Spatiotemporal Pattern Extraction by Spectral Analysis of Vector-Valued Observables

Dimitrios Giannakis,Joanna Slawinska,Zhizhen Zhao,Abbas Ourmazd

doi:10.1007/s00332-019-09548-1

Abstract

We present a data-driven framework for extracting complex spatiotemporal patterns generated by ergodic dynamical systems. Our approach, called vector-valued spectral analysis (VSA), is based on an eigendecomposition of a kernel integral operator acting on a Hilbert space of vector-valued observables of the system, taking values in a space of functions (scalar fields) on a spatial domain. This operator is constructed by combining aspects of the theory of operator-valued kernels for multitask machine learning with delay-coordinate maps of dynamical systems. In contrast to conventional eigendecomposition techniques, which decompose the input data into pairs of temporal and spatial modes with a separable, tensor product structure, the patterns recovered by VSA can be manifestly non-separable, requiring only a modest number of modes to represent signals with intermittency in both space and time. Moreover, the kernel construction naturally quotients out dynamical symmetries in the data and exhibits an asymptotic commutativity property with the Koopman evolution operator of the system, enabling decomposition of multiscale signals into dynamically intrinsic patterns. Application of VSA to the Kuramoto–Sivashinsky model demonstrates significant performance gains in efficient and meaningful decomposition over eigendecomposition techniques utilizing scalar-valued kernels.

Highlights

Spatiotemporal pattern formation is ubiquitous in physical, biological, and engineered systems, ranging from molecular-scale reaction-diffusion systems, to engineering- and geophysical-scale convective flows, and astrophysical flows, among many examples (Cross and Hohenberg 1993; Ahlers et al 2009; Fung et al 2016)
We show that eigenfunctions of kernel integral operators on vector-valued observables, constructed by combining aspects of the theory of operator-valued kernels (Micchelli and Pontil 2005; Caponnetto et al 2008; Carmeli et al 2010) with delay-coordinate maps of dynamical systems (Packard et al 1980; Takens 1981; Sauer et al 1991; Robinson 2005; Deyle and Sugihara 2011): (a) Are superior to conventional algorithms in capturing signals with intermittency in both space and time; (b) Naturally incorporate any underlying dynamical symmetries, eliminating redundant modes and improving physical interpretability of the results; (c) Have a correspondence with Koopman operators, allowing detection of intrinsic dynamical timescales; and, (d) Can be stably approximated via data-driven techniques that provably converge in the asymptotic limit of large data
We have presented a method for extracting spatiotemporal patterns from complex dynamical systems, which combines aspects of the theory of operator-valued kernels for machine learning with delay-coordinate maps of dynamical systems

Summary

Introduction

Spatiotemporal pattern formation is ubiquitous in physical, biological, and engineered systems, ranging from molecular-scale reaction-diffusion systems, to engineering- and geophysical-scale convective flows, and astrophysical flows, among many examples (Cross and Hohenberg 1993; Ahlers et al 2009; Fung et al 2016). The mathematical models for such systems are generally formulated by means of partial differential equations (PDEs), or coupled ordinary differential equations, with dissipation playing an important role in the development of low-dimensional effective dynamics on attracting subsets of the state space (Constantin et al 1989) In light of this property, many pattern-forming systems are amenable to analysis by empirical, data-driven techniques, complementing the scientific understanding gained from first-principles approaches. Modulo sets of μ-measure 0, the elements of H are functions f : X → HY , such that for any dynamical state x ∈ X , f (x) is a scalar (complex-valued) field on Y , square-integrable with respect to ν For every such observable f , the map t → f ( t (x)) describes a spatiotemporal pattern generated by the dynamics.

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Conclusion