High-fidelity Three-Dimensional Flow Field Reconstruction Driven by Turbulent Fluid Dynamics

Hanlin Kong; Jianan Wei; Yuntian Chen; Yujia Wang; Wenguan Wang

Figure 1: Overview of the proposed framework for three-dimensional turbulence super-resolution. a , Set-up and overall framework. Given an LR three-dimensional sequence

X_{\mathrm{L}}

, the model learns a mapping

\mathcal{M}_{\theta}

that produces a representative HR estimate conditioned on the LR input. b , Illustration of the state space model at time step

t

, which captures temporal dynamics of turbulent flows. c , Illustration of the multiscale neural operator for recovering multiscale spatial structure of turbulent flows. d , Illustration of an MoE mechanism with unsupervised adaptation, enabling generalization to unknown coarse-graining.

Methods	Coarse- graining	FIT		MHDT		HBDT
Methods	Coarse- graining	4 $\times$	8 $\times$	4 $\times$	8 $\times$	8 $\times$	16 $\times$
Tricubic	Box/ Favre	$0.1204$	$0.2242$	$0.1838$	$0.3372$	$0.0933$	$0.1666$
RRDB		$0.0460$	$0.1712$	0.0678	$0.2205$	$0.0307$	0.0940
RCAN		0.0415	$0.1753$	$0.0688$	$0.2285$	0.0262	$0.1396$
EDSR		$0.0423$	$0.1693$	$0.0699$	0.2250	$0.0286$	$0.1301$
FNO		$0.0730$	$0.1657$	$0.1166$	$0.2525$	$0.0544$	$0.1224$

Table 1: Numerical comparison of reconstruction metrics nRMSE (Eq. ( 18 )) between the proposed method and baseline methods on the TurbSR3D benchmark .

Figure 2: Point-wise fidelity across three turbulence regimes and representative field quantities. a, Predicted vorticity magnitude

\|\bm{\omega}\|=\|\nabla\times\bm{u}\|

and corresponding residual maps on FIT with

4\times

upscaling. b, Predicted current magnitude

\|\bm{j}\|=\|\nabla\times\bm{b}\|

and residual maps on MHDT with

4\times

upscaling. c, Predicted velocity divergence

\nabla\!\cdot\!\bm{u}

and residual maps on HBDT with

8\times

upscaling.

Figure 3: Statistical fidelity across three turbulence regimes. a, Energy spectra for FIT (

4\times

), MHDT (

4\times

), and HBDT (

8\times

) under spectral-cutoff coarse-graining. b, Probability density functions (PDFs) of representative field quantities: vorticity magnitude

\|\bm{\omega}\|

(FIT,

4\times

), current magnitude

\|\bm{j}\|

(MHDT,

4\times

), and velocity divergence

\nabla\!\cdot\!\bm{u}

(HBDT,

8\times

) under box/Favre coarse-graining.

(\cdot)_{\mathrm{rms}}

denotes the root-mean-square magnitude. c, Joint PDFs of normalized velocity-gradient invariants

(Q^{*},R^{*})

for FIT at

4\times

and

8\times

with box-filtered inputs. Black and colored contours denote DNS and reconstructions, respectively; the magenta dashed curve marks the zero-discriminant boundary. Evaluation details are given in Supplementary Note 3.5 .

Figure 4: Q-criterion structures over time. a , Reference HR flow at six instants

t_{0},\dots,t_{100}

(left to right) on FIT, with a uniform cadence

\Delta t\!=\!2\!\!\times\!\!10^{-3}

. For each time, an isosurface of the

Q

-criterion highlights rotation-dominated regions; surfaces are color-mapped by speed

\|\bm{u}\|

. b and c , Super-resolution experiments at

4\times

and

8\times

: for each SR factor, the top row shows the LR field, and the bottom row shows the reconstruction results by our method, evaluated at the same times as a . The LR

Q

fields are computed from the coarse-grained velocity components, with velocity gradients evaluated using the LR grid spacing.

Figure 5: Temporal coherence of reconstructions. a, Time–space (

t

–

x

) slices of vorticity magnitude

\|\bm{\omega}\|

for a representative FIT sequence. The horizontal axis denotes time

t

and the vertical axis denotes the spatial coordinate

x

. The top row shows the reference (HR) field and reconstructions from different methods, and the bottom row shows error maps. b , Quantitative temporal error measured by tRMSE (lower is better; values reported in units of

\times 10^{-3}

), computed using Eq. ( 19 ), for FIT (

4\times

, left), MHDT (

4\times

, middle), and HBDT (

8\times

, right).

Figure 6: Dynamical consistency in DIT. Reconstructions are evaluated at

4\times

upscaling with spectral-cutoff inputs on DIT dataset (Supplementary Note 2.7 ). a , Reference flow fields at five times from

t=0.2

to

t=0.6

. b,c , Evolution of normalized kinetic energy

K(t)/K(t_{0})

and Taylor-scale Reynolds number

Re_{\lambda}

in the DNS. d , Space–time slices of vorticity magnitude

\|\bm{\omega}\|

for the DNS reference (Ref.), the full model (Ours), and the model trained without

\mathcal{L}_{\mathrm{phy}}

, with reconstruction error maps beneath the predictions. e,f , Reconstruction error (nRMSE) and temporal-increment error (tRMSE); lower values indicate greater fidelity. g , Spatially averaged viscous dissipation normalized by the DNS value at the corresponding time. All displayed values are below unity, so higher ratios indicate closer agreement with DNS. Red and blue curves in e–g denote the full model and the variant without

\mathcal{L}_{\mathrm{phy}}

, respectively.

Figure 7: Atmospheric downscaling on ERA5 reanalysis data. a , Mean Squared Error (MSE) and temporal error (tRMSE, Eq. ( 19 )) computed at

4\times

and

8\times

downscaling factors. b , Kinetic energy spectra

E(k)

at 850 hPa plotted against wavenumber

k

. c , Visualizations of temperature (top) and wind speed (bottom) fields at 850 hPa. The panels display the HR reference (Ref.) alongside reconstructions from the proposed method and SwinIR. Insets display magnified views of local regions.

Figure 8: Generalization to unknown coarse-graining. a , nRMSE of the Box, Gaussian, and Spectral experts and the MoE under three unseen coarse-graining settings: tricubic downsampling, Lanczos filtering, and uniform subsampling with

5\%

additive noise. b , Energy spectrum

E(k)

for the tricubic case. c , Reconstruction of vorticity magnitude

\|\bm{\omega}\|

under tricubic downsampling, showing the HR reference, LR input, and outputs from the MoE and individual experts. d , Joint PDFs of normalized velocity-gradient invariants

(Q^{*},R^{*})

for the tricubic case (Supplementary Note 3.5 ). e , nRMSE of the MoE under uniform subsampling with

5\%

noise as a function of the number of LR sequences used for adaptation. f , Pearson correlation coefficients of reconstruction error fields between pairs of experts for the tricubic case (Supplementary Note 3.8 ).

Figure 9: Generalization to LES inputs in DIT. a , A filtered and downsampled DNS state initializes an independently evolved LES with an SGS closure. LES and DNS fields are paired at matching physical times, separated by

\Delta T=20\delta t

, where

\delta t

is the LES integration time step. b , Vorticity magnitude

\|\bm{\omega}\|

at LES step 160 for DNS, LES, the individual experts and the MoE, with reconstruction error maps beneath the corresponding predictions. c , Reconstruction nRMSE against the paired DNS references over 160 LES steps. d , Spatially averaged viscous dissipation normalized by the DNS value at each corresponding time; the dashed line denotes unity. e , Relative

L_{2}

error of the local energy flux field,

\|\Pi_{\ell}-\Pi_{\ell}^{\mathrm{DNS}}\|_{2}/\|\Pi_{\ell}^{\mathrm{DNS}}\|_{2}

, aggregated over all evaluated times and spatial locations. Filter width

\ell

is normalized by the DNS grid spacing

\Delta x

.

$\lambda_{\mathrm{phy}}$	$\beta$	nRMSE $\downarrow$	SSIM $\uparrow$	PSNR (dB) $\uparrow$
0	0.3	0.0817	0.9402	43.32
0.001	0.3	0.0807	0.9415	43.44
0.01	0.3	0.0773	0.9463	43.82
0.1	0.3	0.1130	0.9022	40.45
0.01	0.1	0.0789	0.9438	43.64
0.01	1	0.0841	0.9372	43.17

Supplementary Table 1: Sensitivity to physical-loss weights on DIT at

4\times

with spectral-cutoff coarse-graining. Best values for each metric are in bold . nRMSE and SSIM are reported to four decimal places and PSNR to two decimal places.

$\lambda_{\mathrm{phy}}$	$\beta$	nRMSE $\downarrow$	SSIM $\uparrow$	PSNR (dB) $\uparrow$
0	0.3	0.1233	0.7957	42.20
0.0001	0.3	0.1232	0.7960	42.20
0.001	0.3	0.1173	0.8093	42.61
0.01	0.3	0.1216	0.7985	42.32
0.1	0.3	0.1810	0.6722	38.66
0.001	0.1	0.1174	0.8090	42.59

Supplementary Table 2: Sensitivity to physical-loss weights on FIT at

8\times

with Gaussian coarse-graining. Best values for each metric are in bold . nRMSE and SSIM are reported to four decimal places and PSNR to two decimal places.

Supplementary Fig. 1: Average parameter counts (M) for eight super-resolution reconstruction methods: Ours, SwinIR, RRDB, EDSR, RCAN, DiffSR, PhySR, and FNO. Each bar shows the equally weighted mean across six configurations: FIT at

4\times

and

8\times

, MHDT at

4\times

and

8\times

, and HBDT at

8\times

and

16\times

, using the parameter counts reported in Supplementary Tables 10 – 12 . Values above the bars are rounded to two decimal places. The proposed method has the lowest average parameter count (3.87 M).

Dataset	Scale	Reference forcing $\mathrm{RMS}_{\mathrm{ref}}(\bm{f})$	After filtering $\mathrm{RMS}(\mathcal{H}_{\sigma}\bm{f})$	RMS retained
FIT	$4\times$	$0.0847$	$1.11\times 10^{-5}$	$0.0131\%$
FIT	$8\times$	$0.0847$	$4.38\times 10^{-5}$	$0.0517\%$
MHDT	$4\times$	$0.0545$	$2.76\times 10^{-5}$	$0.0506\%$
MHDT	$8\times$	$0.0545$	$1.08\times 10^{-4}$	$0.1989\%$

Supplementary Table 3: Forcing RMS before and after high-pass filtering. The same Gaussian filter is used in training and in this diagnostic. Retained percentages are computed from unrounded values relative to the dataset-specific

4\times

pre-filter reference.

Supplementary Fig. 2: Sensitivity to the momentum-loss formulation on FIT at

4\times

super-resolution with box-filtered inputs. Left , Kinetic-energy spectra of the reference and reconstructions using an unfiltered

L_{1}

penalty (“L1 (direct)”), a high-pass squared

L_{2}

penalty (“L2 (high-pass)”), and the implemented high-pass

L_{1}

penalty (“L1 (high-pass)”). The inset highlights differences at high wavenumbers; the black dashed line is a

k^{-5/3}

guide. Right , Relative

L_{1}

errors of the reconstructed energy spectra over

k>128

. Lower values indicate closer agreement with the reference in this high-wavenumber range.

Supplementary Fig. 3: Effect of HCP on the reconstructed energy spectrum. Reference spectrum and predictions from models trained with and without HCP on DIT at

4\times

super-resolution with spectral-cutoff inputs.

Model	Divergence RMS	nRMSE
HCP	$1.52\times 10^{-6}$	0.0779
w/o HCP	$1.41\times 10^{-2}$	0.0773

Supplementary Table 4: Effect of HCP on divergence RMS and nRMSE. Models trained with and without HCP on DIT at

4\times

super-resolution with spectral-cutoff inputs.

System	Velocity $\bm{u}$	Pressure $p$	Extra variable
FIT	✓	✓	–
MHDT	✓	✓	Magnetic field $\bm{b}$
HBDT	✓	✓	Density $\rho$

Supplementary Table 5: Reconstructed variables per turbulent system.

Category	Specification
Reconstructed variables	Velocity $\bm{u}$ (3 components), pressure $p$
Governing model	Incompressible Navier–Stokes with large-scale forcing
Transport coefficient	Kinematic viscosity $\nu=1.85\times 10^{-4}$
Spatial grid spacing	$\Delta x=\Delta y=\Delta z=\pi/512$
Temporal resolution	$\Delta t=2\times 10^{-3}$

Supplementary Table 6: FIT dataset: reconstructed variables, governing equations, and numerical parameters.

Category	Specification
Reconstructed variables	Velocity $\bm{u}$ , magnetic field $\bm{b}$ , pressure $p$
Governing model	Incompressible MHD equations
Transport coefficients	Kinematic viscosity $\nu=1.1\times 10^{-4}$
	Magnetic diffusivity $\eta=1.1\times 10^{-4}$
Spatial grid spacing	$\Delta x=\Delta y=\Delta z=\pi/512$
Temporal resolution	$\Delta t=2.5\times 10^{-3}$

Supplementary Table 7: MHDT dataset: reconstructed variables, governing equations, and numerical parameters.

Category	Specification
Reconstructed variables	Density $\rho$ , velocity $\bm{u}$ , pressure $p$
Governing model	Variable-density, low-Mach-number Navier–Stokes
	with buoyancy production term
Control parameters	Reynolds number $Re_{0}=12500$
	Froude number $Fr=1.0$
	Schmidt number $Sc=1.0$

Supplementary Table 8: HBDT dataset: reconstructed variables, governing equations, and numerical parameters.

System	Samples (train/val/test/total)	Storage
FIT	653/218/218/1089	$\sim$ 70 GB
MHDT	707/236/236/1179	$\sim$ 132 GB
HBDT	822/274/274/1370	$\sim$ 108 GB

Supplementary Table 9: Dataset statistics and splits (samples: train/val/test/total; storage size).

Factor	Coarse-gr.	Method	#Params. (M)	SSIM↑	PSNR↑	nRMSE↓	fRMSE↓
4 $\times$	Box	Tricubic	–	$0.8321$	$42.28$	$0.1204$	$28.56$
		RRDB	$4.2$	$0.9614$	$50.90$	$0.0460$	$11.32$
		RCAN	$4.2$	0.9672	51.92	0.0415	10.23
		EDSR	$4.2$	$0.9661$	$51.77$	$0.0423$	$10.41$
		FNO	$34.0$	$0.9109$	$46.61$	$0.0730$	$17.27$
		PhySR	$3.3$	$0.9590$	$50.66$	$0.0476$	$11.63$

Supplementary Table 10: Full quantitative comparison on FIT. Detailed super-resolution results for

4\times

and

8\times

upscaling under three coarse-graining strategies (box, Gaussian, and spectral cutoff). Performance is evaluated using PSNR (

\uparrow

), SSIM (

\uparrow

), nRMSE (

\downarrow

), and fRMSE (

\downarrow

) (see Section 4.5 ). For each upscaling factor and coarse-graining, the best result is shown in bold . Model sizes are reported as the number of parameters in millions (M).

Factor	Coarse-gr.	Method	#Params. (M)	SSIM↑	PSNR↑	nRMSE↓	fRMSE↓
4 $\times$	Box	Tricubic	–	$0.7643$	$38.39$	$0.1838$	$16.29$
		RRDB	$4.2$	0.9482	47.34	0.0678	5.92
		RCAN	$4.2$	$0.9464$	$47.28$	$0.0688$	$5.99$
		EDSR	$4.3$	$0.9448$	$47.15$	$0.0699$	$6.09$
		FNO	$34.0$	$0.8687$	$42.41$	$0.1166$	$10.28$
		PhySR	$3.5$	$0.9265$	$45.37$	$0.0840$	$7.39$

Supplementary Table 11: Full quantitative comparison on MHDT. Detailed super-resolution results for

4\times

and

8\times

upscaling under three coarse-graining strategies (box, Gaussian, and spectral cutoff). Performance is evaluated using PSNR (

\uparrow

), SSIM (

\uparrow

), nRMSE (

\downarrow

), and fRMSE (

\downarrow

) (see Section 4.5 ). For each upscaling factor and coarse-graining, the best result is shown in bold . Model sizes are reported as the number of parameters in millions (M).

Factor	Coarse-gr.	Method	#Params. (M)	SSIM↑	PSNR↑	nRMSE↓	fRMSE↓
8 $\times$	Favre	Tricubic	–	$0.9226$	$46.87$	$0.0933$	$3.89$
		RRDB	$4.3$	$0.9778$	$57.95$	$0.0307$	$1.38$
		RCAN	$5.1$	0.9800	61.57	0.0262	1.27
		EDSR	$4.7$	$0.9779$	$59.22$	$0.0286$	$1.34$
		FNO	$34.3$	$0.9264$	$51.04$	$0.0544$	$2.14$
		PhySR	$6.1$	$0.9668$	$59.38$	$0.0316$	$1.45$

Supplementary Table 12: Full quantitative comparison on HBDT. Detailed super-resolution results for

8\times

and

16\times

upscaling under three coarse-graining strategies (Favre, Gaussian, and spectral cutoff). Performance is evaluated using PSNR (

\uparrow

), SSIM (

\uparrow

), nRMSE (

\downarrow

), and fRMSE (

\downarrow

) (see Section 4.5 ). For each upscaling factor and coarse-graining, the best result is shown in bold . Model sizes are reported as the number of parameters in millions (M).

Supplementary Fig. 4: Noise robustness on FIT. Mean nRMSE at

4\times

upscaling as a function of the noise magnitude

\sigma_{\xi}

in box-filtered LR inputs. Error bars denote 95% confidence intervals.

Method	Noise magnitude $\sigma_{\xi}$					$\Delta$ nRMSE
Method	$0$	$0.01$	$0.03$	$0.05$	$0.10$	$\Delta$ nRMSE
Ours	0.0345	0.0355	0.0400	0.0453	0.0587	0.0242
SwinIR	0.0447	0.0479	0.0582	0.0684	0.0912	0.0466
PhySR	0.0548	0.0556	0.0603	0.0669	0.0851	0.0302
DiffSR	0.0489	0.0523	0.0692	0.0834	0.1110	0.0621
FNO	0.0878	0.0883	0.0915	0.0976	0.1217	0.0339

Supplementary Table 13: Quantitative comparison of noise robustness on FIT. Mean nRMSE at

4\times

upscaling for box-filtered LR inputs.

\Delta

nRMSE denotes the absolute increase from

\sigma_{\xi}=0

to

0.10

. The best result in each column is shown in bold .

Supplementary Fig. 5: Ablation study of the multiscale neural operator on FIT. The full model is compared with models without the spectral operator

\mathcal{O}_{s}

or the convolutional operator

\mathcal{O}_{c}

at

4\times

upscaling with spectral-cutoff coarse-graining. Left, Velocity nRMSE versus physical wavenumber. Right, Vorticity MAE versus physical wavenumber.

	Velocity nRMSE			Vorticity MAE
Model	Low	Mid	High	Low	Mid	High
Full	0.0122	0.1725	0.3440	2.8059	12.6611	10.2234
w/o $\bm{\mathcal{O}_{s}}$	$0.0207^{\scriptscriptstyle\uparrow 69.9\%}$	$0.1914^{\scriptscriptstyle\uparrow 11.0\%}$	$0.3485^{\scriptscriptstyle\uparrow 1.3\%}$	$4.6619^{\scriptscriptstyle\uparrow 66.1\%}$	$14.3753^{\scriptscriptstyle\uparrow 13.5\%}$	$10.3058^{\scriptscriptstyle\uparrow 0.8\%}$
w/o $\bm{\mathcal{O}_{c}}$	$0.0140^{\scriptscriptstyle\uparrow 14.9\%}$	$0.2137^{\scriptscriptstyle\uparrow 23.9\%}$	$0.3794^{\scriptscriptstyle\uparrow 10.3\%}$	$3.0898^{\scriptscriptstyle\uparrow 10.1\%}$	$15.0576^{\scriptscriptstyle\uparrow 18.9\%}$	$11.2589^{\scriptscriptstyle\uparrow 10.1\%}$

Supplementary Table 14: Scale-resolved errors for the multiscale neural operator ablations. The comparison uses FIT at

4\times

with spectral-cutoff coarse-graining. Entries are mean errors in the Low (

k<64

), Mid (

64\leq k\leq 144

), and High (

k>144

) physical-wavenumber ranges. Arrows give the relative increase from the Full model for the same metric and range.

	Overall	Velocity nRMSE
Spatial design	nRMSE	Low	Mid	High
Ours	0.0615	0.0122	0.1725	0.3440
MS-ESRGAN	0.0739	0.0139	0.2174	0.3803

Supplementary Table 15: Comparison of Ours and MS-ESRGAN spatial designs at closely matched capacity. Ours denotes the proposed multiscale neural operator; MS-ESRGAN denotes the three-dimensional multiscale replacement derived from MS-ESRGAN. Overall nRMSE is evaluated on the complete FIT

4\times

spectral-cutoff test fields. Velocity nRMSE entries are mean shell-wise errors in the Low (

k<64

), Mid (

64\leq k\leq 144

), and High (

k>144

) ranges. The best result in each column is shown in bold .

Supplementary Fig. 6: Comparison of spatial modules on FIT at

4\times

with spectral-cutoff coarse-graining. The proposed multiscale neural operator (MNO) is compared with an MS-ESRGAN-derived three-dimensional multiscale replacement (labelled MS-ESRGAN) at closely matched capacity. Left, shell-wise velocity nRMSE. Right, kinetic-energy spectra; the inset magnifies the high-wavenumber range. Ref. denotes the reference field, and the black dashed line indicates a slope of

-5/3

.

	FIT $4\times$		FIT $8\times$
Model	nRMSE $\downarrow$	tRMSE $\downarrow$	nRMSE $\downarrow$	tRMSE $\downarrow$
SwinIR	0.0421	0.0052	0.1499	0.0103
DiffSR	0.0423	0.0095	0.1440	0.0244
PhySR	0.0476	0.0060	0.1486	0.0106
Ours (w/o SSM)	0.0340	0.0049	0.1371	0.0101
Ours	0.0291	0.0041	0.1195	0.0094

Supplementary Table 16: SSM-removal ablation and baseline comparisons on FIT at

4\times

and

8\times

. Inputs are box-filtered. nRMSE measures field reconstruction error and tRMSE measures error in frame-to-frame increments relative to the reference; lower values are preferred. Values are rounded to four decimal places, and the best value in each column is bold.

Supplementary Fig. 7: Time–space visualization of the SSM-removal ablation on FIT. The left panel shows the reference; the upper row compares Ours, Ours (w/o SSM), SwinIR, and DiffSR, and the lower row shows the corresponding reconstruction errors. Time

t

runs horizontally and position

x

vertically.

Temporal model	Params (M)	nRMSE $\downarrow$	SSIM $\uparrow$	PSNR $\uparrow$	tRMSE $\downarrow$	fRMSE $\downarrow$
State space model	3.7	0.0291	0.9834	55.09	0.0041	7.22
ConvLSTM	3.8	0.0306	0.9819	54.60	0.0044	7.59
Causal Transformer	3.8	0.0314	0.9809	54.36	0.0044	7.79

Supplementary Table 17: Comparison of temporal models on FIT at

4\times

with box-filtered inputs at closely matched parameter counts. Causal Transformer denotes the Transformer decoder baseline; parameter counts refer to the complete networks. Metrics follow Section 4.5 and Supplementary Note 1.9 ; arrows indicate the preferred direction, and the best value for each metric is in bold .

				FIT		MHDT
$d_{s}$	$d_{f}$	Params (M)	GFLOPs	nRMSE $\downarrow$	tRMSE $\downarrow$	nRMSE $\downarrow$	tRMSE $\downarrow$
8	64	0.968	63.468	0.0295	4.23	0.0613	0.90
16	64	1.034	67.864	0.0291	4.14	0.0603	0.88
32	64	1.166	76.655	0.0290	4.11	0.0599	0.87
16	32	0.484	31.789	0.0305	4.33	0.0634	0.92
16	128	2.329	152.899	0.0287	4.07	0.0593	0.86

Supplementary Table 18: Sensitivity of SSM reconstruction to state dimension and feature width on FIT and MHDT at

4\times

.

d_{s}

denotes state dimension and

d_{f}

feature width. Params and GFLOPs count the SSM module alone. All tRMSE values are reported in units of

10^{-3}

and rounded to two decimal places; nRMSE is unscaled.

Supplementary Fig. 8: Vorticity-magnitude reconstruction and error visualization for FIT. Comparison of reconstructed vorticity magnitude

\|\bm{\omega}\|=\|\nabla\times\bm{u}\|

and errors among different super-resolution models under box-filtered coarse-graining. Each column block corresponds to

4\times

(left) and

8\times

(right) upscaling. For each case, three orthogonal slices (y–z, z–x, x–y) are shown for the reference high-resolution field (Ref.), the low-resolution input (LR), the reconstructed fields from our method, FNO, PhySR, RRDB, EDSR, and RCAN, and the tricubic baseline. The right panels visualize the corresponding error maps (Prediction – Reference) using a symmetric color scale. All samples are from held-out test data; color bars denote vorticity magnitude (left) and reconstruction error (right) under identical dynamic ranges across methods.

Supplementary Fig. 9: Reconstruction of the magnetic field curl magnitude for MHDT. Comparison of the reconstructed current magnitude

\|\bm{j}\|=\|\nabla\times\bm{b}\|

field and corresponding voxelwise errors under the box-filtered coarse-graining. Columns correspond to

4\times

(left) and

8\times

(right) super-resolution. Each group shows three orthogonal slices (y–z, z–x, x–y) for the reference high-resolution field (Ref.), the low-resolution input (LR), and reconstructions by our method and baselines (FNO, PhySR, RRDB, EDSR, RCAN, and tricubic interpolation). The right panels visualize error fields (Prediction – Reference) with symmetric color scaling. Color bars denote

\|\bm{j}\|

(left) and reconstruction error (right).

Supplementary Fig. 10: Velocity divergence reconstruction and error visualization for HBDT. Comparison of reconstructed and reference velocity divergence fields

\nabla\cdot\bm{u}

under the Favre-filtered coarse-graining for

8\times

(left) and

16\times

(right) super-resolution reconstruction. Each column block presents three orthogonal slices (y–z, z–x, x–y) of the reference high-resolution field (Ref.), low-resolution input (LR), and reconstructions from our method, FNO, PhySR, RRDB, EDSR, RCAN, and tricubic interpolation. The right panels visualize error fields (Prediction – Reference) with symmetric color scaling. Color bars indicate

\nabla\cdot\bm{u}

(left) and corresponding reconstruction error (right).

Supplementary Fig. 11: Q–R joint probability distributions. Isocontour plots of joint PDFs of normalized

Q^{*}

and

R^{*}

for FIT at

4\times

(top) and

8\times

(bottom) super-resolution reconstruction. Columns show EDSR, DiffSR, RCAN, SwinIR, and tricubic interpolation, from left to right. Black contours denote the reference; colored contours (gray for tricubic interpolation) denote the reconstructed fields. The magenta dashed curve marks

27(R^{*})^{2}+4(Q^{*})^{3}=0

. Evaluation details are given in Supplementary Note 3.5 .

Supplementary Fig. 13: Turbulent energy spectra under box/Favre coarse-graining. Panels show FIT (

4\times

), MHDT (

4\times

), and HBDT (

8\times

) from left to right. Horizontal and vertical axes denote wavenumber

k

and energy spectrum

E(k)

, respectively. Reconstructions from our method, FNO, PhySR, RRDB, SwinIR, DiffSR, EDSR, and RCAN are compared with the reference (Ref.). For MHDT, the spectrum is the sum of kinetic and magnetic contributions. Insets magnify the high-wavenumber range to highlight small-scale recovery. The black dashed line indicates a slope of

-5/3

.

Supplementary Fig. 14: Quantitative distributional similarity across turbulent regimes. Shown are the Jensen–Shannon (JS) divergences between the reconstructed and reference PDFs of three canonical turbulent diagnostics: top , vorticity magnitude

\|\bm{\omega}\|

in FIT; middle , current magnitude

\|\bm{j}\|

in MHDT; bottom , velocity divergence

\nabla\cdot\bm{u}

in HBDT; these correspond to Fig. 3 b.

Factors	Metrics	Ours	SwinIR	FNO	Bicubic
$4\times$	vCoh	0.9925	0.9906	0.9871	0.9743
$4\times$	vRMSE	0.4672	0.5167	0.5925	0.8482
$8\times$	vCoh	0.9533	0.9442	0.9259	0.8745
$8\times$	vRMSE	1.1340	1.2582	1.4174	1.8646

Supplementary Table 19: Cross-level wind diagnostics for ERA5 downscaling. Performance is evaluated using vCoh and vRMSE from Supplementary Eqs. ( S29 )–( S31 ). Higher vCoh and lower vRMSE indicate closer agreement in the spatial organization and amplitude, respectively, of the 850–500 hPa bulk-shear field. The best result is in bold .

Model	SSIM $\uparrow$	nRMSE $\downarrow$	PSNR $\uparrow$	tRMSE $\downarrow$	fRMSE $\downarrow$
Ours	$\mathbf{0.9835}\,{\scriptscriptstyle\pm 0.0001}$	$\mathbf{0.0292}\,{\scriptscriptstyle\pm 0.0001}$	$\mathbf{55.0727}\,{\scriptscriptstyle\pm 0.0178}$	$\mathbf{0.0042}\,{\scriptscriptstyle\pm 0.0000}$	$\mathbf{7.2528}\,{\scriptscriptstyle\pm 0.0225}$
FNO	$0.9163\,{\scriptscriptstyle\pm 0.0040}$	$0.0704\,{\scriptscriptstyle\pm 0.0022}$	$46.9694\,{\scriptscriptstyle\pm 0.3089}$	$0.0072\,{\scriptscriptstyle\pm 0.0004}$	$16.7646\,{\scriptscriptstyle\pm 0.4352}$
PhySR	$0.9585\,{\scriptscriptstyle\pm 0.0001}$	$0.0477\,{\scriptscriptstyle\pm 0.0001}$	$50.6396\,{\scriptscriptstyle\pm 0.0175}$	$0.0062\,{\scriptscriptstyle\pm 0.0002}$	$11.6351\,{\scriptscriptstyle\pm 0.0106}$
RCAN	$0.9674\,{\scriptscriptstyle\pm 0.0007}$	$0.0410\,{\scriptscriptstyle\pm 0.0006}$	$52.0477\,{\scriptscriptstyle\pm 0.1220}$	$0.0064\,{\scriptscriptstyle\pm 0.0001}$	$10.0879\,{\scriptscriptstyle\pm 0.1354}$
EDSR	$0.9657\,{\scriptscriptstyle\pm 0.0002}$	$0.0422\,{\scriptscriptstyle\pm 0.0001}$	$51.7950\,{\scriptscriptstyle\pm 0.0221}$	$0.0065\,{\scriptscriptstyle\pm 0.0001}$	$10.3804\,{\scriptscriptstyle\pm 0.0256}$
RRDB	$0.9641\,{\scriptscriptstyle\pm 0.0028}$	$0.0434\,{\scriptscriptstyle\pm 0.0022}$	$51.4815\,{\scriptscriptstyle\pm 0.5152}$	$0.0068\,{\scriptscriptstyle\pm 0.0004}$	$10.6929\,{\scriptscriptstyle\pm 0.5478}$

Supplementary Table 20: Training-seed uncertainty on FIT at

4\times

with box-filtered inputs. Values are mean

\pm

sample standard deviation over independent training seeds 42, 43, and 44 (

n=3

). Arrows indicate the preferred direction. A standard deviation displayed as

0.0000

is below the reporting precision. The best mean is shown in bold .

Factor	Model	MAE $\downarrow$	MSE $\downarrow$	tRMSE $\downarrow$	vCoh $\uparrow$	vRMSE $\downarrow$
$4\times$	Ours	$\mathbf{0.1202}\,{\scriptscriptstyle\pm 0.0007}$	$\mathbf{0.0762}\,{\scriptscriptstyle\pm 0.0011}$	$\mathbf{0.3146}\,{\scriptscriptstyle\pm 0.0015}$	$\mathbf{0.9925}\,{\scriptscriptstyle\pm 0.0001}$	$\mathbf{0.4625}\,{\scriptscriptstyle\pm 0.0039}$
	FNO	$0.1692\,{\scriptscriptstyle\pm 0.0012}$	$0.1255\,{\scriptscriptstyle\pm 0.0011}$	$0.4038\,{\scriptscriptstyle\pm 0.0015}$	$0.9870\,{\scriptscriptstyle\pm 0.0001}$	$0.5942\,{\scriptscriptstyle\pm 0.0012}$
	SwinIR	$0.1336\,{\scriptscriptstyle\pm 0.0004}$	$0.0947\,{\scriptscriptstyle\pm 0.0005}$	$0.3433\,{\scriptscriptstyle\pm 0.0009}$	$0.9907\,{\scriptscriptstyle\pm 0.0001}$	$0.5162\,{\scriptscriptstyle\pm 0.0013}$
$8\times$	Ours	$\mathbf{0.3491}\,{\scriptscriptstyle\pm 0.0002}$	$\mathbf{0.4451}\,{\scriptscriptstyle\pm 0.0007}$	$\mathbf{0.8041}\,{\scriptscriptstyle\pm 0.0007}$	$\mathbf{0.9529}\,{\scriptscriptstyle\pm 0.0012}$	$\mathbf{1.1363}\,{\scriptscriptstyle\pm 0.0031}$
	FNO	$0.4404\,{\scriptscriptstyle\pm 0.0021}$	$0.6794\,{\scriptscriptstyle\pm 0.0029}$	$1.0002\,{\scriptscriptstyle\pm 0.0011}$	$0.9250\,{\scriptscriptstyle\pm 0.0006}$	$1.4211\,{\scriptscriptstyle\pm 0.0026}$
	SwinIR	$0.3782\,{\scriptscriptstyle\pm 0.0004}$	$0.5333\,{\scriptscriptstyle\pm 0.0006}$	$0.8817\,{\scriptscriptstyle\pm 0.0008}$	$0.9442\,{\scriptscriptstyle\pm 0.0004}$	$1.2576\,{\scriptscriptstyle\pm 0.0002}$

Supplementary Table 21: Training-seed uncertainty on ERA5. Values are mean

\pm

sample standard deviation over independent training seeds 42, 43, and 44 (

n=3

). The best mean within each factor is shown in bold .

Coarse-graining	Expert	SSIM↑	PSNR↑	nRMSE↓	fRMSE↓
Tricubic	Spectral	$0.8679$	$43.38$	$0.1048$	$25.27$
	Box	0.9077	45.02	0.0879	21.44
	Gaussian	$0.8897$	$43.82$	$0.1001$	$24.08$
	MoE (2 experts)	0.9406	47.19	0.0673	16.45
	MoE (3 experts)	0.9333	46.48	0.0731	17.72
Lanczos	Spectral	$0.8581$	$42.78$	$0.1130$	$27.04$

Supplementary Table 22: Full quantitative generalization results on FIT at

4\times

upscaling. We evaluate three unseen test-time coarse-graining settings: tricubic downsampling, Lanczos filtering, and uniform subsampling with

5\%

additive Gaussian noise. We compare three single-expert models (trained with box, Gaussian, or spectral-cutoff coarse-graining) and two MoE variants (2 experts: box+spectral; 3 experts: box+Gaussian+spectral). Higher SSIM/PSNR and lower nRMSE/fRMSE indicate better performance; best results are bold .

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