Organizations: The Hong Kong University of Science and Technology (Guangzhou), Guangzhou, China · SandGold AI Research, Guangzhou, China · The Hong Kong University of Science and Technology, Hong Kong SAR, China · Southern University of Science and Technology, Shenzhen, China
Existing image SR and generic diffusion models transfer poorly to fluid SR: they are sampling-intensive, ignore physical constraints, and often yield spectral mismatch and spurious divergence. We address fluid super-resolution (SR) with ReMD (Residual-Multigrid Diffusion), a physics-consistent diffusion framework. At each reverse step, ReMD performs a multigrid residual correction: the update direction is obtained by coupling data consistency with lightweight physics cues and then correcting the residual across scales; the multiscale hierarchy is instantiated with a multi-wavelet basis to capture both large structures and fine vortical details. This coarse-to-fine design accelerates convergence and preserves fine structures while remaining equation-free. Across atmospheric and oceanic benchmarks, ReMD improves accuracy and spectral fidelity, reduces divergence, and reaches comparable quality with markedly fewer sampling steps than diffusion baselines. Our results show that enforcing physics consistency inside the diffusion process via multigrid residual correction and multi-wavelet multiscale modeling is an effective route to efficient fluid SR. Our code are available on https://github.com/lizhihao2022/ReMD.
Figures & tables
Figure 1: Qualitative comparison on NS ( × 2 ) and ERA5 ( × 4 ). Each block shows (left) a zoomed LR input u0 , (top row) HR reconstructions, and (bottom row) absolute error maps w.r.t. ground truth (shared color scale per block). With only 5 reverse steps, ReMD yields (NS) sharper filaments/coherent fronts and (ERA5) preserved mesoscale shear bands while suppressing ringing/stripe artifacts seen in EDSR/FNO/SwinIR, and achieves lower errors than ResShift despite requiring 5 vs. 15 steps.
Figure 2: Overview of ReMD. Starting from a coarse LR initial solution, the reverse diffusion steps refine the HR estimate ut while a multigrid residual correction module combines data-consistency and physics residuals in the wavelet domain, yielding the final HR field u^ .
Model
NS
ERA5
Ocean
RMSE ↓
PSNR ↑
SSIM ↑
RMSE ↓
PSNR ↑
SSIM ↑
RMSE ↓
PSNR ↑
SSIM ↑
EDSR
2.97E-02
46.89
0.996
9.06E-02
57.13
0.998
1.36E-02
47.48
0.982
FNO
4.42E-02
43.44
0.988
1.35E-01
53.69
0.997
1.67E-02
45.74
0.974
MWT
8.45E-02
37.81
0.970
6.13E-01
40.53
0.968
4.55E-02
37.03
0.852
HiNOTE
8.03E-02
38.25
0.973
2.12E-01
49.76
0.994
1.99E-02
44.21
0.964
Galerkin
5.32E-02
41.83
0.981
3.45E-01
45.51
0.988
3.75E-02
38.70
0.890
Table 1: Quantitative comparison on three fluid SR benchmarks. NS ( 2× ), ERA5 ( 4× ), and Ocean ( 4× ). We report RMSE ↓ , PSNR ↑ , and SSIM ↑ . Bold denotes the best and underline the second-best within each dataset/metric block. ReMD attains the lowest errors and top (or tied) perceptual scores across datasets while using only 2–5 reverse steps, compared with 15 steps for the ResShift baseline.
Figure 3: Error–energy spectrum on ERA5 (, × 4,). Radial average of the Fourier‐domain error (log scale on y) versus frequency (x). Vertical dashed lines mark the LR Nyquist band and the transition toward the HR band. ReMD-5 (red) maintains the lowest error from large to high scales, remaining below ResShift-15 , FNO and image-SR baselines (EDSR, SwinIR), indicating superior spectral fidelity.
Figure 4: Patch-level comparison on NS ( × 4). Left: zoomed LR input. Black boxes highlight frontal bands; gray boxes mark small eddies. ReMD–5 recovers sharper, HR-like fronts and coherent vortices with fewer steps than ResShift–15, while avoiding texture/aliasing artifacts seen in image-SR baselines and blockiness in FNO, and remaining consistent with the LR content.
Model Configuration
RMSE ↓
PSNR ↑
SSIM ↑
w/o Residual correction
1.38E-02
47.37
0.981
w/o smoothing residual
1.40E-02
47.28
0.981
w/o diffusion residual
1.40E-02
47.26
0.981
w/o Spectrum residual
1.41E-02
47.20
0.981
ReMD (Baseline)
1.33E-02
47.71
0.983
Table 2: Ablation study ( Ocean , 4× SR). Removing the MG corrector (“w/o residual correction”) or any physics residual (smoothing, anisotropic diffusion, spectrum) degrades all metrics; dropping the spectrum term yields the largest deterioration.
Metrics
Methods
EDSR
FNO
MWT
HiNOTE
Galerkin
SwinIR
SR3
Resshift
ReMD-2
ReMD-5
RMSE ↓
2.97E-02
4.42E-02
8.45E-02
8.03E-02
5.32E-02
3.63E-02
3.34E-01
2.21E-02
2.11E-02
2.09E-02
PSNR ↑
46.89
43.44
37.81
38.25
41.83
45.15
25.87
49.47
49.84
49.94
Param Count
1367553
684065
272605
420618
703777
3696633
93868033
112366821
118584941
Param(MB)
5.22
5.17
1.75
1.61
2.68
17.85
358.17
428.65
455.55
Training Time (s/epoch)
8.87
4.78
1.72
57.91
146.25
86.26
53.48
64.1
100.33
Table 3: Efficiency on the NS. We report accuracy (RMSE ↓ /PSNR ↑ ), parameter size, and wall-time (training time per epoch; inference time in seconds). Diffusion baselines (SR3, ResShift-15) incur high sampling cost; ReMD-5 attains the best accuracy with fewer steps and lower inference time than ResShift, while ReMD-2 is the fastest with competitive accuracy.
Model
VE ↓
EE ↓
GED ↓
RMSE ( ×8 ) ↓
FNO
5.01E-03
2.57E-05
5.45E-02
3.76E-01
EDSR
6.43E-03
9.41E-05
4.38E-03
3.39E-01
SwinIR
6.16E-03
6.86E-05
5.10E-03
3.34E-01
LIIF
4.99E-03
2.08E-05
6.14E-03
3.32E-01
ResShift
3.26E-03
1.11E-05
4.51E-03
3.49E-01
ReMD (ours)
2.34E-03
3.56E-06
4.24E-03
3.23E-01
Table 4: Physics-related metrics (lower is better ↓ ). VE/EE are vorticity/enstrophy errors on NS2D ( ×4 ). GED is the energy discrepancy on ERA5_uo ( ×4 ). The last column reports RMSE on ERA5_uo ×8 super-resolution.
Figure 5: Time–RMSE trade-off on ERA5_uo ( ×4 SR). Each point varies the sampling steps; lower-left is better.