TY - JOUR
T1 - Data-driven spectral decomposition and forecasting of ergodic dynamical systems
AU - Giannakis, Dimitrios
N1 - Funding Information:
The author would like to thank Tyrus Berry, Peter Constantin, Shuddho Das, John Harlim, Fanghua Lin, Andrew Majda, Lai-Sang Young, and Zhizhen Zhao for stimulating conversations on this work. This research was supported by DARPA grant HR0011-16-C-0116 , NSF grant DMS-1521775 , ONR grant N00014-14-1-0150 , ONR MURI grant 25-74200-F7112 , and ONR YIP grant N00014-16-1-2649 .
Publisher Copyright:
© 2017 The Author(s)
PY - 2019/9
Y1 - 2019/9
N2 - We develop a framework for dimension reduction, mode decomposition, and nonparametric forecasting of data generated by ergodic dynamical systems. This framework is based on a representation of the Koopman and Perron–Frobenius groups of unitary operators in a smooth orthonormal basis of the L2 space of the dynamical system, acquired from time-ordered data through the diffusion maps algorithm. Using this representation, we compute Koopman eigenfunctions through a regularized advection–diffusion operator, and employ these eigenfunctions in dimension reduction maps with projectible dynamics and high smoothness for the given observation modality. In systems with pure point spectra, we construct a decomposition of the generator of the Koopman group into mutually commuting vector fields that transform naturally under changes of observation modality, which we reconstruct in data space through a representation of the pushforward map in the Koopman eigenfunction basis. We also establish a correspondence between Koopman operators and Laplace–Beltrami operators constructed from data in Takens delay-coordinate space, and use this correspondence to provide an interpretation of diffusion-mapped delay coordinates for this class of systems. Moreover, we take advantage of a special property of the Koopman eigenfunction basis, namely that the basis elements evolve as simple harmonic oscillators, to build nonparametric forecast models for probability densities and observables. In systems with more complex spectral behavior, including mixing systems, we develop a method inspired from time change in dynamical systems to transform the generator to a new operator with potentially improved spectral properties, and use that operator for vector field decomposition and nonparametric forecasting.
AB - We develop a framework for dimension reduction, mode decomposition, and nonparametric forecasting of data generated by ergodic dynamical systems. This framework is based on a representation of the Koopman and Perron–Frobenius groups of unitary operators in a smooth orthonormal basis of the L2 space of the dynamical system, acquired from time-ordered data through the diffusion maps algorithm. Using this representation, we compute Koopman eigenfunctions through a regularized advection–diffusion operator, and employ these eigenfunctions in dimension reduction maps with projectible dynamics and high smoothness for the given observation modality. In systems with pure point spectra, we construct a decomposition of the generator of the Koopman group into mutually commuting vector fields that transform naturally under changes of observation modality, which we reconstruct in data space through a representation of the pushforward map in the Koopman eigenfunction basis. We also establish a correspondence between Koopman operators and Laplace–Beltrami operators constructed from data in Takens delay-coordinate space, and use this correspondence to provide an interpretation of diffusion-mapped delay coordinates for this class of systems. Moreover, we take advantage of a special property of the Koopman eigenfunction basis, namely that the basis elements evolve as simple harmonic oscillators, to build nonparametric forecast models for probability densities and observables. In systems with more complex spectral behavior, including mixing systems, we develop a method inspired from time change in dynamical systems to transform the generator to a new operator with potentially improved spectral properties, and use that operator for vector field decomposition and nonparametric forecasting.
KW - Diffusion maps
KW - Dynamic mode decomposition
KW - Ergodic dynamical systems
KW - Kernel methods
KW - Koopman operators
KW - Nonparametric forecasting
KW - Perron–Frobenius operators
KW - Time change
UR - http://www.scopus.com/inward/record.url?scp=85021068849&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85021068849&partnerID=8YFLogxK
U2 - 10.1016/j.acha.2017.09.001
DO - 10.1016/j.acha.2017.09.001
M3 - Article
AN - SCOPUS:85021068849
SN - 1063-5203
VL - 47
SP - 338
EP - 396
JO - Applied and Computational Harmonic Analysis
JF - Applied and Computational Harmonic Analysis
IS - 2
ER -