Physics-Constrained Learning for PDE Systems with Uncertainty Quantified Port-Hamiltonian Models

TL;DR

The paper addresses the challenge of data-driven PDE modeling for nonlinear, flexible systems by introducing GP-dPHS, a physics-constrained learning framework that blends distributed Port-Hamiltonian systems with Gaussian process regression. It learns an unknown Hamiltonian while treating operator parameters as hyperparameters, ensuring energy conservation/dissipation through a dPHS-based kernel and providing principled uncertainty quantification. Data is generated from time-space observations, upsampled spatial points, and derivative information to train a GP that defines a distribution over admissible, energy-consistent PDE models. A nonlinear string PDE example demonstrates accurate mean predictions and credible uncertainty bounds, highlighting robust generalization from sparse data. The approach offers a pathway to energy-aware, reliable PDE modeling with potential applications in safe control of complex physical systems.

Abstract

Modeling the dynamics of flexible objects has become an emerging topic in the community as these objects become more present in many applications, e.g., soft robotics. Due to the properties of flexible materials, the movements of soft objects are often highly nonlinear and, thus, complex to predict. Data-driven approaches seem promising for modeling those complex dynamics but often neglect basic physical principles, which consequently makes them untrustworthy and limits generalization. To address this problem, we propose a physics-constrained learning method that combines powerful learning tools and reliable physical models. Our method leverages the data collected from observations by sending them into a Gaussian process that is physically constrained by a distributed Port-Hamiltonian model. Based on the Bayesian nature of the Gaussian process, we not only learn the dynamics of the system, but also enable uncertainty quantification. Furthermore, the proposed approach preserves the compositional nature of Port-Hamiltonian systems.

Figures (4)

• Figure 1: Gaussian Process Distributed Port-Hamiltonian (GP-dPHS) systems allow to learn a probabilistic spatial-temporal Hamiltonian based on collected data. The Port-Hamiltonian structure enables the inclusion of control inputs as well as dissipation. All sampled models of the GP-dPHS model are physically correct in terms of the evolution of the total energy in the system.
• Figure 2: Example of states observations and the upsampling process.
• Figure 3: Graphical illustration of string PDE system. Observe on the left figure of each time slice; the deflection learned by GP-dPHS (cyan dotted) is close to the ground-truth trajectory on each time step t. The mean prediction of time slice $t = 600$ and five samples of the GP-dPHS model are shown on the right.
• Figure 4: The left figure visualizes the nonlinear Hamiltonian derivative $\delta_q H$. The learned surface (red) is very close to the true surface (blue). The pink surfaces visualize the uncertainty of the Hamiltonian derivative under a 95% prediction uncertainty guarantee. The figure on the right shows the model error for different numbers of training points

Theorems & Definitions (5)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Remark 1