GeoHNNs: Geometric Hamiltonian Neural Networks

TL;DR

GeoHNNs are a novel neural network framework that explicitly encodes geometric principles of physics, such as Riemannian and symplectic geometry, to improve the stability and accuracy of modeling physical systems.

Contribution

The paper introduces GeoHNNs, which incorporate geometric priors into neural networks by parameterizing inertia matrices and preserving phase space volume, advancing physics-informed modeling.

Findings

Outperforms existing models in stability and accuracy

Achieves superior energy conservation in experiments

Effective on systems from oscillators to deformable objects

Abstract

The fundamental laws of physics are intrinsically geometric, dictating the evolution of systems through principles of symmetry and conservation. While modern machine learning offers powerful tools for modeling complex dynamics from data, common methods often ignore this underlying geometric fabric. Physics-informed neural networks, for instance, can violate fundamental physical principles, leading to predictions that are unstable over long periods, particularly for high-dimensional and chaotic systems. Here, we introduce \textit{Geometric Hamiltonian Neural Networks (GeoHNN)}, a framework that learns dynamics by explicitly encoding the geometric priors inherent to physical laws. Our approach enforces two fundamental structures: the Riemannian geometry of inertia, by parameterizing inertia matrices in their natural mathematical space of symmetric positive-definite matrices, and the symplectic geometry of phase space, using a constrained autoencoder to ensure the preservation of phase space volume in a reduced latent space. We demonstrate through experiments on systems ranging from coupled oscillators to high-dimensional deformable objects that GeoHNN significantly outperforms existing models. It achieves superior long-term stability, accuracy, and energy conservation, confirming that embedding the geometry of physics is not just a theoretical appeal but a practical necessity for creating robust and generalizable models of the physical world.