Hamiltonian-based neural networks for systems under nonholonomic constraints

TL;DR

A modified Hamiltonian neural network architecture capable of modeling Hamiltonian systems under holonomic and nonholonomic constraints is developed and a three-network parallel architecture is proposed to simultaneously learn the Hamiltonian of the system, the constraints, and their associated multiplier.

Abstract

There has been increasing interest in methodologies that incorporate physics priors into neural network architectures to enhance their modeling capabilities. A family of these methodologies that has gained traction are Hamiltonian neural networks (HNN) and their variations. These architectures explicitly encode Hamiltonian mechanics both in their structure and loss function. Although Hamiltonian systems under nonholonomic constraints are in general not Hamiltonian, it is possible to formulate them in pseudo-Hamiltonian form, equipped with a Lie bracket which is almost Poisson. This opens the possibility of using some principles of HNNs in systems under nonholonomic constraints. The goal of the present work is to develop a modified Hamiltonian neural network architecture capable of modeling Hamiltonian systems under holonomic and nonholonomic constraints. A three-network parallel architecture is proposed to simultaneously learn the Hamiltonian of the system, the constraints, and their associated multipliers. A rolling disk and a ball on a spinning table are considered as canonical examples to assess the performance of the proposed Hamiltonian architecture. The experiments are then repeated with a noisy training set to study modeling performance under more realistic conditions.

Figures (11)

• Figure 1: Proposed architecture to learn Hamiltonian systems under holonomic and nonholonomic constraints. Three independent networks are used to learn the Hamiltonian, the constraints through $\textbf{A}$ and $\textbf{b}$, and the multipliers in the vector $\boldsymbol{\lambda}$.
• Figure 2: a) The coordinates $x$ and $y$ indicate the position of the point of contact between the disk and the surface, $\phi$ represents the rolling angle of the disk about its symmetry axis, and $\psi$ is the angle of rotation with respect to the vertical axis. b) $\theta$ is the leaning angle of the disk from its fully vertical position.
• Figure 3: Trajectories and deviations for the rolling disk system.
• Figure 4: The position of the ball with respect to the center of the table is given by the vector $\textbf{r}=(x,y)$. The table spins at constant angular velocity $\Omega$ and is considered to be infinite in size.
• Figure 5: Trajectories and deviations for the solid ball on spinning table.