Learning Hamiltonian Dynamics at Scale: A Differential-Geometric Approach
This work addresses the problem of modeling high-dimensional Hamiltonian dynamics for researchers in physics and machine learning, representing an incremental advancement by extending existing methods to higher dimensions.
The paper tackles the challenge of scaling Hamiltonian neural networks to high-dimensional physical systems by introducing the Geometric Reduced-order Hamiltonian Neural Network (RO-HNN), which combines Hamiltonian mechanics with model order reduction, resulting in physically-consistent, stable, and generalizable predictions for complex dynamics.
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.