Augmented physics-informed Hamiltonian networks for dynamical systems under external interactions.

Area of Science:

• Computational Physics
• Dynamical Systems Theory
• Machine Learning for Science

Background:

• Neural networks are powerful tools but underutilized for discovering physical laws.
• Hamiltonian systems are fundamental in physics, but learning their laws under perturbation is challenging.
• Existing models like dissipative Hamiltonian neural networks (DHNNs) have limitations in complex scenarios.

Purpose of the Study:

• To introduce augmented physics-informed Hamiltonian networks (A-PIHNs) for learning physical laws in perturbed Hamiltonian systems.
• To demonstrate A-PIHNs' superior performance compared to existing methods.
• To explore the theoretical underpinnings of A-PIHNs by connecting them to Kolmogorov-Arnold-Moser (KAM) theory.

Main Methods:

• Developed augmented physics-informed Hamiltonian networks (A-PIHNs).
• Employed Helmholtz-Hodge decomposition to identify Hamiltonian and perturbed dynamical systems.
• Treated the A-PIHNs model as a dynamical system, analyzing approximation errors in relation to KAM theory.

Main Results:

• A-PIHNs successfully learned physical laws in Hamiltonian systems with complex perturbations.
• Empirical results showed A-PIHNs achieved superior accuracy and generalization over DHNNs in strong perturbation scenarios.
• The study confirmed that A-PIHNs can approximate Kolmogorov-Arnold-Moser (KAM) theory.

Conclusions:

• A-PIHNs represent a significant advancement in using neural networks to decipher complex physical laws.
• The model's ability to handle perturbations and approximate KAM theory highlights its potential for uncovering new physical phenomena.
• This work underscores the untapped potential of physics-informed neural networks in fundamental scientific discovery.