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PIORF: Physics-Informed Ollivier-Ricci Flow for Long–Range Interactions in Mesh Graph Neural Networks
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Proposes PIORF, a graph rewiring method for mesh GNN simulators that connects Ollivier-Ricci-curvature bottleneck nodes to high-velocity-difference nodes, reducing over-squashing in MGN and related models on fluid benchmarks.
Problem
Mesh-based GNN simulators (e.g., MeshGraphNets) struggle with long-range dependencies and over-squashing in refined mesh regions for fluid dynamics, and existing rewiring methods ignore physical quantities.
Approach
Combine ORC-based node bottleneck selection with physics (velocity difference) to pick partner nodes, adding bidirectional edges in a single pass with one hyperparameter (pooling ratio), avoiding greedy iterative rewiring.
Key results
- PIORF on MGN reduces CylinderFlow velocity RMSE from 48.8 to 41.6 (x10^3) and pressure from 36.7 to 28.9 (x10^3).
- On EAGLE, PIORF with MGN achieves a 14.5% velocity RMSE improvement over MGN (1,950 vs 2,280) while BORF times out.
- Across models and datasets, the paper reports up to 26.2% improvement over baselines.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add PIORF rewiring | MGN · CylinderFlow | Velocity RMSE (×10^3): 48.8\scriptsize{\;±\;}{5.6} -> \textbf{41.6\scriptsize{\;±\;}{3.9}} | Helped |
| Add PIORF rewiring | MGN · CylinderFlow | Pressure RMSE (×10^3): 36.7\scriptsize{\;±\;}{2.4} -> \textbf{28.9\scriptsize{\;±\;}{1.5}} | Helped |
| Add PIORF rewiring | MGN · AirFoil | Velocity RMSE (×10^3): 10,261\scriptsize{\;±\;}{832} -> \textbf{7,743\scriptsize{\;±\;}{584}} | Helped |
| Add PIORF rewiring | MGN · AirFoil | Pressure RMSE (×10^3): 3,043,186\scriptsize{\;±\;}{282,514} -> \textbf{2,245,858\scriptsize{\;±\;}{142,452}} | Helped |
| Add PIORF rewiring | MGN · AirFoil | Density RMSE (×10^3): 29.4\scriptsize{\;±\;}{2.7} -> \textbf{22.5\scriptsize{\;±\;}{1.4}} | Helped |
| Add PIORF rewiring | MGN · EAGLE | Velocity RMSE (×10^3): 2,280\scriptsize{\;±\;}{135} -> 1,950\scriptsize{\;±\;}{28} | Helped |
| Add PIORF rewiring | MGN · EAGLE | Pressure RMSE (×10^3): 10,893\scriptsize{\;±\;}{632} -> 9,449\scriptsize{\;±\;}{167} | Helped |
| Use pressure instead of velocity to select rewiring nodes (ablation) | PIORF · CylinderFlow | Velocity RMSE (×10^3): 41.6\scriptsize{\;±\;}{3.9} -> 80.9\scriptsize{\;±\;}{13.4} | Hurt |
| Select the second node randomly instead of using physical context (ablation) | PIORF · CylinderFlow | Velocity RMSE (×10^3): 41.6\scriptsize{\;±\;}{3.9} -> 43.4\scriptsize{\;±\;}{2.4} | Hurt |
| Remove edges instead of adding them (ablation) | PIORF · CylinderFlow | Velocity RMSE (×10^3): 41.6\scriptsize{\;±\;}{3.9} -> 42.0\scriptsize{\;±\;}{2.3} | Hurt |
| Add edges and remove an edge (ablation) | PIORF · CylinderFlow | Velocity RMSE (×10^3): 41.6\scriptsize{\;±\;}{3.9} -> 49.0\scriptsize{\;±\;}{7.5} | Hurt |
| Use velocity L2 distance as edge weights (ablation) | PIORF · CylinderFlow | Velocity RMSE (×10^3): 41.6\scriptsize{\;±\;}{3.9} -> 53.2\scriptsize{\;±\;}{8.6} | Hurt |
Surprising findings
- Adding edges alone outperforms adding plus removing edges; removal of highest-positive-curvature edges did not help.
- Weighted edges by velocity L2 distance in ORC computation did not improve performance.
Limitations
- Depends on the choice of physical quantity for rewiring (velocity vs pressure), which must be tuned per dataset.
- Pooling ratio delta must be tuned per dataset; 9% increases variance and 1% can underperform MGN.
- Only tested on fluid dynamics mesh simulations; extension to other physics not evaluated.
Experimental setup
Datasets: CylinderFlow AirFoil EAGLE CylinderFlow-Tiny
Benchmarks: CylinderFlow AirFoil EAGLE
Metrics: RMSE (rollout-all) computation time total effective resistance
Baselines: MGN DIGL SDRF FoSR BORF BSMS Graph Transformer HMT
Builds on: MeshGraphNets Ollivier-Ricci curvature BORF SDRF FoSR DIGL
Closest papers by meaning
- PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systems (ICLR 2025 · similarity 0.88)
- MeshMask: Physics-Based Simulations with Masked Graph Neural Networks (ICLR 2025 · similarity 0.87)
- Learning Distributions of Complex Fluid Simulations with Diffusion Graph Networks (ICLR 2025 · similarity 0.87)
- Prometheus: Out-of-distribution Fluid Dynamics Modeling with Disentangled Graph ODE (ICML 2024 · similarity 0.86)
- Towards Universal Mesh Movement Networks (NeurIPS 2024 · similarity 0.86)
- UGM2N: An Unsupervised and Generalizable Mesh Movement Network via M-Uniform Loss (NeurIPS 2025 · similarity 0.86)
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