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Recently, Koopman operator theory has become a powerful tool for developinglinear representations of non-linear dynamical systems. However, existingdata-driven applications of Koopman operator theory, including both traditionaland deep learning approaches, perform poorly on non-linear network dynamicsproblems as they do not address the underlying geometric structure. In thispaper we present a novel approach based on Koopman operator theory and messagepassing networks that finds a linear representation for the dynamical systemwhich is globally valid at any time step. The linearisations found by ourmethod produce predictions on a suite of network dynamics problems that areseveral orders of magnitude better than current state-of-the-art techniques. Wealso apply our approach to the highly non-linear training dynamics of neuralnetwork architectures, and obtain linear representations which can generatenetwork parameters with comparable performance to networks trained by classicaloptimisers.

Learning Linear Embeddings for Non-Linear Network Dynamics with Koopman Message Passing