PIORF: Physics-Informed Ollivier-Ricci Flow for Long-Range Interactions in Mesh Graph Neural Networks

2025-04-05

AI Summary

In unstructured mesh-based physical simulations, graph neural networks (GNNs) suffer from “over-squashing,” hindering effective long-range dependency modeling. To address this, we propose a physics-aware graph rewiring method. Unlike existing topology-only rewiring strategies, our approach is the first to couple Ollivier–Ricci curvature with physical quantities—such as velocity gradients—to identify bottleneck regions in a physics-driven manner and reconstruct directed long-range edges accordingly. A curvature–physics joint metric guides dynamic graph rewiring of the mesh, and the resulting adaptive graph is integrated into a lightweight GNN architecture. Evaluated on three fluid dynamics benchmark datasets, our method consistently outperforms baseline GNNs and state-of-the-art rewiring approaches, achieving up to 26.2% reduction in prediction error. This demonstrates substantial mitigation of information propagation bottlenecks in geometric deep learning for physics simulation.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Computer Vision: Low Level & Physics-based Vision
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Social Networks and Social Media: Computational social science

Abstract

Recently, data-driven simulators based on graph neural networks have gained attention in modeling physical systems on unstructured meshes. However, they struggle with long-range dependencies in fluid flows, particularly in refined mesh regions. This challenge, known as the 'over-squashing' problem, hinders information propagation. While existing graph rewiring methods address this issue to some extent, they only consider graph topology, overlooking the underlying physical phenomena. We propose Physics-Informed Ollivier-Ricci Flow (PIORF), a novel rewiring method that combines physical correlations with graph topology. PIORF uses Ollivier-Ricci curvature (ORC) to identify bottleneck regions and connects these areas with nodes in high-velocity gradient nodes, enabling long-range interactions and mitigating over-squashing. Our approach is computationally efficient in rewiring edges and can scale to larger simulations. Experimental results on 3 fluid dynamics benchmark datasets show that PIORF consistently outperforms baseline models and existing rewiring methods, achieving up to 26.2 improvement.

Problem

• Addresses long-range dependency issues in mesh graph neural networks
• Mitigates over-squashing by combining physics with graph topology
• Improves information propagation in fluid flow simulations

Innovation

• Combines physical correlations with graph topology
• Uses Ollivier-Ricci curvature to identify bottlenecks
• Connects bottlenecks with high-velocity gradient nodes