Graph Attention Hamiltonian Neural Networks: A Lattice System Analysis Model Based on Structural Learning

The paper informally asserts properties of the learned attention matrix A—most notably that symmetry of A implies an even-symmetric potential—without a precise definition of the trajectory-to-attention map or a proof. It also uses ad hoc thresholding and ratio inferences supported only by examples. The model solution supplies a concrete, identifiable map from trajectories to A and rigorously proves: (i) exact edge recovery under a gap; (ii) even-symmetric potential implies symmetric A, while the converse is false in general unless additional identifiability and modeling conditions hold; (iii) order-wise interaction ratios can be recovered on a 1D chain under normalization and shared shape assumptions. These results reconcile and formalize the paper’s empirical claims while correcting an over-strong converse claim.