Bayesian Transfer Operators in Reproducing Kernel Hilbert Spaces

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.

💡 Research Summary

The paper presents a unified Bayesian framework that combines kernel-based dynamic mode decomposition (DMD) with Gaussian process (GP) regression to address two major challenges in kernel Koopman methods: computational scalability and hyper‑parameter/dictionary learning. By interpreting the Koopman and Perron–Frobenius operators as random variables in a reproducing kernel Hilbert space (RKHS), the authors derive a probabilistic model where the lifted observables are treated as GP‑distributed functions. The prior over the embedded Perron–Frobenius operator is a zero‑mean GP with a covariance constructed from a kernel on the state space and a cross‑covariance operator that captures measurement noise.

To make the approach tractable for large datasets, the authors adopt the variational free‑energy (VFE) method, a sparse GP technique introduced by Titsias (2009). This reduces the cubic complexity of exact kernel DMD (O(N³) for N snapshots) to O(NM²) or O(M³) by introducing a set of M « N pseudo‑inputs (inducing points) that form a compact dictionary. The resulting posterior mean has the same algebraic form as the classic Extended DMD (EDMD) estimator, while the posterior covariance provides a natural quantification of uncertainty for multi‑step predictions.

The paper also discusses how the Bayesian formulation automatically incorporates hyper‑parameter optimization (e.g., kernel length‑scale, noise variance) through maximization of the variational lower bound, eliminating the need for separate cross‑validation or heuristic tuning. Moreover, the sparse dictionary learning is achieved by optimizing the locations of the inducing points, effectively learning a compact set of basis functions that best explain the data.

Experimental results on synthetic chaotic maps, fluid‑flow snapshots, and noisy robotic arm trajectories demonstrate that the proposed Bayesian Transfer Operator (BTO) method achieves comparable or superior prediction accuracy to standard kernel DMD while requiring significantly less computation time. The uncertainty estimates produced by the posterior covariance reliably envelope the true prediction errors, highlighting the practical benefit of the probabilistic treatment.

In summary, the authors provide a theoretically sound and computationally efficient bridge between Koopman operator theory, kernel methods, and Gaussian process regression. Their Bayesian Transfer Operator framework not only scales kernel DMD to high‑dimensional, noisy datasets but also delivers calibrated uncertainty quantification and principled hyper‑parameter learning, paving the way for more robust data‑driven modeling of nonlinear dynamical systems.