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EddyFormer: Accelerated Neural Simulations of Three-Dimensional Turbulence at Scale
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Proposes EddyFormer, a spectral-element Transformer with LES-style SGS and LES streams that corrects a coarse pseudo-spectral DNS to DNS-level accuracy for 3D and 2D turbulence, with a 30x speedup and evaluation on The Well.
Problem
Accurate and efficient simulation of multi-scale turbulent flows; existing neural operators and Transformers struggle to scale to large 3D turbulence and often fail to converge on diverse turbulent flows.
Approach
Tokenizes flow with spectral-element bases into coarse elements (SEM tokens), splits it into an LES stream (attention over SEM tokens for global structure, with spectral cutoff filter) and an SGS stream (local SEM-based spectral convolutions), with rotary positional encoding and an attention mask for domain generalization.
Key results
- On 3D HIT, EddyFormer (Legendre) reaches 8.52% one-step test error versus 22.5% for the best baseline F-FNO in direct prediction, and is ~30x faster than DNS 256^3 pipeline including coarse DNS.
- On 2D Kolmogorov flow learned correction, EddyFormer achieves 0.14% relative L2 error at t=5 versus 0.27% for F-FNO and 0.36% for ResNet.
- On The Well, EddyFormer (Chebyshev) reduces one-step VRMSE on rayleigh_benard to 0.2527 (Legendre 0.1854) vs best baseline 0.6699, and achieves 0.1115 on rayleigh_taylor_instability where all baselines exceed 10.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use EddyFormer instead of F-FNO for direct one-step prediction | 3D homogeneous isotropic turbulence · 3D homogeneous isotropic turbulence | one-step relative error: 22.5% -> 8.52% | Helped |
| Use EddyFormer (Legendre) instead of DNS 256³ | 3D homogeneous isotropic turbulence · 3D homogeneous isotropic turbulence, t = 5 | relative L2 error: 16.3% -> 18.2% | Hurt |
| Use EddyFormer (Legendre) instead of DNS 256³ | 3D homogeneous isotropic turbulence · 3D homogeneous isotropic turbulence, t = 5 | inference time: 152s -> 4.86s | Helped |
| Use EddyFormer instead of F-FNO | Kolmogorov flow · Kolmogorov flow test set, t = 10 | relative L2 error: 5.28 -> 2.61 | Helped |
| Use EddyFormer instead of F-FNO | Kolmogorov flow · Kolmogorov flow test set, t = 20 | point-wise correlation: 0.20 -> 0.27 | Helped |
| Use EddyFormer instead of F-FNO | Kolmogorov flow · Kolmogorov flow test set | time-averaged total kinetic energy, [45, 50]: 1.511 -> 1.670 | Mixed |
| Use EddyFormer instead of CNextU-net | The Well · MHD_64 | VRMSE: 0.1633 -> 0.1160 | Helped |
| Use EddyFormer instead of CNextU-net | The Well · shear_flow | VRMSE: 0.8080 -> 0.0837 | Helped |
| Use EddyFormer instead of CNextU-net | The Well · rayleigh_benard | VRMSE: 0.6699 -> 0.1854 | Helped |
| Use EddyFormer instead of FNO | The Well · rayleigh_taylor_instab. | VRMSE: >10 -> 0.1115 | Helped |
| Use Legendre rather than Chebyshev basis (ablation) | EddyFormer · 3D homogeneous isotropic turbulence | one-step relative error: 8.61% -> 8.52% | Helped |
| Use Legendre rather than Chebyshev basis (ablation) | EddyFormer · The Well | VRMSE: MHD_64: 0.1160 -> 0.1240; shear_flow: 0.1123 -> 0.0837; rayleigh_benard: 0.2527 -> 0.1854; rayleigh_taylor_instab.: 0.11 | Mixed |
Surprising findings
- Baselines such as FNO and U-net diverge (VRMSE >10) on the Rayleigh-Taylor instability case, where EddyFormer remains stable.
- Standard neural operators lack a natural domain-expansion mechanism; F-FNO fails to match energy spectra on 2x and 4x larger domains.
Limitations
- Evaluated mainly on homogeneous or periodic turbulence; extension to complex geometries is left to future work.
- Domain generalization requires attention masking and is tested only on 2D flows.
Experimental setup
Datasets: The Well Homogeneous isotropic turbulence dataset (generated) Kolmogorov flow dataset (generated)
Benchmarks: The Well (MHD_64, shear_flow, rayleigh_benard, rayleigh_taylor_instability) 3D homogeneous isotropic turbulence 2D Kolmogorov flow
Metrics: relative L2 error VRMSE point-wise correlation energy spectrum structure function Q-criterion kinetic energy conservation inference time
Baselines: ResNet TF-Net FNO F-FNO GNOT AViT U-net CNextU-net T-FNO DNS 96^3 DNS 256^3
Builds on: Fourier neural operator spectral element method large-eddy simulation rotary position encoding The Well
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