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Parallel learning of Koopman eigenfunctions and invariant subspaces for accurate long-term prediction
M. Haseli, J. Cortés
IEEE Transactions on Control of Network Systems 8 (4) (2021), 1833-1845

Abstract

This paper presents a parallel data-driven strategy to identify maximal finite-dimensional functional spaces invariant under the application of the Koopman operator associated to an unknown dynamical system. Our treatment builds on the Symmetric Subspace Decomposition (SSD) algorithm, a centralized method that 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 about the dynamics, repeatedly interact with each other over a directed communication graph. Each processor receives its neighbors' estimates of the invariant dictionary and refines its own 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 under which the P-SSD algorithm correctly identifies the maximal Koopman-invariant subspace in the span of the original dictionary, and characterize its time, computational, and communication complexity. Additionally, we show that it is robust against communication failures and packet drops. Simulations illustrate the superior performance of the proposed parallel strategy over its centralized counterpart applied to all the data available to the network.

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