Koopman-Equivariant Gaussian Processes

2025-02-10

AI Summary

This work addresses the problem of probabilistic modeling and prediction for dynamical systems that exhibit nonlinear dependence on initial conditions but linear time responses. We propose KoGP—the first Gaussian process (GP) model incorporating Koopman equivariance—to enable trustworthy predictions and probabilistic representation learning. Methodologically, we design a Koopman-equivariant kernel and integrate variational inference with inducing-point approximations, enforcing equivariance constraints directly at the trajectory level to yield analytically tractable predictive uncertainty. Compared to standard GPs and kernel-based methods, KoGP matches or improves prediction accuracy across multiple dynamical system benchmarks while substantially enhancing generalization and uncertainty calibration. Our core contribution is the first principled integration of Koopman equivariance into the GP framework, establishing a novel paradigm for probabilistic dynamical system modeling that unifies physical interpretability with statistical rigor.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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 extit{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.

Problem

• Forecasting dynamical systems accurately
• Learning Koopman operator representations probabilistically
• Enhancing generalization with Koopman equivariance

Innovation

• Koopman-Equivariant Gaussian Processes
• Variational inference for scalability
• Trajectory-based equivariance enhancement