Hamiltonian neural networks with automatic symmetry detection

Area of Science:

• Computational Physics
• Machine Learning
• Dynamical Systems

Background:

• Hamiltonian neural networks (HNNs) integrate physical knowledge into data-driven models for Hamiltonian systems.
• Preserving symplectic structure is a key feature of HNNs.
• Incorporating symmetries into HNNs requires specialized methods.

Purpose of the Study:

• To enhance Hamiltonian neural networks (HNNs) by incorporating a Lie algebra framework for symmetry detection and embedding.
• To enable simultaneous learning of symmetry group actions and total energy within the neural network.
• To improve the accuracy and physical realism of data-driven models for complex dynamical systems.

Main Methods:

• Developed an enhanced HNN architecture utilizing a Lie algebra framework.
• Implemented methods to detect and embed symmetries directly into the neural network.
• Applied the enhanced HNN to model a pendulum on a cart and a two-body astrodynamics problem.

Main Results:

• Successfully integrated symmetry detection and embedding into HNNs via Lie algebra.
• Demonstrated simultaneous learning of symmetry group actions and system energy.
• Validated the approach on benchmark physical systems, showing improved modeling capabilities.

Conclusions:

• The Lie algebra-enhanced HNN provides a powerful framework for data-driven learning of physical systems with symmetries.
• This method preserves fundamental physical properties like energy and symmetry, leading to more robust models.
• The approach is applicable to various fields, including robotics, celestial mechanics, and molecular dynamics.