  We propose an $\alpha$-separable graph Hamiltonian network ($\alpha$-SGHN)
that reveals complex interaction patterns between particles in lattice systems.
Utilizing trajectory data, $\alpha$-SGHN infers potential interactions without
prior knowledge about particle coupling, overcoming the limitations of
traditional graph neural networks that require predefined links. Furthermore,
$\alpha$-SGHN preserves all conservation laws during trajectory prediction.
Experimental results demonstrate that our model, incorporating structural
information, outperforms baseline models based on conventional neural networks
in predicting lattice systems. We anticipate that the results presented will be
applicable beyond the specific onsite and inter-site interaction lattices
studied, including the Frenkel-Kontorova model, the rotator lattice, and the
Toda lattice.
