GeoHNNs: Geometric Hamiltonian Neural Networks
This work addresses the challenge of creating robust and generalizable models for physical systems, which is crucial for applications in physics and engineering, though it is incremental as it builds on prior physics-informed neural networks.
The authors tackled the problem of modeling physical dynamics with neural networks that often ignore underlying geometric principles, leading to unstable predictions. They introduced GeoHNN, a framework that explicitly encodes geometric priors, and demonstrated it significantly outperforms existing models in long-term stability, accuracy, and energy conservation on systems like coupled oscillators and high-dimensional deformable objects.
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.