  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 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.
