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Title: Kuranet: Systems of coupled oscillators that learn to synchronize
Networks of coupled oscillators are some of the most studied objects in the theory of dynamical systems. Two important areas of current interest are the study of synchrony in highly disordered systems and the modeling of systems with adaptive network structures. Here, we present a single approach to both of these problems in the form of "KuraNet", a deep-learning-based system of coupled oscillators that can learn to synchronize across a distribution of disordered network conditions. The key feature of the model is the replacement of the traditionally static couplings with a coupling function which can learn optimal interactions within heterogeneous oscillator populations. We apply our approach to the eponymous Kuramoto model and demonstrate how KuraNet can learn data-dependent coupling structures that promote either global or cluster synchrony. For example, we show how KuraNet can be used to empirically explore the conditions of global synchrony in analytically impenetrable models with disordered natural frequencies, external field strengths, and interaction delays. In a sequence of cluster synchrony experiments, we further show how KuraNet can function as a data classifier by synchronizing into coherent assemblies. In all cases, we show how KuraNet can generalize to both new data and new network scales, making it easy to work with small systems and form hypotheses about the thermodynamic limit. Our proposed learning-based approach is broadly applicable to arbitrary dynamical systems with wide-ranging relevance to modeling in physics and systems biology.  more » « less
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  1. Abstract

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In his seminal 1950s works Krein realized that in addition to their compressing properties, network realizations can be used to embed the data back into the state space of the underlying continuum problems. In more recent works of the authors Krein's ideas gave rise to so-called nite-dierence Gaussian quadrature rules (FDGQR), allowing to approximately map the ROM state-space representation to its full order continuum counterpart on a judicially chosen grid. Thus, the state variables can be accessed directly from the transfer function without solving the full problem and even explicit knowledge of the PDE coecients in the interior, i.e., the FDGQR directly learns" the problem from its transfer function. This embedding property found applications in PDE solvers, inverse problems and unsupervised machine learning. Here we show a generalization of this approach to dissipative PDE problems, e.g., electromagnetic and acoustic wave propagation in lossy dispersive media. 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