Machine learning symmetry discovery for integrable Hamiltonian dynamics.

Area of Science:

• Physics
• Dynamical Systems
• Computational Science

Background:

• Discovering continuous symmetries is fundamental in physics, often relying on analytical methods.
• Identifying Lie-algebraic structures from trajectory data presents a significant computational challenge.

Purpose of the Study:

• To introduce a data-driven machine learning symmetry discovery (MLSD) framework.
• To automatically identify continuous symmetry generators and their Lie-algebraic structure from phase-space data.

Main Methods:

• MLSD parameterizes conserved quantities using neural networks.
• It learns structure coefficients by enforcing Poisson-bracket closure.
• A weak independence regularizer aids the learning process.

Main Results:

• MLSD successfully recovered non-Abelian algebras SO(4) and SU(3) for integrable systems.
• The framework was validated on the 3D Kepler problem and isotropic harmonic oscillator.
• Symmetries were identified directly from canonical coordinate trajectories.

Conclusions:

• The MLSD framework offers a novel data-driven approach for symmetry discovery in physics.
• This method is effective for integrable systems with well-defined conserved quantities.
• Future work will explore extending MLSD to chaotic and mixed phase-space regimes.