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

TL;DR

PIORF addresses the over-squashing problem in mesh-based graph neural networks for fluid dynamics by integrating physical signals with topology through Ollivier-Ricci curvature. The method identifies bottlenecks via node-level curvature $\gamma_i$ and connects low-curvature nodes to high-velocity regions with bidirectional edges, enabling long-range information flow in refined meshes. Key contributions include physics-informed rewiring, a single-pass and scalable edge-addition scheme, and successful extension to temporal mesh graphs with substantial gains (up to $26.2\%$) across multiple benchmarks and architectures. The approach advances realistic CFD simulations on unstructured meshes by improving accuracy and scalability, with potential applicability to dynamic meshes and broader physics-informed graph modeling tasks.

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.

Figures (13)

• Figure 1: Visualization of PIORF rewiring in CylinderFlow-Tiny. (a) Blue areas indicate potential bottlenecks. Red circles () denote critical bottleneck nodes. (b) The black circle () denotes the highest velocity node. PIORF connects bottleneck nodes () with high-velocity nodes ().
• Figure 2: The radar plot shows the percentage improvement over MGN for each method on 3 datasets. The radial distance indicates the magnitude of improvement. PIORF consistently outperforms other methods with substantial gains particularly in AirFoil (24.5% for Velocity) and CylinderFlow (21.3% for Pressure).
• Figure 3: Structural analyses of mesh graphs: (a) Correlation between ORC and node degree in training dataset of CylinderFlow, revealing potential information bottlenecks. (b) Node degree distribution across datasets, showing the prevalence of degree-6 nodes in uniform regions. (c) Non-uniform mesh refinement near boundary conditions.
• Figure 4: Comparison of 2D cross-sectional velocity magnitude contours for CylinderFlow (a)-(d) and AirFoil (e)-(h) at the last time step with the largest cumulative error. It is most similar to ground truth when PIORF is applied. The closer the color is to red, the faster the velocity. The black boxes () highlight regions where PIORF shows particular accuracy in predicting complex flow structures. PIORF consistently achieves the closest match to ground truth on both datasets. More rollout images can be found in \ref{['app:contour']}.
• Figure 5: Sensitivity to pooling ratio $\delta$. The dashed lines represent RMSE of MGN without rewiring.