SympNets: Intrinsic structure-preserving symplectic networks for identifying Hamiltonian systems

Area of Science:

• Dynamical Systems
• Machine Learning
• Computational Physics

Background:

• Hamiltonian systems are fundamental in physics but challenging to learn from data.
• Existing methods often struggle with complex dynamics or require extensive training.
• The need for efficient and accurate models for Hamiltonian systems is critical.

Purpose of the Study:

• To propose Symplectic Networks (SympNets) for identifying Hamiltonian systems from data.
• To introduce two classes of SympNets: LA-SympNets and G-SympNets.
• To prove universal approximation theorems for SympNets.

Main Methods:

• SympNets are built using linear, activation, and gradient modules.
• Two architectures, LA-SympNets and G-SympNets, are defined.
• Universal approximation theorems are proven for SympNets' ability to approximate symplectic maps.

Main Results:

• SympNets effectively learn both separable and non-separable Hamiltonian systems.
• Experiments on pendulum and three-body problems show strong generalization with small network sizes.
• SympNets significantly outperform baseline models in speed and accuracy.

Conclusions:

• SympNets offer a powerful and efficient tool for learning Hamiltonian dynamics.
• An extended version handles irregularly sampled data, acting as a universal model.
• SympNets represent a significant advancement in data-driven discovery of physical laws.