Kernel Methods for Stochastic Dynamical Systems with Application to Koopman Eigenfunctions: Feynman-Kac Representations and RKHS Approximation

2026-03-01

AI Summary

This work addresses the challenge of effectively extending kernel-based methods—originally developed for deterministic dynamical systems—to stochastic differential equations (SDEs) for approximating eigenfunctions of the Koopman operator. By leveraging the Feynman–Kac path integral representation, the study unifies three distinct kernel constructions—variational principles, Green’s function convolutions, and resolvent operators—into a coherent framework for stochastic systems with diffusion, establishing a corresponding reproducing kernel Hilbert space (RKHS) approximation scheme. Theoretically, under uniform ellipticity, these three approaches are shown to be equivalent, revealing that diffusion enhances numerical conditioning through elliptic regularization. The analysis further provides error bounds that separate RKHS approximation error from Monte Carlo sampling error. Numerical experiments on the Ornstein–Uhlenbeck process, nonlinear SDEs, and high-dimensional systems demonstrate the method’s efficacy, showing that moderate diffusion significantly improves numerical stability.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We extend the unified kernel framework for transport equations and Koopman eigenfunctions, developed in previous work by the authors for deterministic systems, to stochastic differential equations (SDEs). In the deterministic setting, three analytically grounded constructions-Lions-type variational principles, Green's function convolution, and resolvent operators along characteristic flows--were shown to yield identical reproducing kernels. For stochastic systems, the Koopman generator includes a second-order diffusion term, transforming the first-order hyperbolic transport equation into a second-order elliptic-parabolic PDE. This fundamental change necessitates replacing the method of characteristics with probabilistic representations based on the Feynman--Kac formula. Our main contributions include: (i) extension of all three kernel constructions to stochastic systems via Feynman--Kac path-integral representations; (ii) proof of kernel equivalence under uniform ellipticity assumptions; (iii) a collocation-based computational framework incorporating second-order differential operators; (iv) error bounds separating RKHS approximation error from Monte Carlo sampling error; (v) analysis of how diffusion affects numerical conditioning; and (vi) connections to generator EDMD, diffusion maps, and kernel analog forecasting. Numerical experiments on Ornstein--Uhlenbeck processes, nonlinear SDEs with varying diffusion strength, and multi-dimensional systems validate the theoretical developments and demonstrate that moderate diffusion can improve numerical stability through elliptic regularization.

Problem

• stochastic dynamical systems
• Koopman eigenfunctions
• kernel methods
• Feynman-Kac formula
• RKHS approximation

Innovation

• Feynman-Kac representation
• reproducing kernel Hilbert space (RKHS)
• Koopman eigenfunctions
• stochastic differential equations
• elliptic regularization