Moreover, we also compare our SDE constructing method with the conventional Mori–Zwanzig projected operator method, which is used to derive a generalised Langevin equation for dimensionality-reduced systems, to assess the applicability of the obtained SDE model derived by the presented method.