Data-Driven Reduction for a Class of Multiscale Fast-Slow Stochastic Dynamical Systems
Data-Driven Reduction for a Class of Multiscale Fast-Slow Stochastic Dynamical Systems
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DOI：
10.1137/151004896
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发表时间：
2016-01-01
影响因子：
2.1
通讯作者：
Kevrekidis, Ioannis G.
中科院分区：
文献类型：
--
作者：
Dsilva, Carmeline J.;Talmon, Ronen;Kevrekidis, Ioannis G.
多时间尺度随机动力系统在科学和工程中普遍存在，将此类系统及其模型简化为仅其慢分量通常对于科学计算和进一步分析至关重要。这些系统通常不能以明确的分析模型的形式提供，而只能作为一个数据集来观察，这个数据集体现了几个时间尺度上的动态。我们专注于应用和适应数据挖掘和流形学习技术来检测一类这样的多尺度数据中的慢分量。传统的数据挖掘方法是基于指标（因此，几何），这是不告知的多尺度性质的底层系统动态，这样的方法不能成功地恢复慢变量。在这里，我们提出了一种方法，它利用本地的几何形状和本地噪声动态的数据集内，通过一个度量，这是不敏感的快速变量和更一般的比简单的统计平均。我们的分析方法提供了成功恢复的基础慢变量的条件，以及指导选择的方法参数的经验协议。有趣的是，恢复的基础变量是规范不变的，它们对测量仪器/观测函数不敏感。
Multi-time-scale stochastic dynamical systems are ubiquitous in science and engineering, and the reduction of such systems and their models to only their slow components is often essential for scientific computation and further analysis. Rather than being available in the form of an explicit analytical model, often such systems can only be observed as a data set which embodies dynamics on several time scales. We focus on applying and adapting data-mining and manifold learning techniques to detect the slow components in a class of such multiscale data. Traditional data-mining methods are based on metrics (and thus, geometries) which are not informed of the multiscale nature of the underlying system dynamics; such methods cannot successfully recover the slow variables. Here, we present an approach which utilizes both the local geometry and the local noise dynamics within the data set through a metric which is both insensitive to the fast variables and more general than simple statistical averaging. Our analysis of the approach provides conditions for successfully recovering the underlying slow variables, as well as an empirical protocol guiding the selection of the method parameters. Interestingly, the recovered underlying variables are gauge invariant-they are insensitive to the measuring instrument/observation function.