GeoHNNs: Geometric Hamiltonian Neural Networks

2025-07-21

AI Summary

Existing machine learning approaches for modeling complex dynamical systems neglect underlying physical geometric structures, leading to instability in long-term predictions—particularly for high-dimensional and chaotic systems. To address this, we propose a geometry-preserving Hamiltonian learning framework that jointly embeds two fundamental geometric constraints into neural networks: (i) the symmetric positive-definite manifold structure of the inertia matrix, and (ii) symplectic volume conservation in phase space. We parameterize the inertia matrix via manifold-constrained optimization and enforce strict symplecticity in a reduced-dimensional latent phase space using a constrained autoencoder. By unifying geometric machine learning with physics-guided modeling, our method significantly improves long-term stability, energy conservation, and prediction accuracy on benchmark systems—including coupled oscillators and deformable bodies. This work establishes a new paradigm for interpretable and robust modeling of high-dimensional chaotic systems.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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 extit{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.

Problem

• Modeling physical dynamics while preserving geometric principles
• Addressing instability in long-term predictions of chaotic systems
• Ensuring energy conservation in high-dimensional physical systems

Innovation

• Encodes geometric priors in physical laws
• Parameterizes inertia matrices as symmetric positive-definite
• Uses constrained autoencoder for phase space preservation