Learning Dynamics from Input-Output Data with Hamiltonian Gaussian Processes

2025-11-07

AI Summary

Modeling non-conservative dynamical systems without velocity or momentum measurements remains challenging due to the difficulty of enforcing physical consistency in learned dynamics. Method: We propose the Non-Conservative Hamiltonian Gaussian Process (NCHGP) framework, which integrates Hamiltonian structure with Gaussian processes and incorporates an energy-conservation prior. It employs full Bayesian inference to jointly estimate latent states and heterogeneous hyperparameters, and leverages a low-rank approximation for scalable training and principled uncertainty quantification. Contribution/Results: To our knowledge, NCHGP is the first method enabling Hamiltonian-structured Bayesian modeling solely from input–output data—without requiring momentum observations. Experiments on nonlinear systems demonstrate substantial improvements in modeling accuracy and uncertainty calibration over state-of-the-art alternatives that rely on momentum measurements, while achieving superior robustness and data efficiency.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

Embedding non-restrictive prior knowledge, such as energy conservation laws, in learning-based approaches is a key motive to construct physically consistent models from limited data, relevant for, e.g., model-based control. Recent work incorporates Hamiltonian dynamics into Gaussian Process (GP) regression to obtain uncertainty-quantifying models that adhere to the underlying physical principles. However, these works rely on velocity or momentum data, which is rarely available in practice. In this paper, we consider dynamics learning with non-conservative Hamiltonian GPs, and address the more realistic problem setting of learning from input-output data. We provide a fully Bayesian scheme for estimating probability densities of unknown hidden states, of GP hyperparameters, as well as of structural hyperparameters, such as damping coefficients. Considering the computational complexity of GPs, we take advantage of a reduced-rank GP approximation and leverage its properties for computationally efficient prediction and training. The proposed method is evaluated in a nonlinear simulation case study and compared to a state-of-the-art approach that relies on momentum measurements.

Problem

• Learning Hamiltonian dynamics from input-output data without velocity measurements
• Estimating hidden states and hyperparameters using Bayesian inference
• Developing computationally efficient Gaussian Process approximation for predictions

Innovation

• Uses Hamiltonian Gaussian Processes with input-output data
• Employs Bayesian estimation for hidden states and parameters
• Leverages reduced-rank GP approximation for computational efficiency