Koopman-Equivariant Gaussian Processes

TL;DR

This paper introduces a Gaussian process framework for dynamical systems that leverages Koopman equivariance, enabling accurate, scalable forecasting and uncertainty quantification with improved generalization over existing kernel methods.

Contribution

It proposes a novel GP model incorporating Koopman equivariance for better generalization and tractability in learning dynamical systems with linear responses.

Findings

Achieves comparable or better forecasting accuracy than kernel-based methods.

Provides a tractable probabilistic approach to uncertainty quantification.

Enables large-scale regression through variational inference with inducing points.

Abstract

Credible forecasting and representation learning of dynamical systems are of ever-increasing importance for reliable decision-making. To that end, we propose a family of Gaussian processes (GP) for dynamical systems with linear time-invariant responses, which are nonlinear only in initial conditions. This linearity allows us to tractably quantify forecasting and representational uncertainty, simultaneously alleviating the challenge of computing the distribution of trajectories from a GP-based dynamical system and enabling a new probabilistic treatment of learning Koopman operator representations. Using a trajectory-based equivariance -- which we refer to as \textit{Koopman equivariance} -- we obtain a GP model with enhanced generalization capabilities. To allow for large-scale regression, we equip our framework with variational inference based on suitable inducing points. Experiments demonstrate on-par and often better forecasting performance compared to kernel-based methods for learning dynamical systems.