Entropic transfer operators for data-driven analysis of dynamical systems

用于动力系统数据驱动分析的熵传递算子

• 批准号: 521064440
• 负责人: Professor Dr. Bernhard Schmitzer
• 金额: --
• 依托单位: Institut für Informatik (IfI)
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/521064440?language=en
• 关键词: Entropic; transfer; operators; data; driven

The analysis of complex dynamical systems plays a central role in many fields, e.g. atmospheric flows in meteorology, turbulent flows in engineering, and large molecules in biochemistry. Systems of interest are typically chaotic, stochastic, and high-dimensional, but sometimes an approximate low-dimensional description exists, e.g. by metastable conformations of a molecule and their transition rates. Methods for systematic identification of such low-dimensional approximations are therefore highly relevant. Mathematically dynamical systems can be modelled via their transfer- and Koopman operators. They provide information on time-scale separation, metastable states, and for dimensionality reduction. In practice the operators must be approximated numerically and estimated from data. A broad range of corresponding methods has been developed and successfully applied in the natural sciences. But discretization and choice of basis functions remain challenging, in particular in high dimensions. We have recently introduced entropic transfer operators, a novel way to estimate an explicit discrete Markov matrix from Lagrangian data. Discretization artefacts and finiteness of the available data are remedied by blurring with entropic optimal transport. The latter is becoming an increasingly popular and numerically mature tool for data analysis. We have shown that in the limit of increasing available data the discrete operator converges to a blurred version of the true operator and allows to recover features of the true operator above the blur length scale. The method is mesh free and the only parameter, the blur length scale, allows for a trade-off of available data and resolution scale of the analysis. Consequently we believe that it will be a valuable addition to the toolbox of data-driven dynamic system analysis. We will generalize the method in various directions, derive convergence rates, and investigate its potential for combination with other recent methods from dynamic system analysis.

复杂动力系统的分析在许多领域中起着核心作用，例如气象学中的大气流动，工程中的湍流，以及生物化学中的大分子。感兴趣的系统通常是混沌的、随机的和高维的，但有时存在近似的低维描述，例如通过分子的亚稳态构象及其跃迁速率。因此，系统识别这种低维近似的方法是高度相关的。在数学上，动态系统可以通过它们的传递算子和库普曼算子来建模。它们提供关于时间尺度分离、亚稳态和降维的信息。在实践中，算子必须在数值上近似，并从数据中估计。一系列相应的方法已被开发并成功应用于自然科学。但是离散化和基函数的选择仍然具有挑战性，特别是在高维情况下。我们最近介绍了熵转移算子，一种新的方法来估计显式离散拉格朗日数据的马尔可夫矩阵。离散化文物和有限的可用数据进行补救模糊熵最佳运输。后者正在成为一个越来越受欢迎和数字成熟的数据分析工具。我们已经表明，在增加可用数据的限制下，离散算子收敛到真实算子的模糊版本，并允许恢复模糊长度尺度以上的真实算子的特征。该方法是无网格的，唯一的参数，模糊长度尺度，允许权衡可用的数据和分辨率尺度的分析。因此，我们相信，这将是一个有价值的补充工具箱的数据驱动的动态系统分析。我们将在各个方向推广的方法，推导出收敛速度，并探讨其与其他最近的方法从动态系统分析相结合的潜力。