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Improving Long-Range Interactions in Graph Neural Simulators via Hamiltonian Dynamics

Tai Hoang, Alessandro Trenta, Alessio Gravina, Niklas Freymuth, Philipp Becker, Davide Bacciu, Gerhard Neumann

TL;DR

IGNS introduces an information-preserving graph neural simulator based on port-Hamiltonian dynamics to address long-range interactions and rollout error in neural physical simulators. The method couples a Hamiltonian latent core with a symplectic integrator, a $l$-step warmup, geometric mesh encoding, and a multi-step loss to enable accurate, stable long-horizon predictions across irregular meshes. Theoretical results establish universality and non-vanishing gradient propagation under the Hamiltonian core, providing a principled basis for long-range information flow. Empirically, IGNS and its time-varying variant consistently outperform state-of-the-art GNSs across six physics tasks, including oscillatory and non-conservative dynamics, highlighting practical benefits for faster, physically consistent neural simulation.

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 Simulators (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 to reduce 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.

Improving Long-Range Interactions in Graph Neural Simulators via Hamiltonian Dynamics

TL;DR

IGNS introduces an information-preserving graph neural simulator based on port-Hamiltonian dynamics to address long-range interactions and rollout error in neural physical simulators. The method couples a Hamiltonian latent core with a symplectic integrator, a -step warmup, geometric mesh encoding, and a multi-step loss to enable accurate, stable long-horizon predictions across irregular meshes. Theoretical results establish universality and non-vanishing gradient propagation under the Hamiltonian core, providing a principled basis for long-range information flow. Empirically, IGNS and its time-varying variant consistently outperform state-of-the-art GNSs across six physics tasks, including oscillatory and non-conservative dynamics, highlighting practical benefits for faster, physically consistent neural simulation.

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 Simulators (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 to reduce 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.

Paper Structure

This paper contains 36 sections, 7 theorems, 40 equations, 10 figures, 14 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Partial Differential Equations for Physical Modeling
  4. Remark.
  5. Message Passing for Model Learning
  6. The igns
  7. Port-Hamiltonian Formalism
  8. Warmup phase
  9. Geometrical Information Encoding
  10. Multi-step loss
  11. Theoretical Properties
  12. Experiments
  13. Results and Discussions
  14. Conclusion
  15. Related Work
  16. ...and 21 more sections

Key Result

Theorem 1

Let $\Psi_{\theta}$ be the functional that maps the initial condition ${\bm{x}}_0,\dot{{\bm{x}}}_0$ to the solution at time $\tau$ to the ODE defined eq:port_hamiltonian and eq:hamiltonian. In other words, $\Psi_\theta$ represents igns. Then, for any map $F:\mathbb{R}^{n\times d}\rightarrow\mathbb{R

Figures (10)

  • Figure 1: A high-level overview of the proposed igns. The model takes as input the initial node state $\bar{{\bm{X}}}$ and performs an $l$-step warmup phase (left), enabling each node to incorporate broader spatial context before the rollout begins. The enriched state is then used to initialize the rollout ${\bm{X}}^{(0)} = \bar{{\bm{X}}}^{(l)}$ (middle). During the simulation phase (right), the system evolves according to the port-Hamiltonian dynamics of \ref{['eq:port_hamiltonian']}, while the multi-step loss $\mathcal{L}_{\text{multi-step}}$ supervises all intermediate predictions, ensuring stable and accurate simulations.
  • Figure 2: Three novel tasks. (a)Plate Deformation: a flat plate deformed subject to varying numbers and magnitudes of point forces. (b)Sphere Cloth: a cloth mesh impacted by a falling sphere, producing elastic deformation with contact dynamics. (c)Wave Balls: surface waves on water generated by three balls moving linearly from different initial positions.
  • Figure 3: Ablations: (a) data efficiency on final-state-only Plate Deformation task; (b) warm-up steps on full-roll-out Plate Deformation task; (c) Sphere Cloth (SC) vs. Wave Balls (WB) with $T=100$ (Wave Balls MSE $\times 100$ for visibility). All results are computed using $4$ seeds.
  • Figure 4: Dataset overview. (a) Plate Deformation: a flat plate deformed subject to varying numbers and magnitudes of point forces, simulated with linear elasticity. (b) Sphere Cloth: a cloth mesh impacted by a falling sphere, producing elastic deformation with contact dynamics. (c) Impact Plate: an elastic plate under impact with a rigid ball. (d) Wave Balls: surface waves on water generated by three balls moving linearly from different initial positions. (e) Kuramoto-Sivashinsky: chaotic spatio-temporal evolution of a scalar field governed by the KS equation. (f) Cylinder Flow: incompressible fluid flow around cylindrical obstacles with vortex shedding.
  • Figure 5: Plate Deformation qualitative comparison on three test samples with predictions from MGN-rewiring, MGN and igns. The arrows indicate the forces (with magnitude propotion to its length) applied to the plate at specific postions.
  • ...and 5 more figures

Theorems & Definitions (10)

  • Theorem 1: Universality of igns
  • Theorem 2
  • Theorem 3: Theorem 3.1 of Lanthaler2023-NeuralOscillators
  • Theorem : Universality of igns, \ref{['thm:universality']} of \ref{['sec:theoretical_properties']}
  • proof
  • Remark 1
  • Theorem 4: Eigenvalues of an Oscillatory System
  • proof
  • Lemma 1: Lemma 1 of galimberti2021-unified_hdgn
  • Theorem : \ref{['thm:phdgn_sensitivity']} of \ref{['sec:theoretical_properties']} and Theorem 2.3 of gravina_phdgn