  We present a parallel data-driven strategy to identify finite-dimensional
functional spaces invariant under the Koopman operator associated to an unknown
dynamical system. We build on the Symmetric Subspace Decomposition (SSD)
algorithm, a centralized method that under mild conditions on data sampling
provably finds the maximal Koopman-invariant subspace and all Koopman
eigenfunctions in an arbitrary finite-dimensional functional space. A network
of processors, each aware of a common dictionary of functions and equipped with
a local set of data snapshots, repeatedly interact over a directed
communication graph. Each processor receives its neighbors' estimates of the
invariant dictionary and refines its estimate by applying SSD with its local
data on the intersection of the subspaces spanned by its own dictionary and the
neighbors' dictionaries. We identify conditions on the network topology to
ensure the algorithm identifies the maximal Koopman-invariant subspace in the
span of the original dictionary, characterize its time, computational, and
communication complexity, and establish its robustness against communication
failures.
