Controlled oscillation modeling using port-Hamiltonian neural networks

TL;DR

This paper introduces a novel approach for modeling controlled dynamical systems using port-Hamiltonian neural networks combined with a second-order discrete gradient method, improving accuracy over traditional Runge-Kutta discretizations.

Contribution

It proposes integrating a second-order discrete gradient method into port-Hamiltonian neural networks to enhance power-preserving discretizations in dynamical system modeling.

Findings

Discrete gradient method outperforms Runge-Kutta in experiments.

Method effectively models systems with different dynamical behaviors.

Regularizing the Jacobian impacts training stability.

Abstract

Learning dynamical systems through purely data-driven methods is challenging as they do not learn the underlying conservation laws that enable them to correctly generalize. Existing port-Hamiltonian neural network methods have recently been successfully applied for modeling mechanical systems. However, even though these methods are designed on power-balance principles, they usually do not consider power-preserving discretizations and often rely on Runge-Kutta numerical methods. In this work, we propose to use a second-order discrete gradient method embedded in the learning of dynamical systems with port-Hamiltonian neural networks. Numerical results are provided for three systems deliberately selected to span different ranges of dynamical behavior under control: a baseline harmonic oscillator with quadratic energy storage; a Duffing oscillator, with a non-quadratic Hamiltonian offering amplitude-dependent effects; and a self-sustained oscillator, which can stabilize in a controlled limit cycle through the incorporation of a nonlinear dissipation. We show how the use of this discrete gradient method outperforms the performance of a Runge-Kutta method of the same order. Experiments are also carried out to compare two theoretically equivalent port-Hamiltonian systems formulations and to analyze the impact of regularizing the Jacobian of port-Hamiltonian neural networks during training.