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Wavelet Diffusion Neural Operator
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Proposes WDNO, a diffusion model operating in the wavelet domain with multi-resolution training, for PDE simulation, control, and zero-shot super-resolution.
Problem
Diffusion models for PDE simulation and control struggle with abrupt changes (shocks, discontinuities) and fail to generalize to finer resolutions than seen in training.
Approach
Perform classifier-free-guided diffusion over wavelet coefficients of whole trajectories (rather than raw space-time grids), and train a super-resolution diffusion model on downsampled data pairs exploiting approximate scale invariance so that it generalizes zero-shot to finer grids; control uses objective-guided denoising.
Key results
- On 1D compressible Navier-Stokes, WDNO achieves MSE 0.2195 versus 5.5228 for DDPM and 0.2575 for FNO.
- On 2D incompressible fluid control, WDNO reduces the smoke-leakage objective J to 0.0679 versus 0.3066 for the next best method (BPPO), about 78% reduction.
- On 1D Burgers' control, WDNO achieves J=0.0205, slightly better than SL (0.0235) and DDPM (0.0272).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use wavelet-domain generation instead of the original DDPM approach (ablation) | DDPM · 1D Navier-Stokes | MSE: 5.5228 -> 0.2195 | Helped |
| replace wavelet transform with Fourier transform (ablation) | WDNO · 1D Navier-Stokes | MSE: 0.2195 -> 3.0258 | Hurt |
| use FNO as the noise prediction model (ablation) | WDNO · 1D Navier-Stokes | MSE: 0.2195 -> 145.0469 | Hurt |
| add objective guidance during denoising (λ = 120000) (ablation) | WDNO · 1D Burgers' control | J: 0.3360 -> 0.0205 | Helped |
| train FNO on mixed-resolution rather than individual-resolution data (ablation) | FNO · 1D Burgers' original resolution | MSE: 4.94e-04 -> 4.72e-04 | Helped |
| add multi-resolution training for zero-shot super-resolution (ablation) | WDNO · 1D Burgers' zero-shot super-resolution, 3 steps | MSE: FNO (linear) 0.05610; WDNO (linear) 0.00035 | Helped |
| increase DDIM sampling steps from 20 to 50 (ablation) | WDNO · 1D Burgers' simulation | MSE: 0.00022 -> 0.00014 | Helped |
| increase DDIM η from 0.2 to 1 (ablation) | WDNO · 1D Burgers' simulation | MSE: 0.00020 -> 0.00014 | Helped |
| increase DDIM η from 0.2 to 1 (ablation) | WDNO · 2D incompressible fluid control | Results: 0.2285 -> 0.0205 | Helped |
| increase training samples from 0.2 times to 0.4 times the current dataset (ablation) | WDNO · 1D Navier-Stokes | MSE | Mixed |
| use multi-resolution training in the original space-time domain rather than the wavelet domain (ablation) | WDNO · 1D Burgers' zero-shot super-resolution | MSE | Hurt |
Surprising findings
- On 1D Burgers' simulation, WDNO (MSE 0.00014) is nearly tied with DDPM (0.00013), so most simulation gains over DDPM come on the other systems.
- Diffusion in the Fourier domain gives some gain over DDPM but is far worse than wavelet-domain diffusion, and an FNO used as the noise predictor performs very poorly (MSE 145.05).
Limitations
- Applicable only to static, uniform grid data because of the wavelet transform and U-Net denoiser.
- No real-world experiments; no physics-informed loss incorporated.
- Inference time and memory grow with super-resolution level.
Experimental setup
Datasets: 1D Burgers' equation (PDEBench-style setup) PDEBench 1D advection PDEBench 1D compressible Navier-Stokes 2D incompressible fluid (smoke control) ERA5 temperature
Benchmarks: 1D Burgers' equation 1D advection 1D compressible Navier-Stokes 2D incompressible fluid ERA5
Metrics: MSE MAE L_infinity error control objective J relative L2 error
Baselines: DDPM WNO MWT FNO CNN OFormer U-Net PID SL SAC BC BPPO
Builds on: DDPM classifier-free guidance DDIM Fourier Neural Operator Wavelet Neural Operator
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