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

arXiv: https://arxiv.org/abs/2604.23606 · HTML: https://mlsanity.com/paper/2604.23606/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 47 references that are in the map.
- First posted on arXiv 26 Apr 2026.

## Builds on (most cited first, 18 of 18)

- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Neural Relational Inference for Interacting Systems](https://mlsanity.com/paper/1802.04687/) (2018)
- [Hamiltonian Graph Networks with ODE Integrators](https://mlsanity.com/paper/1909.12790/) (2019)
- [Combinatorial Optimization with Physics-Inspired Graph Neural Networks](https://mlsanity.com/paper/2107.01188/) (2021)
- [EqMotion: Equivariant Multi-agent Motion Prediction with Invariant Interaction Reasoning](https://mlsanity.com/paper/2303.10876/) (2023)
- [AI Poincar'e: Machine Learning Conservation Laws from Trajectories](https://mlsanity.com/paper/2011.04698/) (2020)
- [Neural ODE control for classification, approximation and transport](https://mlsanity.com/paper/2104.05278/) (2021)
- [Learning Poisson systems and trajectories of autonomous systems via Poisson neural networks](https://mlsanity.com/paper/2012.03133/) (2020)
- [Symplectic Learning for Hamiltonian Neural Networks](https://mlsanity.com/paper/2106.11753/) (2021)
- [Nonseparable Symplectic Neural Networks](https://mlsanity.com/paper/2010.12636/) (2020)
- [Finite-data error bounds for Koopman-based prediction and control](https://mlsanity.com/paper/2108.07102/) (2021)
- [Deep Neural Network Modeling of Unknown Partial Differential Equations in Nodal Space](https://mlsanity.com/paper/2106.03603/) (2021)
- [Physics guided neural networks for modelling of non-linear dynamics](https://mlsanity.com/paper/2205.06858/) (2022)
- [Wide and Deep Neural Networks Achieve Optimality for Classification](https://mlsanity.com/paper/2204.14126/) (2022)
- [Discovering Conservation Laws using Optimal Transport and Manifold Learning](https://mlsanity.com/paper/2208.14995/) (2022)
- [CRYSPNet: Crystal Structure Predictions via Neural Network](https://mlsanity.com/paper/2003.14328/) (2020)
- [Link Prediction via Graph Attention Network](https://mlsanity.com/paper/1910.04807/) (2019)

## Abstract (arXiv)

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 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.