Learning Linear Embeddings for Non-Linear Network Dynamics with Koopman Message Passing
This addresses the challenge of accurately predicting non-linear network dynamics, which is incremental as it builds on existing Koopman operator theory by incorporating geometric structure.
The paper tackles the problem of modeling non-linear network dynamics by developing a linear representation using Koopman operator theory and message passing networks, achieving predictions several orders of magnitude better than state-of-the-art techniques and generating neural network parameters with performance comparable to classical optimizers.
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.