Research map / Probabilistic and scientific modeling / Dynamical systems
Learning Distributions of Complex Fluid Simulations with Diffusion Graph Networks
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Proposes graph-based diffusion models (DGN) and a latent variant (LDGN, with a VGAE on a multi-scale graph) that directly sample equilibrium fluid-flow states from short simulation trajectories on unstructured meshes.
Problem
Obtaining full probability distributions and flow statistics of unsteady, chaotic fluid systems on unstructured meshes normally requires long, expensive simulations; existing GNN surrogates predict mean flows or trajectories and generative GNN models scale poorly to thousands of nodes.
Approach
Applies DDPM directly on mesh nodes with a multi-scale (U-Net-style) GNN denoiser built on Guillard coarsening, and a latent variant (LDGN) that diffuses in a VGAE latent graph with a condition encoder, learning equilibrium distributions from short, incomplete trajectories.
Key results
- On the Ellipse task the LDGN achieved the highest sample accuracy (R^2), closely followed by the DGN; on EllipseFlow the DGN was best, followed by LDGN.
- LDGN samples had on average 3x reduced high-frequency error relative to DGN, and LDGN was reported as 8x faster than DGN at sampling while requiring about 55% more training time.
- On the Wing task, LDGN generated 3,000 samples in 2.43 minutes on a single GPU versus about 2,989 minutes of ground-truth simulation on 8 CPU threads.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use LDGN instead of DGN | DGN · Wing | generation speed: 8× faster | Helped |
| use LDGN instead of DGN | DGN · EllipseFlow-InDist and Wing-Test | high-frequency error: 3x reduced high-frequency error | Helped |
| use LDGN instead of DGN | DGN · Ellipse, EllipseFlow, and Wing | Wasserstein-2 distributional accuracy | Helped |
| use a multi-scale processor instead of a single-scale processor (ablation) | DGN · Ellipse | sample and distributional accuracy | Helped |
| use a multi-scale processor instead of a single-scale processor (ablation) | DGN · EllipseFlow and Wing | feasible solution generation | Helped |
| increase the number of scales (ablation) | DGN · EllipseFlow | model performance | Helped |
| use five scales instead of two scales (ablation) | DGN · EllipseFlow | computational efficiency: 25% | Helped |
| train the GM-GNN on shorter rather than complete trajectories | GM-GNN · Ellipse-InDist | Wasserstein-2 node distance | Hurt |
| use GM-GNN instead of DGN or LDGN on complete trajectories | DGN and LDGN · Ellipse | Wasserstein-2 node distance | Helped |
| use VGAE instead of LDGN | LDGN · EllipseFlow | distributional accuracy | Hurt |
Surprising findings
- The GM-GNN achieves the lowest node-level W2 on the Ellipse task when trained on complete trajectories, yet its graph-level W2 is consistently high because it ignores spatial correlations.
- The DGN and LDGN distributional accuracy is barely affected when training trajectories are shortened below a full oscillation period, while most baselines degrade sharply.
Limitations
- Generated samples are not temporally correlated; the model lacks temporal conditioning.
- Performance degrades as the variance of ground-truth trajectories increases.
- Sampling requires many denoising steps (50 steps used with accelerated sampling), and the single-scale DGN fails on larger meshes.
Experimental setup
Datasets: Ellipse EllipseFlow Wing
Benchmarks: Ellipse EllipseFlow Wing
Metrics: coefficient of determination (R^2) Wasserstein-2 distance (node-level and graph-level) graph Fourier high-frequency energy
Baselines: Vanilla GNN Bayesian GNN Gaussian regression GNN Gaussian mixture GNN VGAE VAE
Builds on: DDPM Improved DDPM (learned variance) Guillard coarsening latent diffusion models U-Net
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