Machine Learning Symmetry Discovery for Integrable Hamiltonian Dynamics

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 $\mathfrak{s}\mathfrak{o}(4)$ and $\mathfrak{s}\mathfrak{u}(3)$) 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.