PFEM-GP-dpHs : a finite element framework for combining Gaussian processes and infinite-dimensional port-Hamiltonian systems

TL;DR

This paper introduces PFEM-GP-dpHs, a finite element framework that combines Gaussian processes with infinite-dimensional port-Hamiltonian systems to efficiently learn and model nonlinear distributed systems.

Contribution

It proposes a novel combination of finite element methods and Gaussian processes for learning port-Hamiltonian systems, reducing computational complexity and focusing on nonlinear dynamics.

Findings

Successfully applied to a nonlinear wave equation with unknown parameters

Reduces numerical complexity of Gaussian process-based system learning

Enables modeling of nonlinear distributed port-Hamiltonian systems

Abstract

In order to learn distributed port-Hamiltonian systems (dpHs) using Gaussian processes (GPs), the partitioned finite element method (PFEM) is combined with the Gp-dpHs method. By following a late lumping approach, the discretization of the functional hyperparameters of the GP prior over the Hamiltonian functional is chosen independently from the discretization of the dpHs, thus reducing the numerical complexity of our method. We next model the mean of the GP prior of the Hamiltonian as a quadratic form, enabling the GP kernel to focus on the nonlinear part of a given dpHs. We illustrate our method on a nonlinear one dimensional wave equation with unknown physical parameters (tension and linear mass).