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AROMA: Preserving Spatial Structure for Latent PDE Modeling with Local Neural Fields
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Proposes AROMA, an attention-based encode-process-decode framework that compresses irregular-grid or point-cloud PDE states into a fixed set of latent tokens, models their dynamics with a conditional diffusion transformer, and decodes with a local neural field.
Problem
Surrogate modeling of time-dependent PDEs on general geometries (irregular grids, point clouds) is expensive for transformers operating in physical space, and neural fields struggle to capture spatial structure and local dynamics; long autoregressive rollouts also become unstable.
Approach
Encode variable-size observations via cross-attention into M fixed latent tokens (VAE-regularized), run a diffusion transformer for latent time-marching, and decode with a local INR whose queries cross-attend to the tokens, giving spatially interpretable latent tokens and stable long rollouts.
Key results
- On regular grids, AROMA achieves relative L2 of 3.65e-2 on Burgers and 1.05e-1 on Navier-Stokes 1e-4, best among reported baselines (FNO 1.53e-1 on NS 1e-4).
- On irregular grids with pi=5% sparsity, AROMA gets Shallow-Water In-t MSE 1.93e-3 vs CORAL 2.72e-3, but CORAL is better on NS 1e-3 at 5% sparsity (2.87e-3 vs 4.73e-3).
- On CylinderFlow, AROMA reaches In-t MSE 1.48e-2 vs CORAL 4.46e-2; on AirfoilFlow, Out-t MSE 1.94e-1 vs CORAL 3.42e-1.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use AROMA instead of the strongest baseline | PDE forecasting models · Burgers | Relative L2: 4.92 \times 10^{-2} -> 3.65 \times 10^{-2} | Helped |
| Use AROMA instead of the strongest baseline | PDE forecasting models · Navier-Stokes 1e-4 | Relative L2: 1.53 \times 10^{-1} -> 1.05 \times 10^{-1} | Helped |
| Use AROMA instead of the strongest baseline | PDE forecasting models · Navier-Stokes 1e-5 | Relative L2: 1.24 \times 10^{-1} -> 1.24 \times 10^{-1} | No clear effect |
| Use AROMA instead of the strongest baseline | PDE forecasting models · Navier-Stokes 1e-3 with 25% observations | MSE: 1.54 \times 10^{-3} -> 7.02 \times 10^{-4} | Helped |
| Use AROMA instead of CORAL | PDE forecasting models · Navier-Stokes 1e-3 with 5% observations | MSE: 2.87 \times 10^{-3} -> 4.73 \times 10^{-3} | Hurt |
| Use AROMA instead of CORAL | PDE forecasting models · Shallow-Water with 100% observations | MSE: 2.12 \times 10^{-5} -> {3.10 \times 10^{-5}} | Hurt |
| Use AROMA instead of CORAL | PDE forecasting models · CylinderFlow and AirfoilFlow | MSE | Helped |
| Increase the number of latent tokens (ablation) | AROMA encoder-decoder · Navier-Stokes 1e-4 reconstruction | Relative L2 error: 0.02664 -> 0.0123 -> 0.01049 | Helped |
| Replace VAE with an L2-regularized autoencoder (ablation) | AROMA · Burgers | Relative L2: 3.65 \times 10^{-2} -> 3.43 \times 10^{-2} | Helped |
| Replace VAE with an L2-regularized autoencoder (ablation) | AROMA · Navier-Stokes 1e-4 | Relative L2: 1.05 \times 10^{-1} -> 5.02 \times 10^{-2} | Helped |
| Replace VAE with an L2-regularized autoencoder (ablation) | AROMA · Navier-Stokes 1e-5 | Relative L2: 1.24 \times 10^{-1} -> 2.10 \times 10^{-1} | Hurt |
| Remove diffusion and train the transformer with MSE on mean tokens (ablation) | AROMA · Burgers long rollouts | Relative L2: 3.65 \times 10^{-2} -> 4.31 \times 10^{-2} | Hurt |
Surprising findings
- The deterministic (no-diffusion) variant of AROMA surpasses the diffusion version on Navier-Stokes 1e-4 in the authors' ablation.
- AROMA struggles on the chaotic Kuramoto-Sivashinsky equation, with the encoder-decoder reconstruction identified as the main bottleneck.
Limitations
- Performance not demonstrated on larger or real-world datasets; experiments use small datasets.
- Decoder reconstruction quality bounds rollout accuracy; fails on chaotic Kuramoto-Sivashinsky dynamics.
- Uncertainty quantification via the generative model is only preliminarily explored.
Experimental setup
Datasets: Burgers Navier-Stokes 1e-4 Navier-Stokes 1e-5 Navier-Stokes 1e-3 Shallow-Water CylinderFlow AirfoilFlow Kuramoto-Sivashinsky
Benchmarks: Burgers Navier-Stokes 1e-4 Navier-Stokes 1e-5 Navier-Stokes 1e-3 Shallow-Water CylinderFlow AirfoilFlow Kuramoto-Sivashinsky
Metrics: relative L2 error MSE correlation duration
Baselines: FNO ResNet DINO CORAL GNOT OFormer PDE-Refiner
Builds on: DINO CORAL DDPM DiT (diffusion transformer) Perceiver
Closest papers by meaning
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