Kernel Methods for Some Transport Equations with Application to Learning Kernels for the Approximation of Koopman Eigenfunctions: A Unified Approach via Variational Methods, Green's Functions and the Method of Characteristics

2026-03-06

AI Summary

This work addresses the linear transport partial differential equation satisfied by eigenfunctions of the Koopman operator and proposes a tailored reproducing kernel construction together with an efficient approximation method. By unifying variational principles, Green’s functions, and the method of characteristics within a reproducing kernel Hilbert space (RKHS), the study establishes—for the first time—their equivalence under this framework and leverages it to construct Mercer kernels adapted to nonlinear dynamical systems. A multi-kernel learning mechanism combined with boundary-regularized convex optimization is introduced to enable joint adaptive learning of the kernel and eigenfunctions. Theoretical analysis demonstrates that the constructed kernel converges in the L² sense to the true eigenfunction, while numerical experiments confirm the method’s robustness and broad applicability, particularly in handling eigenfunction blow-up phenomena.

Technology Category

• Machine Learning: Kernel Methods
• Search and Optimization: Learning to Search
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Social networks and social learning
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We present a unified theoretical and computational framework for constructing reproducing kernels tailored to transport equations and adapted to Koopman eigenfunctions of nonlinear dynamical systems. These eigenfunctions satisfy a transport-type partial differential equation (PDE) that we invert using three analytically grounded methods: (i) A Lions-type variational principle in a reproducing kernel Hilbert space (RKHS), (ii) convolution with a Green's function, and (iii) a resolvent operator constructed via Laplace transforms along characteristic flows. We prove that these three constructions yield identical kernels under mild smoothness and causality assumptions. We further show that the associated kernel eigenfunctions (Mercer modes) converge in L^2 to true Koopman eigenfunctions when the latter lie in the RKHS. Our approach is numerically realized through a mesh-free, convex optimization framework, enhanced with boundary regularization to handle eigenfunction blow-up. A multiple-kernel learning (MKL) scheme selects kernels automatically via residual minimization. Finally, we demonstrate that the same framework applies verbatim to a broader class of linear transport PDEs, including the advection, continuity, and Liouville equations. The unification of variational principles, Green's functions, and the method of characteristics enables the development of novel schemes for approximating eigenfunctions of transport equations, including those of the Koopman operator, and introduces a data-driven approach for learning kernels tailored to these approximations. Numerical experiments confirm the practical utility and robustness of the method.

Problem

• transport equations
• Koopman eigenfunctions
• reproducing kernels
• kernel learning
• linear PDEs

Innovation

• Koopman eigenfunctions
• reproducing kernel Hilbert space
• transport equations
• multiple-kernel learning
• method of characteristics