Data-efficient Kernel Methods for Learning Hamiltonian Systems

2025-09-21

AI Summary

This work addresses the challenges of modeling Hamiltonian systems and preserving conservation laws under data scarcity. We propose a nonparametric learning framework based on kernel methods that jointly infers trajectories and the underlying Hamiltonian function as a dynamics-constrained functional regression problem. We introduce two novel kernel estimators—one-step and two-step—that guarantee exact conservation of energy and other invariants while enabling high-accuracy predictions. Theoretically, we derive a priori error bounds for the estimator; methodologically, the framework naturally extends to general autonomous dynamical systems. Experiments on benchmark Hamiltonian systems—including the harmonic oscillator, double pendulum, and three-body problem—demonstrate that our approach significantly outperforms conventional sequential modeling strategies. Notably, it maintains stable, high-fidelity long-term predictions even under extremely sparse initial conditions (<10 samples), highlighting its robustness and efficiency in low-data regimes.

Technology Category

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

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

Hamiltonian dynamics describe a wide range of physical systems. As such, data-driven simulations of Hamiltonian systems are important for many scientific and engineering problems. In this work, we propose kernel-based methods for identifying and forecasting Hamiltonian systems directly from data. We present two approaches: a two-step method that reconstructs trajectories before learning the Hamiltonian, and a one-step method that jointly infers both. Across several benchmark systems, including mass-spring dynamics, a nonlinear pendulum, and the Henon-Heiles system, we demonstrate that our framework achieves accurate, data-efficient predictions and outperforms two-step kernel-based baselines, particularly in scarce-data regimes, while preserving the conservation properties of Hamiltonian dynamics. Moreover, our methodology provides theoretical a priori error estimates, ensuring reliability of the learned models. We also provide a more general, problem-agnostic numerical framework that goes beyond Hamiltonian systems and can be used for data-driven learning of arbitrary dynamical systems.

Problem

• Learning Hamiltonian systems directly from data using kernel methods
• Achieving accurate predictions with limited data while preserving conservation properties
• Providing a general framework for data-driven learning of dynamical systems

Innovation

• Kernel-based methods for Hamiltonian system identification
• One-step joint inference of trajectories and Hamiltonians
• Data-efficient predictions with theoretical error guarantees