Learning Dynamics from Input-Output Data with Hamiltonian Gaussian Processes

TL;DR

This work advances physics-informed learning by enabling uncertainty-quantified dynamics modeling from input-output data using non-conservative Hamiltonian Gaussian Processes. It preserves energy-related structure through a gradient-based Hamiltonian represented by a reduced-rank GP, achieving computational efficiency with complexity O(M) and linear-in-parameters learning. The authors develop a fully Bayesian inference scheme based on pmcmc with particle Gibbs, offering closed-form parameter updates and MH-based hyperparameter learning for kernel and structural terms, while providing dense posterior characterizations of hidden states and energies. Demonstrations on nonlinear simulations show physically consistent predictions and competitive accuracy compared to state-of-the-art momentum-based methods, even when only IO data are available. The approach holds promise for data-efficient, uncertainty-aware model-based control in practical settings where velocity or momentum measurements are unavailable.

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.

Figures (5)

• Figure 1: Test system & Hamiltonian.
• Figure 2: Flow maps following from the true and learned Hamiltonians. The reduced-rank Hamiltonian gp yields similar approximation accuracy while allowing for computationally efficient prediction. Despite only having access to input-output data, the proposed method (right) enables learning a Hamiltonian gp with accuracy comparable to methods with access to full-state measurements.
• Figure 3: True and estimated system behavior in the training data set (left), as well as density estimates for the gp kernel and structural hyperparameters $\boldsymbol{\vartheta} = \{ \boldsymbol{\vartheta}_{\mathrm{K}}, {\vartheta}_{\mathrm{S}} \}$ (right). From input-output data, the proposed method infers densities of unknown hidden states, the current system energy, and hyperparameters in a fully Bayesian fashion.
• Figure 4: True system behavior and forward predictions of the learned Hamiltonian gp model. The proposed method provides a probabilistic Hamiltonian system, and---despite learning only from input-output data---each sampled model yields a physically consistent prediction that resembles the actual system behavior.
• Figure 5: Flow maps following from the true and learned Hamiltonians of the non-harmonic oscillator system.

Theorems & Definitions (2)

• remark 1
• remark 2