Identifying the nonlinear string dynamics with port-Hamiltonian neural networks

Hybrid machine learning combines physical knowledge with data-driven models to enhance interpretability and performance. In this context, Port-Hamiltonian Systems (PHS), which generalize Hamiltonian mechanics to describe open, non-autonomous dynamical systems, have been successfully integrated with neural networks under the name Port-Hamiltonian Neural Networks (PHNNs). While the ability of PHNNs to identify Hamiltonian ordinary differential equation (ODE) systems has already been demonstrated, their application to learning Hamiltonian partial differential equation (PDE) systems remains largely unexplored. This limitation restricts their use in musical acoustics, where instruments are typically modeled as distributed parameter systems governed by PDEs. In this work, we demonstrate how to learn the nonlinear string dynamics from data in a physically-consistent framework through a PHNN extension to PDEs. By constructing structured neural network architectures based on PHS, we can recover both the Hamiltonian governing the string and the dissipation affecting it. This approach outperforms baseline, non-physics-informed methods in terms of both accuracy and interpretability. Numerical experiments using synthetic data demonstrate the ability of the proposed PHNN model to identify and emulate the nonlinear dynamics of the system.

Controlled oscillation modeling using port-Hamiltonian neural networks

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.