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Diffusion-Based Hierarchical Graph Neural Networks for Simulating Nonlinear Solid Mechanics
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Proposes ROBIN, a diffusion-based hierarchical GNN (AMG-coarsened AMPN) simulator with rolling diffusion-batched inference (ROBI) for nonlinear solid mechanics, achieving higher accuracy and up to 10x faster inference than standard diffusion simulators.
Problem
Learned mesh-based simulators struggle to capture global deformation (bending, long-range correlations) and accumulate errors over long rollouts, while diffusion-based simulators are expensive at inference due to iterative denoising.
Approach
Combines DDPM-based residual denoising with a hierarchical GNN built on algebraic multigrid coarsening (V-cycle, shared layers), and a rolling batched inference scheme (ROBI) that overlaps denoising steps across consecutive physical time steps so each time step needs one model call after warm-up.
Key results
- On BendingBeam, ROBIN (1/20) achieves 29.00e-3 displacement RMSE versus 121.53e-3 for HCMT and 189.52e-3 for MGN.
- ROBIN default (1/20) matches conventional diffusion inference accuracy (20/20: 28.90e-3) at 15.03 s vs 162.43 s on BendingBeam, roughly an order of magnitude faster.
- On ImpactPlate ROBIN reaches 12.33e-3 RMSE vs 19.57e-3 for HCMT; on DeformingPlate 4.98e-3 vs 8.04e-3 for HCMT.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace the strongest baseline HCMT with ROBIN | nonlinear solid-mechanics simulators · BendingBeam | displacement RMSE: 121.53 $\pm$ 1.87 -> 29.00 $\pm$ 1.00 | Helped |
| replace the strongest baseline HCMT with ROBIN | nonlinear solid-mechanics simulators · ImpactPlate | displacement RMSE: 19.57 $\pm$ 0.38 -> 12.33 $\pm$ 0.96 | Helped |
| replace the strongest baseline HCMT with ROBIN | nonlinear solid-mechanics simulators · DeformingPlate | displacement RMSE: 8.04 $\pm$ 0.13 -> 4.98 $\pm$ 0.33 | Helped |
| use rolling diffusion-batched inference (1/20) instead of conventional diffusion inference (20/20) | ROBIN · BendingBeam | rollout time: 162.43 $\pm$ 0.89 -> 15.03 $\pm$ 0.04 | Helped |
| use rolling diffusion-batched inference (1/20) instead of conventional diffusion inference (20/20) | ROBIN · ImpactPlate | displacement RMSE: 14.43 $\pm$ 1.93 -> 12.33 $\pm$ 0.96 | Helped |
| reduce diffusion length from the full schedule to 10 or 5 diffusion steps (ablation) | ROBIN · BendingBeam, ImpactPlate, and DeformingPlate | displacement RMSE | Hurt |
| remove diffusion and train the AMPN as a one-step MSE predictor (ablation) | ROBIN · BendingBeam, ImpactPlate, and DeformingPlate | displacement RMSE | Hurt |
| remove hierarchical message passing (ablation) | ROBIN · BendingBeam, ImpactPlate, and DeformingPlate | displacement RMSE | Hurt |
| replace residual prediction with direct state prediction (ablation) | ROBIN · BendingBeam, ImpactPlate, and DeformingPlate | displacement RMSE | Hurt |
| replace shared message-passing layers with 15 unshared layers (ablation) | ROBIN · BendingBeam | displacement RMSE | Hurt |
| replace AMPN with HCMT while retaining diffusion and ROBI (ablation) | ROBIN · BendingBeam, ImpactPlate, and DeformingPlate | displacement RMSE | Hurt |
| truncate denoising at step 1 instead of 20 (ablation) | ROBIN · BendingBeam | displacement RMSE: 29.00 $\pm$ 1.00 -> 64.23 $\pm$ 11.86 | Hurt |
Surprising findings
- Partially denoised rollouts with a single denoising step per physical step (1/1) still outperform all baselines in displacement RMSE on BendingBeam.
- Default ROBIN is slightly more accurate than full conventional diffusion inference on ImpactPlate, attributed to anchoring low-frequency components.
Limitations
- Lacks SO(3) equivariance, which could matter for orientation-sensitive predictions.
- Only DDPM formulation was studied; other diffusion formulations and schedules were not explored.
- Experiments are limited to nonlinear solid mechanics; fluid dynamics and other domains are untested.
Experimental setup
Datasets: BendingBeam ImpactPlate DeformingPlate BendingBeamLarge
Benchmarks: BendingBeam ImpactPlate DeformingPlate BendingBeamLarge
Metrics: RMSE gradient RMSE rollout wall-clock time
Baselines: MeshGraphNet HCMT BSMS PDE-Refiner
Builds on: PDE-Refiner MeshGraphNets Hierarchical Contact Mesh Transformer BSMS DDPM
Closest papers by meaning
- MaNGO — Adaptable Graph Network Simulators via Meta-Learning (NeurIPS 2025 · similarity 0.87)
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- EvoMesh: Adaptive Physical Simulation with Hierarchical Graph Evolutions (ICML 2025 · similarity 0.85)
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- PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systems (ICLR 2025 · similarity 0.84)
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