Hamiltonian Graph Inference Networks: Joint structure discovery and dynamics prediction for lattice Hamiltonian systems from trajectory data

TL;DR

HGIN is a novel neural network that jointly infers the interaction topology and predicts long-term dynamics of lattice Hamiltonian systems from trajectory data, handling both separable and non-separable cases with heterogeneous nodes.

Contribution

HGIN introduces a joint structure learning and trajectory prediction framework that works for complex Hamiltonian systems, surpassing existing methods in accuracy and interpretability.

Findings

HGIN significantly reduces energy and trajectory prediction errors by 6 to 13 orders of magnitude.

It accurately recovers interaction graphs, including parity of pair potentials.

The method is effective on benchmarks with long-range interactions and heterogeneous nodes.

Abstract

Lattice Hamiltonian systems underpin models across condensed matter, nonlinear optics, and biophysics, yet learning their dynamics from data is obstructed by two unknowns: the interaction topology and whether node dynamics are homogeneous. Existing graph-based approaches either assume the graph is given or, as in $\alpha$-separable graph Hamiltonian network, infer it only for separable Hamiltonians with homogeneous node dynamics. We introduce the Hamiltonian Graph Inference Network (HGIN), which jointly recovers the interaction graph and predicts long-time trajectories from state data alone, for both separable and non-separable Hamiltonians and under heterogeneous node dynamics. HGIN couples a structure-learning module -- a learnable weighted adjacency matrix trained under a Hamilton's-equations loss -- with a trajectory-prediction module that partitions edges into physically distinct subgraphs via $k$-means clustering, assigning each subgraph its own encoder and thereby breaking the parameter-sharing bottleneck of conventional GNNs. On three benchmarks -- a Klein--Gordon lattice with long-range interactions and two discrete nonlinear Schr\"odinger lattices (homogeneous and heterogeneous) -- HGIN reduces long-time energy prediction error and trajectory prediction error by six to thirteen orders of magnitude relative to baselines. A symmetry argument on the Hamiltonian loss further shows that the learned weights encode the parity of the underlying pair potential, yielding an interpretable readout of the system's interaction structure.