Machine Learning Symmetry Discovery for Integrable Hamiltonian Dynamics

TL;DR

This paper introduces a machine learning framework that identifies continuous symmetries and their algebraic structures directly from phase-space data of integrable Hamiltonian systems, successfully recovering known symmetry algebras.

Contribution

The paper presents a novel neural network-based method for discovering symmetry generators and Lie algebras from trajectory data in Hamiltonian dynamics, validated on benchmark systems.

Findings

Successfully recovers $rak{so}(4)$ and $rak{su}(3)$ algebras from data

Learns conserved quantities and symmetry structures directly from trajectories

Demonstrates effectiveness on integrable systems with canonical coordinates

Abstract

We propose a data-driven Machine-Learning Symmetry Discovery (MLSD) framework for identifying continuous symmetry generators and their Lie-algebraic structure directly from phase-space trajectory data expressed in canonical coordinates. MLSD parameterizes candidate conserved quantities with neural networks and learns antisymmetric structure coefficients by enforcing Poisson-bracket closure, supplemented by a weak independence regularizer. We validate MLSD on two integrable benchmark systems -- the three-dimensional Kepler problem and the three-dimensional isotropic harmonic oscillator -- recovering the expected non-Abelian algebras (respectively $\mathfrak{so}(4)$ and $\mathfrak{su}(3)$) up to basis transformations. This work focuses on integrable benchmark dynamics, where global conserved quantities are well-defined and admit compact representations learnable from canonical-coordinate trajectories. Extending symmetry discovery to mixed or chaotic phase-space regimes is an important direction for future work.