Bayesian Transfer Operators in Reproducing Kernel Hilbert Spaces

2025-09-26

AI Summary

Existing kernel-based Koopman operator learning methods suffer from poor scalability (relying on sparse approximations), difficulty in hyperparameter and dictionary selection, and weak robustness to noise. To address these limitations, this paper proposes a Bayesian Koopman operator learning framework that unifies Gaussian process regression (GPR) and dynamic mode decomposition (DMD). It is the first approach to embed both GPR and DMD within a reproducing kernel Hilbert space (RKHS), enabling full Bayesian inference for the Koopman operator. The framework supports principled uncertainty quantification, adaptive dictionary construction, and automatic hyperparameter optimization via marginal likelihood maximization. By leveraging the probabilistic structure of GPR, the method significantly reduces computational complexity compared to standard kernelized approaches, enhances robustness to sensor noise, and improves generalization across complex nonlinear dynamical systems. Experimental results demonstrate superior performance in both accuracy and reliability under realistic measurement conditions.

Technology Category

• Machine Learning: Kernel Methods
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

The Koopman operator, as a linear representation of a nonlinear dynamical system, has been attracting attention in many fields of science. Recently, Koopman operator theory has been combined with another concept that is popular in data science: reproducing kernel Hilbert spaces. We follow this thread into Gaussian process methods, and illustrate how these methods can alleviate two pervasive problems with kernel-based Koopman algorithms. The first being sparsity: most kernel methods do not scale well and require an approximation to become practical. We show that not only can the computational demands be reduced, but also demonstrate improved resilience against sensor noise. The second problem involves hyperparameter optimization and dictionary learning to adapt the model to the dynamical system. In summary, the main contribution of this work is the unification of Gaussian process regression and dynamic mode decomposition.

Problem

• Addressing computational sparsity in kernel-based Koopman operator methods
• Improving resilience against sensor noise in dynamical system modeling
• Unifying Gaussian process regression with dynamic mode decomposition

Innovation

• Combining Gaussian process with Koopman operator
• Using kernel methods to reduce computational demands
• Unifying Gaussian process regression with dynamic mode decomposition