IGNS: Improving Long-Range Interactions in Graph Neural Simulators

Abstract

Learning to simulate complex physical systems from data has emerged as a promising way to overcome the limitations of traditional numerical solvers, which often require prohibitive computational costs for high-fidelity solutions. Recent Graph Neural Simulators (GNSs) accelerate simulations by learning dynamics on graph-structured data, yet often struggle to capture long-range interactions and suffer from error accumulation under autoregressive rollouts.

To address these challenges, we propose Information-preserving Graph Neural Simulator (IGNS) , a graph-based neural simulator built on the principles of Hamiltonian dynamics. This structure guarantees preservation of information across the graph, while extending to port-Hamiltonian systems allows the model to capture a broader class of dynamics, including non-conservative effects. IGNS further incorporates a warmup phase to initialize global context, geometric encoding to handle irregular meshes, and a multi-step training objective that facilitates PDE matching, where the trajectory produced by integrating the port-Hamiltonian core aligns with the ground-truth trajectory, thereby reducing rollout error.

To evaluate these properties systematically, we introduce new benchmarks that target long-range dependencies and challenging external forcing scenarios. Across all tasks, IGNS consistently outperforms state-of-the-art GNSs, achieving higher accuracy and stability under challenging and complex dynamical systems.

Method

IGNS builds on four key components: (i) Port-Hamiltonian formalism to capture energy-conserving and non-conservative behavior, (ii) Warmup phase to establish global context, (iii) Geometric encoding for irregular meshes, and (iv) Multi-step objective to reduce error accumulation.

Port-Hamiltonian Dynamics

We parameterize the Hamiltonian $H_\theta(t, \mathbf{X})$ and evolve the joint state according to:

$$ \dot{\mathbf{x}}_i = \mathbf{J}\nabla_{\mathbf{x}_i} H_\theta(t, \mathbf{X}) - \begin{bmatrix}0\\ \mathbf{D}_\theta \nabla_{\mathbf{p}_i} H_\theta(t, \mathbf{X})\end{bmatrix} + \begin{bmatrix}0\\ \mathbf{r}_\theta(t, \mathbf{X})\end{bmatrix}, \quad \mathbf{J}=\begin{bmatrix}0&\mathbf{I}\\ - \mathbf{I}&0\end{bmatrix} $$

where $\mathbf{x}_i=[\mathbf{q};\mathbf{p}]^T_i$ represents generalized coordinates and momenta, and $\mathbf{J}$ encodes the symplectic structure. The damping and forcing terms enable modeling of non-conservative systems while maintaining the energy-preserving structure. We parameterize the Hamiltonian as:

$$ H_\theta(t, \mathbf{X}) = \sum_{i \in \mathcal{V}} \tilde{\sigma} \left( \mathbf{W}(t) \mathbf{x}_i + \sum_{j \in \mathcal{N}_i} \mathbf{V}(t) \mathbf{x}_{j} \right)^T \cdot \mathbf{1}_d $$

where $\tilde{\sigma}$ is an anti-derivative of a non-linear activation, $\mathbf{1}_d$ is a row vector of ones, and $\mathbf{W}(t)$, $\mathbf{V}(t)$ are block diagonal learnable weighted matrices that ensure separation into the $\mathbf{q}$ and $\mathbf{p}$ components.

Warmup Phase

To capture long-range interactions from the start, we perform $l$ rounds of message passing without advancing time. This broadcasts information across the graph up to radius $l$ from each node, providing global context that is preserved throughout the rollout thanks to the energy-conserving core.

Geometric Encoding

We embed geometric features directly into edge attributes to capture spatial structure on irregular meshes. Each edge feature encodes displacement vectors ($\mathbf{s}_{ij}=\text{pos}_j - \text{pos}_i$) and distances ($d_{ij}=\|\text{pos}_j - \text{pos}_i\|_2$), enabling the model to leverage mesh geometry while preventing overfitting to specific discretizations.

Multi-Step Loss

We train with a multi-step objective that supervises entire rollout windows:

$$ \mathcal{L}_{\text{multi-step}} = \sum_{\tau=1}^{K}\Big(\|\widehat{\mathbf{q}}^{(t+\tau)}-\mathbf{q}^{(t+\tau)}\|_2^2 + \|\widehat{\mathbf{p}}^{(t+\tau)}-\mathbf{p}^{(t+\tau)}\|_2^2\Big) $$

This enhances trajectory-level consistency and reduces error accumulation compared to one-step training followed by autoregressive rollout.

Tasks & Results

Click on each task to view demonstrations

Plate Deformation

Sphere-Cloth

Impact Plate

Wave Balls

Cylinder Flow

Kuramoto-Sivashinsky

BibTeX

@inproceedings{
  hoang2026igns,
  title={Improving Long-Range Interactions in Graph Neural Simulators via Hamiltonian Dynamics},
  author={Hoang, Tai and Trenta, Alessandro and Gravina, Alessio and 
          Freymuth, Niklas and Becker, Philipp and Bacciu, Davide and Neumann, Gerhard},
  booktitle={The Fourteenth International Conference on Learning Representations},
  year={2026},
  url={https://openreview.net/forum?id=x66u6TEDUw}
}