Technology Category
Machine Learning: Calibration & Uncertainty QuantificationReasoning under Uncertainty: Relational Probabilistic ModelsIntelligent Robots: State Estimation
Koopman-Equivariant Gaussian Processes
🤖 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
Application Category
📝 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
Research questions and friction points this paper is trying to address.
Forecasting dynamical systems accurately
Learning Koopman operator representations probabilistically
Enhancing generalization with Koopman equivariance
Innovation
Methods, ideas, or system contributions that make the work stand out.
Koopman-Equivariant Gaussian Processes
Variational inference for scalability
Trajectory-based equivariance enhancement
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