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

2026-04-26

AI Summary

This work proposes the Hamiltonian Graph Inference Network (HGIN) for lattice Hamiltonian systems, which simultaneously infers unknown interaction topologies and heterogeneous node dynamics solely from trajectory data while enabling long-term evolution prediction. The method innovatively integrates interpretable learning of a weighted adjacency matrix with a physics-informed trajectory prediction module based on subgraph partitioning: edge sets are clustered via k-means, and distinct encoders are assigned to each subgraph, thereby overcoming the parameter-sharing limitation of conventional GNNs. Structural interpretability is further enhanced through loss constraints derived from Hamiltonian equations and symmetry principles. Evaluated on Klein–Gordon and two variants of discrete nonlinear Schrödinger lattice systems, HGIN achieves 6–13 orders of magnitude improvement over baseline methods in long-term energy conservation and trajectory prediction accuracy.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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 $α$-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 Schrödinger 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.

Problem

• Hamiltonian systems
• graph inference
• structure discovery
• dynamics prediction
• heterogeneous dynamics

Innovation

• Hamiltonian Graph Inference Network
• structure discovery
• heterogeneous dynamics
• non-separable Hamiltonians
• interpretable graph learning