Entropic transfer operators for data-driven analysis of dynamical systems

复杂动力系统的分析在许多领域起着核心作用，例如气象学中的大气流动、工程中的湍流和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，但有时也存在近似的低维描述，例如通过分子的亚稳构象及其跃迁速率。因此，系统地识别这种低维近似的方法是高度相关的。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。它们提供了关于时间尺度分离、亚稳态和降维的信息。在实际操作中，必须对操作符进行数值近似和从数据中估计。在自然科学中已经发展出一系列相应的方法并成功地应用。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。我们最近引入了熵转移算子，这是一种从拉格朗日数据估计显式离散马尔可夫矩阵的新方法。离散伪影和可用数据的有限性是通过熵最优传输模糊来弥补的。后者正在成为一种日益流行和数字成熟的数据分析工具。我们已经证明，在增加可用数据的极限下，离散算子收敛到真算子的模糊版本，并允许在模糊长度尺度以上恢复真算子的特征。该方法是无网格的，唯一的参数，模糊长度尺度，允许权衡可用数据和分析的分辨率尺度。因此，我们相信它将是数据驱动的动态系统分析工具箱的一个有价值的补充。我们将在各个方向上推广该方法，推导收敛速率，并研究其与动态系统分析中其他最新方法结合的潜力。

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.