Learning Passive Continuous-Time Dynamics with Multistep Port-Hamiltonian Gaussian Processes

2025-09-30

AI Summary

How can physically consistent continuous-time dynamics and a posterior distribution over Hamiltonian functions be learned from noisy, irregularly sampled trajectory data while strictly enforcing energy conservation and passivity constraints? This paper proposes the multi-step port-Hamiltonian Gaussian process (MS-PHS GP): it places a Gaussian process prior on the Hamiltonian function and encodes physical constraints—derived from a variable-step, multi-step numerical integrator—into linear functionals. This enables closed-form conditional inference over both the vector field and the Hamiltonian surface, without latent variables, thereby inherently guaranteeing energy balance and port-Hamiltonian structure. Leveraging finite-sample error analysis and discretization-aware constraints, the method achieves significantly improved vector field recovery accuracy on benchmark systems—including the harmonic oscillator, van der Pol oscillator, and Duffing oscillator—and provides well-calibrated uncertainty quantification over the Hamiltonian.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments

Abstract

We propose the multistep port-Hamiltonian Gaussian process (MS-PHS GP) to learn physically consistent continuous-time dynamics and a posterior over the Hamiltonian from noisy, irregularly-sampled trajectories. By placing a GP prior on the Hamiltonian surface $H$ and encoding variable-step multistep integrator constraints as finite linear functionals, MS-PHS GP enables closed-form conditioning of both the vector field and the Hamiltonian surface without latent states, while enforcing energy balance and passivity by design. We state a finite-sample vector-field bound that separates the estimation and variable-step discretization terms. Lastly, we demonstrate improved vector-field recovery and well-calibrated Hamiltonian uncertainty on mass-spring, Van der Pol, and Duffing benchmarks.

Problem

• Learning physically consistent continuous-time dynamics from noisy data
• Estimating Hamiltonian posterior with energy balance constraints
• Improving vector-field recovery on nonlinear dynamical systems

Innovation

• Learning dynamics with multistep port-Hamiltonian Gaussian processes
• Encoding integrator constraints as finite linear functionals
• Enforcing energy balance and passivity by design