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

TL;DR

This paper introduces IGNS, a Hamiltonian-based graph neural simulator that effectively captures long-range interactions and reduces error accumulation in simulating complex physical systems.

Contribution

We propose a novel Hamiltonian-based graph neural simulator, IGNS, with features like information preservation, port-Hamiltonian extension, and specialized training for improved long-range interaction modeling.

Findings

IGNS outperforms existing GNSs in accuracy and stability.

The model effectively captures long-range dependencies.

It demonstrates robustness in complex dynamical systems.

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 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. Our project page: https://thobotics.github.io/neural_pde_matching.