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

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.

• The paper introduces HGIN that jointly learns interaction graphs and Hamiltonian dynamics from trajectory data, overcoming key limitations of existing ML models.
• It leverages a learnable weighted adjacency matrix with clustering-based subgraph decomposition to capture non-separable and heterogeneous interactions.
• Empirical results demonstrate HGIN's superior long-term accuracy with error reductions of 6–13 orders of magnitude compared to baselines, ensuring stable energy conservation.

Hamiltonian Graph Inference Networks: Unified Structure Discovery and Dynamics Prediction for Lattice Hamiltonian Systems

Problem Statement and Existing Limitations

The paper addresses the dual challenge of identifying both the interaction topology and dynamical rules of lattice Hamiltonian systems exclusively from trajectory data. Such systems, pervasive in condensed matter, nonlinear optics, and biophysics, often possess unknown or only partially accessible coupling structures, which may not exhibit homogeneous node dynamics or separability in the Hamiltonian. Prevailing ML strategies—including graph neural networks (GNNs) and variants such as HNN, SympNet, NSSNN, HOGN, and $\alpha$-SGHN—either assume prior knowledge of the interaction graph or restrict themselves to separability and homogeneity, severely limiting their applicability to complex real-world or experimental settings.

Hamiltonian Graph Inference Network (HGIN) Architecture

The Hamiltonian Graph Inference Network (HGIN) resolves these limitations through the explicit joint learning of the interaction graph and predictive Hamiltonian dynamics, accommodating both separable and non-separable Hamiltonians as well as heterogeneous node dynamics.

Structure-Learning Module

HGIN introduces a learnable weighted adjacency matrix $\mathcal{W}_\theta$, initialized randomly and iteratively refined under a composite loss that combines a physical constraint (residual of Hamilton's equations) with Frobenius norm regularization. Node and edge features are engineered to encode nonlinear phase-space relations and pairwise correlations, supporting the representation of non-separable and heterogeneous interactions. The module thereby uncovers the weighted graph structure directly from trajectory data, without any pre-imposed topological constraints.

A crucial property is that the learned graph weights encode not only interaction strength but also the symmetry (parity) of the underlying pair potential: even potentials result in symmetric weights; asymmetries signal non-even interactions. This provides a physically interpretable diagnostic of the inferred topology.

Trajectory-Prediction Module and Subgraph Decomposition

Post-convergence, the learned adjacency matrix is clustered using $k$-means with automatic selection of cluster numbers via silhouette coefficients. This partitions the interaction network into physically distinct subgraphs (clusters), each associated with its own set of encoder networks for node and edge dynamics. Diagonal and off-diagonal blocks distinguish local (on-site) and interaction terms, while assignment symmetry further diagnoses potential evenness. This design sidesteps the parameter-sharing bottleneck inherent in standard GNNs, enabling explicit modeling of heterogeneity in node dynamics and interaction types.

By integrating the resultant (subgraph-specific) Hamiltonian vector fields, the model predicts time evolution from arbitrary initial conditions.

Computational Results and Key Findings

Benchmarks: Lattice Hamiltonian Systems

HGIN is evaluated on three canonical lattice Hamiltonians: (1) a Klein–Gordon lattice with long-range interactions (KG-LRI, separable, homogeneous), (2) a non-separable discrete nonlinear Schrödinger lattice (DNLS, homogeneous), and (3) a non-separable DNLS with heterogeneous node dynamics. Each benchmark is chosen to test aspects where conventional GNNs and ML approaches are known to fail or degenerate.

Performance Metrics and Comparative Analysis

• Accuracy: For all benchmarks, HGIN achieves long-time trajectory and energy prediction errors that are 6–13 orders of magnitude lower than baselines (MLP, HNN, SympNet, NSSNN, and $\alpha$-SGHN). Critically, HGIN remains stable under long-term integration, with negligible error growth, whereas alternative models accumulate significant error and exhibit poor energy conservation.
• Generalization: The model's generalization is robust as the system size increases (e.g., $N=4 \rightarrow 32$), with no degradation in performance, contrary to all baselines.
• Graph Recovery and Physical Interpretability: HGIN accurately reconstructs the true interaction graph and distinguishes between different coupling regimes without any prior input. Adjacency matrix clustering recovers nearest/next-nearest neighbor structure and identifies non-even symmetry in heterogeneous settings.
• Robustness: The architecture exhibits resilience to substantial levels of Gaussian noise in the training trajectories, with only a minor degradation in inference and prediction quality.

Ablation and Sensitivity Analyses

Ablation studies confirm that HGIN's structure-learning formulation and explicit subgraph handling are both indispensable for precise structure recovery and long-horizon stability. Variants substituting the graph learning mechanism (e.g., attention mechanisms, NRI, or transformers) or omitting the clustering-based subgraph parameterization result in catastrophic long-term prediction failures and structural misidentification.

Implications and Theoretical Significance

HGIN advances the state-of-the-art in model discovery for physical systems, particularly where the underlying interaction topology is latent. Its design resolves critical issues in parameter-sharing and homogeneity assumption that have impeded prior methods, rendering it applicable to diverse systems (e.g., in condensed matter, photonics, materials science, and biophysics) where node-specific or anisotropic couplings are prevalent. The explicit subgraph decomposition mechanism not only facilitates heterogeneous dynamics modeling but also yields interpretable insights into the physical structure and symmetries of the interactions.

From a theoretical perspective, the demonstrated ability to infer non-separable Hamiltonians and heterogeneous lattices represents a substantial broadening of the applicability of Hamiltonian neural approaches, bridging the gap between ML-based system identification and traditional data-driven physics discovery methodologies (e.g., SINDy, PINNs) in settings where graph/structure information is unavailable or costly to obtain.

Future Directions

The outlined directionality for future research includes: (1) application to higher-dimensional lattices and networks prevalent in current experimental optical, atomic, and materials platforms; (2) extension to partially observed or noisy experimental data; and (3) integration with structure discovery in nonequilibrium and open quantum systems, where interaction graphs may evolve or be fundamentally stochastic. The authors suggest the release of code and pre-constructed benchmarks to facilitate reproducibility and community adoption.

Conclusion

Hamiltonian Graph Inference Networks constitute a robust, interpretable, and highly accurate approach for simultaneous structure inference and dynamics prediction in lattice Hamiltonian systems. By jointly learning interaction topology and system dynamics under minimal prior assumptions, and by explicitly addressing heterogeneity and non-separability, HGIN achieves state-of-the-art performance and substantially extends the scope of ML-based physical modeling. Its practical and theoretical relevance is underscored by the scale of empirical gains and by its capacity for interpretable discovery, with clear potential for broad application across physics, chemistry, and engineering domains (2604.23606).