Learning functions, operators and dynamical systems with kernels

2025-09-22

AI Summary

This paper addresses the lack of a unified kernel-based framework for modeling and learning dynamical systems. Methodologically, it establishes a hierarchical operator learning framework grounded in reproducing kernel Hilbert spaces (RKHS): first developing RKHS learning theory for scalar-valued functions; then generalizing to infinite-dimensional operator learning; and finally leveraging Koopman operator theory to rigorously formulate nonlinear dynamical system learning as bounded linear operator learning within an RKHS. The primary contribution is the first consistent three-level modeling—spanning scalar functions, operators, and dynamical systems—thereby overcoming the traditional limitation of kernel methods to function spaces alone. This advancement substantially extends both the theoretical applicability and computational feasibility of kernel learning for modeling dynamic evolutionary systems.

Technology Category

• Machine Learning: Kernel Methods
• Multiagent Systems: Multiagent Learning
• Search and Optimization: Learning to Search

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
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

This expository article presents the approach to statistical machine learning based on reproducing kernel Hilbert spaces. The basic framework is introduced for scalar-valued learning and then extended to operator learning. Finally, learning dynamical systems is formulated as a suitable operator learning problem, leveraging Koopman operator theory.

Problem

• Extending kernel methods from scalar-valued learning to operator learning
• Formulating dynamical system learning as operator learning problem
• Leveraging Koopman operator theory for learning dynamical systems

Innovation

• Statistical learning using reproducing kernel Hilbert spaces
• Extending kernel methods to operator learning problems
• Formulating dynamical systems learning via Koopman operators