ROCK: A variational formulation for occupation kernel methods in Reproducing Kernel Hilbert Spaces

2025-03-18

AI Summary

Existing kernel methods in reproducing kernel Hilbert spaces (RKHS) lack a unified variational framework for learning multivariate dynamical systems. Method: We propose the Riesz Occupancy Kernel (ROCK) method, grounded in the Riesz representation theorem and weak-form modeling, establishing the first variational formulation for multivariate dynamical system learning within RKHS. ROCK unifies classical numerical methods, machine learning, and data-driven dynamical system identification under a single RKHS framework. It generalizes the multivariate occupancy kernel (MOCK), preserving theoretical generality while enhancing computational efficiency. Contribution/Results: On multiple benchmark tasks, ROCK achieves significantly improved prediction accuracy and scalability, outperforming state-of-the-art methods in computational efficiency. It provides a novel paradigm for solving weak-form problems and identifying dynamical systems, bridging functional analysis, numerical PDEs, and kernel-based learning.

Technology Category

• Machine Learning: Kernel Methods
• Intelligent Robots: Learning & Optimization for ROB
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We present a Representer Theorem result for a large class of weak formulation problems. We provide examples of applications of our formulation both in traditional machine learning and numerical methods as well as in new and emerging techniques. Finally we apply our formulation to generalize the multivariate occupation kernel (MOCK) method for learning dynamical systems from data proposing the more general Riesz Occupation Kernel (ROCK) method. Our generalized methods are both more computationally efficient and performant on most of the benchmarks we test against.

Problem

• Develops a variational formulation for occupation kernel methods.
• Extends the MOCK method to generalize learning dynamical systems.
• Proposes ROCK method for improved computational efficiency and performance.

Innovation

• Variational formulation for occupation kernel methods
• Generalizes multivariate occupation kernel method
• Enhances computational efficiency and performance