Velocity-Inferred Hamiltonian Neural Networks: Learning Energy-Conserving Dynamics from Position-Only Data

TL;DR

This paper presents a method to train Hamiltonian Neural Networks using only position data by transforming the Hamiltonian into a form involving velocity, enabling energy-conserving dynamics modeling without momentum measurements.

Contribution

The authors introduce a theoretical framework that allows Hamiltonian Neural Networks to learn from position-only data by substituting momentum with velocity under certain conditions.

Findings

Successfully applied to classical physics systems like pendulums and multi-body problems.

Achieved stable, energy-conserving long-term predictions with position-only data.

Demonstrated the method's effectiveness across multiple canonical examples.

Abstract

Data-driven modeling of physical systems often relies on learning both positions and momenta to accurately capture Hamiltonian dynamics. However, in many practical scenarios, only position measurements are readily available. In this work, we introduce a method to train a standard Hamiltonian Neural Network (HNN) using only position data, enabled by a theoretical result that permits transforming the Hamiltonian $H(q,p)$ into a form $H(q, v)$. Under certain assumptions, namely, an invertible relationship between momentum and velocity, we formally prove the validity of this substitution and demonstrate how it allows us to infer momentum from position alone. We apply our approach to canonical examples including the spring-mass system, pendulum, two-body, and three-body problems. Our results show that using only position data is sufficient for stable and energy-consistent long-term predictions, suggesting a promising pathway for data-driven discovery of Hamiltonian systems when momentum measurements are unavailable.