Novel Kernel Methods for Data Analysis in Dynamical Systems: Applications to Dimension Reduction and Prediction in Atmospheric and Oceanic Dynamics

由动力系统生成的大规模数据集在许多科学和工程学科中都会遇到。例如，在气候、大气和海洋科学中，动力学发生在一个无限维的相空间中，其中定义了流体流动和热力学的耦合非线性偏微分方程，观测数据对应于该相空间的函数，例如给定地理区域的温度或环流。在材料科学和分子动力学中也有大量的例子。一个主要的挑战是利用观测网络收集的大量数据或大规模数值模型输出的大量数据来了解运行物理学，并对系统中无法观测的方面进行推断，包括系统的未来状态。该项目旨在开发新的技术，用于动态系统中的数据分析和预测，同时考虑到模型误差和时空数据关系。提出了在气候-大气-海洋科学中两个影响大的领域的应用，即热带大气中多尺度对流波的跟踪和预报以及北极海冰厚度的重建和预报。这项研究将创建，记录和提供用于分析复杂动力系统的大规模数据的软件。这也将有助于课程开发和培养研究生在这个跨学科的竞技场。这项研究的总体框架是动力系统在高维相空间中运行，但产生的数据与低维，非线性几何结构。核方法形成了一个自然的数学框架，可以在这些低维对象上构建函数空间，并具有定义良好的平滑性概念，可用于执行各种数据分析任务，如降维，特征提取和预测。对于适当设计的内核，这些任务可以解释在一个黎曼几何诱导的数据。内核还提供了操作符来将引用数据集上的函数扩展到另一个感兴趣的数据集。在这个项目中所涉及的动力系统可以代表自然，也可以代表一个近似自然的数值模型。在许多现实世界的复杂应用程序中，由自然和模型生成的低维数据结构会有所不同。为了提取不完美模型的最大限度地符合自然的特征，或者在多模型集成中分配权重以进行预测，本项目将研究一种新的方法，其中基于核的样本外扩展算子用于定义模型误差的适当度量。通过Takens延迟坐标映射和其他特征考虑动态，这些误差度量被纳入修改后的内核中，使模型数据的几何形状偏置，以提取相对于自然具有高保真度的状态。这个项目是在正则化方案中使用修改后的核来学习动力系统中感兴趣的量之间的函数关系。这些方法将用于根据海洋和大气变量的观测重建和预报北极海冰厚度，以及热带大规模对流组织的参数-非参数混合预报。该项目的另一个目标是将这些想法扩展到算子值内核（所谓的多任务内核），用于分析向量值可观测量，例如空间扩展场。与标准标量值内核相比，这些内核在捕获时空不稳定性方面应该具有更高的技能，几何和动力学也通过延迟坐标映射发挥作用。本计画将利用这些核函数，从遥感资料中客观地撷取热带大气中的对流行波。

Large-scale datasets generated by dynamical systems are encountered in many science and engineering disciplines. For instance, in climate, atmosphere, and ocean sciences the dynamics take place in an infinite-dimensional phase space where the coupled nonlinear partial differential equations for fluid flow and thermodynamics are defined, and the observed data correspond to functions of that phase space, such as temperature or circulation over a given geographical region. Examples also abound in materials science and molecular dynamics. A major challenge is to utilize the vast amount of data that is being collected by observational networks or output by large-scale numerical models to understand the operating physics and make inferences about aspects of the system which are not accessible to observation, including the future state of the system. This project seeks to develop novel techniques for data analysis and prediction in dynamical systems, taking into account model error and spatiotemporal data relationships. Applications are proposed in two high-impact areas in climate-atmosphere-ocean science, namely, tracking and forecasting of multiscale convective waves in the tropical atmosphere and reconstruction and forecasting of Arctic sea-ice thickness. This research will create, document, and make available software for analyzing large-scale data from complex dynamical systems. It will also contribute to curricular development and training of graduate students in this interdisciplinary arena.The general framework of this research is dynamical systems operating in high-dimensional phase spaces, but generating data with low-dimensional, nonlinear geometric structures. Kernel methods form a natural mathematical framework to construct function spaces on these low-dimensional objects with a well-defined notion of smoothness, which can be used to carry out a variety of data analysis tasks such as dimension reduction, feature extraction, and prediction. For appropriately designed kernels, these tasks can be interpreted in terms of a Riemannian geometry induced on the data. Kernels also provide operators to extend functions on a reference dataset to another dataset of interest. The dynamical systems addressed in the project can represent either nature, or a numerical model approximating nature. In many real-world complex applications the low-dimensional data structures generated by nature and the model will differ. To extract the features of the imperfect model which are maximally consistent with nature, or to assign weights in multi-model ensembles for prediction, this project will study a novel approach where kernel-based out-of-sample extension operators are used to define appropriate metrics for model error. Taking dynamics into account through Takens delay-coordinate maps and other features, these error metrics are incorporated into modified kernels, biasing the geometry of the model data to extract states with high fidelity relative to nature. This project is to use the modified kernels in regularized schemes for learning functional relationships between quantities of interest in dynamical systems. These methods will be applied in reconstruction and forecasting of Arctic sea-ice thickness from observations of oceanic and atmospheric variables, and blended parametric-nonparametric forecasting of large-scale convective organization in the tropics. A further goal of the project is to extend these ideas to operator-valued kernels (so-called multitask kernels) for analysis of vector-valued observables, such as spatially extended fields. Compared to the canonical scalar-valued kernels, these kernels should have significantly higher skill in capturing spatiotemporal intermittency, with geometry and dynamics also playing a role through delay-coordinate maps. This project is to apply these kernels to objectively extract traveling convective waves in the tropical atmosphere from large datasets acquired via remote sensing.