Learning Hamiltonian Dynamics at Scale: A Differential-Geometric Approach

2025-09-29

AI Summary

To address the poor scalability of Hamiltonian Neural Networks (HNNs) in high-dimensional physical systems—and the inherent trade-off between energy conservation and computational efficiency—this paper proposes the Reduced-Order Hamiltonian Neural Network (RO-HNN). RO-HNN is the first framework to tightly couple a symplectic autoencoder with a geometric Hamiltonian network: the former learns a low-dimensional symplectic submanifold within a differential-geometric framework, rigorously preserving phase-space geometry; the latter models symplectic dynamics on this manifold while incorporating symmetry preservation and model-order reduction. Experiments across diverse high-dimensional physical systems—including gravitational N-body problems and nonlinear wave equations—demonstrate that RO-HNN achieves high-accuracy, numerically stable, and strongly generalizable dynamical predictions. It significantly improves long-term simulation fidelity while reducing computational cost by an order of magnitude, thereby extending the applicability of HNNs to complex, large-scale physical modeling.

Technology Category

• Machine Learning: Learning with Manifolds
• Intelligent Robots: Learning & Optimization for ROB
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

By embedding physical intuition, network architectures enforce fundamental properties, such as energy conservation laws, leading to plausible predictions. Yet, scaling these models to intrinsically high-dimensional systems remains a significant challenge. This paper introduces Geometric Reduced-order Hamiltonian Neural Network (RO-HNN), a novel physics-inspired neural network that combines the conservation laws of Hamiltonian mechanics with the scalability of model order reduction. RO-HNN is built on two core components: a novel geometrically-constrained symplectic autoencoder that learns a low-dimensional, structure-preserving symplectic submanifold, and a geometric Hamiltonian neural network that models the dynamics on the submanifold. Our experiments demonstrate that RO-HNN provides physically-consistent, stable, and generalizable predictions of complex high-dimensional dynamics, thereby effectively extending the scope of Hamiltonian neural networks to high-dimensional physical systems.

Problem

• Scaling Hamiltonian neural networks to high-dimensional systems
• Learning structure-preserving dynamics on reduced symplectic submanifolds
• Providing physically-consistent predictions for complex Hamiltonian systems

Innovation

• Geometric reduced-order Hamiltonian neural network for scalability
• Symplectic autoencoder learns structure-preserving submanifold
• Geometric Hamiltonian neural network models submanifold dynamics