Learning Linear Embeddings for Non-Linear Network Dynamics with Koopman
  Message Passing
King Fai Yeh Paris Flood William Redman Pietro Liò
Recently, Koopman operator theory has become a powerful tool for developing
linear representations of non-linear dynamical systems. However, existing
data-driven applications of Koopman operator theory, including both traditional
and deep learning approaches, perform poorly on non-linear network dynamics
problems as they do not address the underlying geometric structure. In this
paper we present a novel approach based on Koopman operator theory and message
passing networks that finds a linear representation for the dynamical system
which is globally valid at any time step. The linearisations found by our
method produce predictions on a suite of network dynamics problems that are
several orders of magnitude better than current state-of-the-art techniques. We
also apply our approach to the highly non-linear training dynamics of neural
network architectures, and obtain linear representations which can generate
network parameters with comparable performance to networks trained by classical
optimisers.