Machine Learning Symmetry Discovery for Integrable Hamiltonian Dynamics

TL;DR

The paper tackles the challenge of extracting continuous Lie-group symmetries from classical trajectory data by introducing MLSD, a data-driven pipeline that learns conserved quantities $G_i$ via neural networks and enforces Poisson-bracket closure through antisymmetric structure coefficients $f^{ijk}$. By combining Hamiltonian learning with an algebra-closure objective and a weak independence regularizer, MLSD identifies both Abelian and non-Abelian symmetry algebras from integrable Hamiltonian dynamics expressed in canonical coordinates. The framework is validated on the 3D Kepler problem and the 3D isotropic harmonic oscillator, successfully recovering SO(4) and SU(3) algebras, respectively, using Killing-form analysis and basis transformations to confirm the algebras. The results demonstrate a practical end-to-end approach for symmetry discovery from data, with clear pathways for extension to more complex dynamics and potential quantum/many-body applications.

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.

Figures (8)

• Figure 1: The Machine Learning Symmetry Discovery (MLSD) framework takes time evolution data from a classical system and feeds the canonical coordinates into neural networks to predict a physical quantity. By optimizing the loss function, the predicted observables converge to conserved quantities, and the corresponding structure coefficients reveal the underlying symmetry group.
• Figure 2: Hamiltonian learning of the 3D Kepler problem: orbit comparison. Two representative examples (a, b) showing ground truth orbits (gray) versus learned trajectories (blue) integrated over one orbital period $T$. The red star marks the central body at the origin. Blue arrowheads indicate the velocity direction at the end of integration. Right panels show the corresponding energy drift $|\Delta H|$ as a function of normalized time $t/T$, demonstrating that errors accumulate due to imperfect Hamiltonian learning. All trajectories are integrated using the velocity-Verlet symplectic integrator.
• Figure 3: MLSD for the 3D Kepler problem: training convergence and symmetry dimension identification. (a--e) Training loss $\mathcal{L}_G$ versus epoch for symmetry dimensions $n = 4, 6, 8, 10, 12$, with 15 independent random seeds per dimension. Insets show the distribution of final converged losses. (f) Mean converged loss versus symmetry dimension $n$, with error bars indicating standard deviation across seeds. The vertical dashed red line marks the true symmetry dimension $n = 6$, which achieves the lowest loss, correctly identifying the $\mathfrak{so}(4)$ symmetry of the 3D Kepler problem.
• Figure 4: The trained structure coefficient of Kepler problem.
• Figure 5: Hamiltonian learning of the 3D harmonic oscillator: orbit comparison. Two representative examples (a, b) showing ground truth orbits (gray) versus learned trajectories (blue) integrated over one period $T$. The red star marks the equilibrium position at the origin. Blue arrowheads indicate the velocity direction at the end of integration. Right panels show the corresponding energy drift $|\Delta H|$ as a function of normalized time $t/T$, demonstrating error accumulation from imperfect Hamiltonian learning. All trajectories are integrated using the velocity-Verlet symplectic integrator.