Learning Hamiltonian Dynamics at Scale: A Differential-Geometric Approach

Embedding physical intuition into network architectures allows the learning of dynamics that enforce fundamental properties, such as energy conservation laws, thereby leading to physically-plausible predictions. Yet, scaling these models to intrinsically high-dimensional dynamical systems remains a significant challenge. This paper introduces 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.

Predicting how physical systems, such as robots, deformable materials, and fluids, evolve over time is crucial for analysis and design, but challenging because their dynamics are highly complex, nonlinear, and often (very) high-dimensional. Machine learning models can learn these dynamics from data. However, standard neural networks often produce physically-inconsistent predictions, especially over long time horizons. Models such as Hamiltonian Neural Networks (HNNs) partially address this by embedding physical laws into the networks, but fail to scale beyond low-dimensional systems. We introduce Reduced-order Hamiltonian Neural Networks (RO-HNNs), that can learn the dynamics of high-dimensional physical systems while preserving their underlying structure. Given data from a system, our method learns a low-dimensional representation that preserves the most important motion features, and models the system dynamics in this reduced space using a HNN. Geometric and physical network constraints guarantee preservation of the system’s nonlinear, physical structure, and thus stability and conservation laws. We evaluate our approach on multiple simulated systems: a coupled pendulum, particle vortex, and deformable cloth with hundreds of degrees of freedom. RO-HNNs produce more accurate, stable, and physically-consistent predictions than existing methods while remaining computationally efficient, demonstrating a promising direction for scaling machine learning models to realistic large-scale physical systems.