Hamiltonian-based neural networks for systems under nonholonomic constraints

TL;DR

This paper develops a modified Hamiltonian neural network architecture to model systems with nonholonomic constraints, extending the applicability of physics-informed neural networks to more complex mechanical systems.

Contribution

It introduces a three-network architecture that learns Hamiltonian functions, constraints, and multipliers for systems under holonomic and nonholonomic constraints.

Findings

Successfully models a rolling disk system.

Accurately predicts a ball on a spinning table.

Performs well even with noisy training data.

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.