Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Boumediene Hamzi, Houman Owhadi, Umesh Vaidya

TL;DR

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.

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.

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

TL;DR

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.

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.
Paper Structure (53 sections, 19 theorems, 201 equations, 4 figures, 3 tables)

This paper contains 53 sections, 19 theorems, 201 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Our contributions.
  3. Mathematical Foundations
  4. Lions’s Variational Approach and RKHS
  5. Lions's Variational Approach for Transport PDEs
  6. A Unified Framework for Kernel Construction: From Koopman PDEs to RKHS Representations
  7. Recovery of Eigenfunctions and Kernel Construction via Green’s Function and Characteristics
  8. Green’s Function Method
  9. Method of Characteristics.
  10. Spectral Decomposition of the Kernel and Koopman Modes
  11. Connection to Lions’s Variational Framework
  12. Equivalence of the Different Approaches
  13. Variational Approximation of Koopman Eigenfunctions
  14. Lions-Based Variational Optimization
  15. Lions’s Variational Principle.
  16. ...and 38 more sections

Key Result

Theorem 3.1

Let $\Omega \subset \mathbb{R}^d$ be a bounded domain with $C^{1,\alpha}$ boundary, and let $f \in C^1(\Omega; \mathbb{R}^d)$ generate a complete flow $s_t$. Fix $\lambda \in \mathbb{C}$ and define the transport operator Assume the following: Then the retarded Green’s function $G(x, \xi)$ solving with causality imposed by the flow from $\Sigma$, exists and is smooth off the diagonal $x = \xi$.

Figures (4)

  • Figure 1: Comparison of Koopman eigenfunction approximations using singular vs. RBF kernels, both with boundary penalties.
  • Figure 2: Approximation of Koopman eigenfunction $\phi$ using MKL. Top row: initial, learned, and rescaled $\phi$. Bottom row: true $\phi$, pointwise rescaled error.
  • Figure 3: Approximation of second Koopman eigenfunction $\phi_{\lambda_2}(x)$ using MKL. Top row: initial, learned, and rescaled $\phi$. Bottom row: true $\phi$, pointwise rescaled error.
  • Figure 4: Surface and heatmap visualizations of the Koopman eigenfunction $\phi_{\lambda{+}}(x)$.

Theorems & Definitions (38)

  • Theorem 3.1: Existence and Regularity of Green’s Function Kernel
  • proof
  • Theorem 3.2: Resolvent Kernel
  • proof
  • Lemma 3.3: Mercer Eigenfunctions Equal Koopman Modes
  • proof
  • Theorem 3.4: Spectral Convergence of Mercer Modes to Koopman Eigenfunctions
  • proof
  • Theorem 3.5: Spectral Expansion and Koopman Eigenfunctions
  • proof
  • ...and 28 more