Controlled oscillation modeling using port-Hamiltonian neural networks

TL;DR

This work proposes to use a second-order discrete gradient method embedded in the learning of dynamical systems with port-Hamiltonian neural networks and shows how the use of this discrete gradient method outperforms the performance of a Runge-Kutta method of the same order.

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.

Figures (12)

• Figure 1: Architecture of the two PHNN models considered in this work. White boxes with orange contour denote fixed algebraic operations whereas orange boxes indicate the trainable parameters.
• Figure 2: Training and inference diagram for the continuous models $\bm f_{\theta}$.
• Figure 3: Training and inference diagram for the discrete models $\bm g_{\theta}$.
• Figure 4: Training points, test initial points and two complete test trajectories for each of the three oscillatory systems. Note that in the case of the harmonic and Duffing oscillator, the applied control shifted the equilibrium point from $(p,q)=(0,0)$ whereas in the case of the self-sustained oscillator, it stabilizes the trajectories in a limit cycle.
• Figure 5: Schematic sampling and dataset construction procedure. A trajectory generated at sampling frequency ${sr}_{gen}$ over a duration $D=D_{infer}=\beta T_0$ is shown as white dot markers. From this trajectory, a training point, highlighted with a green star marker, is uniformly sampled from a subset of samples, shown as black dot markers, obtained at frequency ${sr}_{train}$ and restricted to $t\leq\alpha T_0$. The training horizon $D_{train}$ and the inference horizon $D_{infer}$ are indicated by arrows, with the vertical dashed line marking the end of the training interval.

Theorems & Definitions (5)

• Theorem 1
• Definition 1: Passivity, khalil_nonlinear_2002
• Proposition 1
• Proposition 2
• Proposition 3