TY - JOUR
T1 - Spatiotemporal Pattern Extraction by Spectral Analysis of Vector-Valued Observables
AU - Giannakis, Dimitrios
AU - Ourmazd, Abbas
AU - Slawinska, Joanna
AU - Zhao, Zhizhen
N1 - Funding Information:
D.G. acknowledges support from NSF EAGER Grant 1551489, ONR YIP Grant N00014-16-1-2649, NSF Grant DMS-1521775, and DARPA Grant HR0011-16-C-0116. J.S. and A.O. acknowledge support from NSF EAGER Grant 1551489. Z.Z. received support from NSF Grant DMS-1521775. We thank Shuddho Das for stimulating conversations.
Publisher Copyright:
© 2019, The Author(s).
PY - 2019/10/1
Y1 - 2019/10/1
N2 - 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.
AB - 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.
KW - Dynamical symmetries
KW - Dynamical systems
KW - Kernel methods
KW - Koopman operators
KW - Spatiotemporal patterns
KW - Spectral decomposition
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U2 - 10.1007/s00332-019-09548-1
DO - 10.1007/s00332-019-09548-1
M3 - Article
AN - SCOPUS:85065716938
SN - 0938-8974
VL - 29
SP - 2385
EP - 2445
JO - Journal of Nonlinear Science
JF - Journal of Nonlinear Science
IS - 5
ER -