Scale-Split Neural Operator for Memory- and Data-Efficient 3D Turbulence Prediction

Shaoxiang Qin; Yucheng Zhao; Zongyi Li; Liangzhu Leon Wang; Xiongye Xiao

cs.LGSep 30, 2026

Scale-Split Neural Operator for Memory- and Data-Efficient 3D Turbulence Prediction

Authors: Shaoxiang Qin, Yucheng Zhao, Zongyi Li, Liangzhu Leon Wang, Xiongye Xiao

Organizations: University of Tennessee, Knoxville · McGill University · New York University · Concordia University

Abstract

Neural surrogates have emerged as fast alternatives to the numerical simulation of three-dimensional turbulence. However, training them at high resolution remains challenging, since the memory of full-field models grows with the resolution. In addition, full-resolution training data are expensive to simulate and store, and therefore scarce. We introduce ScaleSplit-NO (Scale-Split Neural Operator), which exploits the scale structure of turbulence with two neural operators: a Parent predicts the global coarse field at the next time step, and a Child predicts full-resolution local patches conditioned on this prediction. Neither model operates on the full-resolution field. The Child is pretrained alone and then attached to the Parent's coarse prediction through zero-initialized connections. On two complex high-resolution turbulence benchmarks, ScaleSplit-NO surpasses all competing baselines in both prediction accuracy and data efficiency. On the higher-resolution dataset JHTDB256 ( $256^3$ ), its normalized mean squared error (NMSE) is 53% lower than that of the strongest baseline, and its training memory is 79% lower than that of the most memory-efficient baseline. We further demonstrate its effectiveness for urban wind prediction in a real district of Montreal on a $500\times150\times500$ grid, reducing one-step NMSE by 65.8% relative to the baseline. Moreover, swapping in a Parent trained on additional coarse fields improves prediction without retraining the Child, providing further accuracy gains at a small storage cost.

Figures & tables

Figure 1: Accuracy versus training memory. Test error (NMSE) with limited training data against the peak GPU memory during training (Table 1 ). Lower left is better.

Figure 2: Separate supervision and composed prediction. (a) The Parent is trained to predict the coarse target $Du_{t+1}$ from the coarse input $Du_{t}$ . (b) The Child is pretrained to predict the target patch $T_{j}u_{t+1}$ from the input patch $T_{j}u_{t}$ . (c) The coarse prediction of the frozen Parent is interpolated and cropped to the condition $c_{j}$ , which enters the Child through zero-initialized connections (diamonds). Fine-tuning updates the backbone and these connections, and the loss $\ell$ is the only place where the target field enters. At inference all weights are fixed, and the predicted patches $\hat{y}_{j}$ are assembled into the predicted field $\hat{u}_{t+1}=F(u_{t})$ .

		NMSE, low data (1/8)				NMSE, full data				Training
		One step		Rollout		One step		Rollout		Memory
	Method	$\times 10^{-3}$		$\times 10^{-2}$		$\times 10^{-3}$		$\times 10^{-2}$		GiB
MHD64	U-Net	15.5		2.83		5.97		1.30		4.6
	FNO	97.6	( $\uparrow$ 529%)	17.2	( $\uparrow$ 507%)	29.5	( $\uparrow$ 394%)	4.96	( $\uparrow$ 283%)	5.5	( $\uparrow$ 20%)
	MG-TFNO	10.3	( $\downarrow$ 33%)	2.06	( $\downarrow$ 27%)	5.98	(0%)	1.27	( $\downarrow$ 2%)	10.6	( $\uparrow$ 130%)
	EddyFormer	8.83	( $\downarrow$ 43%)	1.99	( $\downarrow$ 30%)	5.88	( $\downarrow$ 2%)	1.32	( $\uparrow$ 2%)	17.0	( $\uparrow$ 268%)

Table 1: Test error (NMSE) and training memory. U-Net serves as the reference. Parentheses give the change in error and memory relative to U-Net, (

\downarrow

better) or (

\uparrow

worse) . Bold : best result. Thick underline : best baseline. Dashed underline : second-best baseline.

Figure 3: Visualization of 3D turbulence predictions and errors. The figure shows one test pair of each dataset for one-step prediction in the low-data setting. Each panel draws a 3D field as surfaces of constant value (isosurfaces) at the levels in its legend. Velocity panels show the $x$ -velocity of the ground truth and of the prediction of ScaleSplit-NO. Error panels show the magnitude of the velocity error $\|\hat{u}_{t+1}-u_{t+1}\|$ of ScaleSplit-NO and of the strongest baseline of each dataset.

Figure 4: Test error versus the amount of stored training data. (a1, a2) Test error of ScaleSplit-NO and the two strongest baselines of each dataset for different amounts of training data. (b1, b2) Test error of ScaleSplit-NO when a Child trained on the labeled fraction of the data is kept fixed and combined with Parents trained on additional coarse fields. In (b1, b2), gray dashed lines connect models whose Parent and Child are trained on the same data, and the relative training data size also counts the additional coarse fields.

Appendix figures & tables22 assets

Supplementary material from the paper’s appendix.

Appendix

	MHD64		JHTDB256
Method	Low data (1/8)	Full data	Low data (1/8)	Full data
U-Net	11.3	7.01	9.94	8.14
FNO	29.0	15.9	30.3	19.8
MG-TFNO	9.37	7.06	16.4	15.3
EddyFormer	8.58	6.94	19.5	18.8
P3D	11.2	7.10	5.07	4.62

Appendix

Table 4: One-step test NRMSE (

\times 10^{-2}

) with 1/8 and with all of the training data.

Figure 7: MHD64, x-velocity of all methods. One-step prediction (left) and rollout step 2 (right) with the normalized errors.

Figure 8: MHD64, all seven channels at one step. Ground truth and normalized errors of the two strongest baselines, MG-TFNO and EddyFormer, and of ScaleSplit-NO.

Figure 9: MHD64, complete field at one step. Isosurfaces of the x-velocity at one and two standard deviations of the ground truth, and of the velocity error magnitude at 0.10 and 0.16, for MG-TFNO, EddyFormer, and ScaleSplit-NO.

Figure 10: JHTDB256, x-velocity of U-Net, FNO, MG-TFNO, and ScaleSplit-NO. One-step prediction (left) and rollout step 15 (right) with the normalized errors.

Figure 11: JHTDB256, x-velocity of EddyFormer, P3D, ReViT, and ScaleSplit-NO. One-step prediction (left) and rollout step 15 (right) with the normalized errors, on the same scales as Figure 10 .

Figure 12: JHTDB256, all four channels at one step. Ground truth and normalized errors of the two strongest baselines, U-Net and P3D, and of ScaleSplit-NO.

Figure 13: JHTDB256, complete field at one step. Isosurfaces of the x-velocity at one and two standard deviations of the ground truth, and of the velocity error magnitude at 0.15 and 0.30, for U-Net, P3D, and ScaleSplit-NO.

Figure 14: Urban wind at one step. Isosurfaces of the fluctuation of the velocity component $v$ about its temporal mean at $\pm 1$ and $\pm 2$ , and of the absolute error of $v$ at 0.3 and 0.6, for U-Net and ScaleSplit-NO. The NMSE refers to $v$ in the fluid region of this test pair.

Figure 15: MHD64, velocity spectra at one step. Energy spectra of the ground truth and of all predictions (left) and of the prediction errors (right).

Figure 16: JHTDB256, velocity spectra at one step. Energy spectra of the ground truth and of all predictions (left) and of the prediction errors (right).

Method	MHD64	JHTDB256
U-Net	$21.45\pm 0.34$	$17.19\pm 0.86$
FNO	$99.81\pm 2.23$	$99.50\pm 10.61$
MG-TFNO	$9.67\pm 0.34$	$32.23\pm 3.91$
EddyFormer	$12.45\pm 0.23$	$41.10\pm 0.02$
P3D	$21.15\pm 0.94$	$4.07\pm 0.04$
ReViT	$93.89\pm 0.13$	$32.96\pm 0.09$

Appendix

Table 5: One-step test NMSE (

\times 10^{-3}

) of all methods with 1/8 of the data over three repeated runs at 20% of the training cost (seeds 101, 202, 303), mean

\pm

sample standard deviation.

	NMSE
ScaleSplit-NO final configuration	$6.88\pm 0.11$
w/o pretraining and zero init.	$7.22\pm 0.23$
w/o Child fine-tuning	$7.23\pm 0.04$
Fine-tuning with coarse target	$12.04\pm 0.48$
Residual correction	$7.33\pm 0.14$
Joint Parent–Child training	$11.81\pm 0.02$

Appendix

Table 6: One-step test NMSE (

\times 10^{-3}

) of the ablation variants of Table 5 on MHD64 with 1/8 of the data over three repeated runs at 20% of the training cost, mean

\pm

sample standard deviation.

Dataset	Child	1/8	1/4	1/2	1	Coarse target
		Parent
MHD64	1/8	6.20	5.68	5.10	4.99	2.63
	1/4		5.53	4.94	4.82	2.44
	1/2			4.79	4.77	2.48
	1				4.52	2.35
JHTDB256	1/8	1.254	1.248	1.238	1.231	1.182

Appendix

Table 7: One-step test NMSE (

\times 10^{-3}

) of a fixed Child with Parents trained on more data, and with the coarse target as condition. Rows: training data of the Child. Columns: training data of the Parent. Both are given as fractions of the full training set.

Dataset	Setting	NMSE
MHD64	Final: global coarse grid $=$ Parent grid $24^{3}$ , local Child grid $16^{3}$	6.20
	Global coarse grid $=$ Parent grid $24^{3}\rightarrow 16^{3}$	6.20	( $\uparrow$ 0.1%)
	Global coarse grid $=$ Parent grid $24^{3}\rightarrow 32^{3}$	6.42	( $\uparrow$ 3.5%)
	Local Child grid $16^{3}\rightarrow 12^{3}$	6.30	( $\uparrow$ 1.6%)
	Local Child grid $16^{3}\rightarrow 24^{3}$	7.06	( $\uparrow$ 13.9%)
JHTDB256	Final: global coarse grid $64^{3}$ , Parent grid $32^{3}$ , local Child grid $16^{3}$	1.25

Appendix

Table 8: Sizes of the Parent and the Child with 1/8 of the data. One-step test NMSE (

\times 10^{-3}

), with the change relative to the final configuration of each dataset in parentheses, (

\uparrow

worse) .

Overlap	NMSE ( $\times 10^{-3}$ )	Voxels	Time (s)
0%	1.29	1.0 $\times$	3.44
33%	1.25	3.4 $\times$	3.46
50%	1.25	8.0 $\times$	3.50

Appendix

Table 9: Overlap of the Parent patches on JHTDB256 with 1/8 of the data. Bold : the default configuration.

	Parent		Child
	MHD64	JHTDB256	MHD64	JHTDB256
Input	$24^{3}$ field	$32^{3}$ patch	$16^{3}$ patch	$16^{3}$ patch
Fourier blocks	4	4	8	8
Hidden width	32	32	64	64
Modes per axis	12	16	8	8
Parameters ( $10^{6}$ )	56.6	134.2	134.3	134.3

Appendix

Table 10: Backbone configurations. Modes are the number of Fourier modes per axis. The parameters of the Child include the condition pathway.

Dataset	Stage	Epochs	Learning rate	Batch
MHD64	Parent training	300	$2\times 10^{-3}$	1 field
	Child pretraining	50	$1\times 10^{-3}$	8 patches
	Child fine-tuning	300	$2\times 10^{-3}$	8 patches
JHTDB256	Parent training	300	$2\times 10^{-3}$	8 patches
	Child pretraining	100	$5\times 10^{-4}$	16 patches
	Child fine-tuning	500	$2\times 10^{-3}$	16 patches

Appendix

Table 11: Training settings of the three stages.

Method	Width	Depth	Modes	Epochs	Learning rate	Batch
U-Net	64 / 16	4 levels	–	300	$2\times 10^{-4}$	2 / 1
FNO	48 / 12	4 layers	16 / 12	400	$1\times 10^{-3}$	1
MG-TFNO	64 / 40	4 layers	32 / 24	500 / 200	$1\times 10^{-3}$	1 field
EddyFormer	32	4 layers	13	10k–80k steps	$1\times 10^{-3}$	1
P3D	P3D-L / P3D-B		–	1,000 / 4,000	$2\times 10^{-4}$	8 / 4
ReViT	48	1-2-4-2-1	–	300	$1\times 10^{-3}$	2 / 1

Appendix

Table 12: Baseline configurations on MHD64 / JHTDB256. Width is the base channel count of U-Net, the hidden width of FNO, MG-TFNO, and EddyFormer, and the embedding dimension of ReViT. EddyFormer is trained for 10k steps with 1/8 of the data and 80k steps with all of the data.

	MHD64				JHTDB256
	Input	Batch	Memory	Time	Input	Batch	Memory	Time
Method			GiB	h			GiB	h
U-Net	$64^{3}$	2	4.6	4.6	$256^{3}$	1	31.7	41.3
FNO	$64^{3}$	1	5.5	4.6	$256^{3}$	1	35.0	41.4
MG-TFNO	$32^{3}$	8	10.6	20.1	$128^{3}$	1	19.8	32.7
EddyFormer	$64^{3}$	1	17.0	16.7	$256^{3}$	1	26.4	22.9

Appendix

Table 13: Training samples. Input is the grid size of one training sample, batch the number of samples per update, memory the peak GPU memory allocated during one training update, and time the training time with all of the data. For ScaleSplit-NO the first row gives the maximum memory and the total time of its three stages, which are trained one after another and listed below it. MG-TFNO splits each field into eight patches and computes one loss on the prediction assembled from all of them, so every update uses one whole field. For MG-TFNO, Input is the size of one patch without padding,

48^{3}

and

144^{3}

with periodic padding, and Batch the number of patches processed together.

	MHD64		JHTDB256
Method	Overlap	Time (ms)	Overlap	Time (s)
U-Net	–	28	–	0.31
FNO	–	15	–	0.41
MG-TFNO	–	55	–	0.71
EddyFormer	–	341	–	0.38
P3D	–	35	–	0.41

Appendix

Table 14: Inference time. Time for one complete field. For ScaleSplit-NO the time is given for several overlaps of the Child patches.

Figure 17: Training memory versus batch size. Peak GPU memory of one training update, with the input size of one sample in the legend. ScaleSplit-NO is shown for each of its three training stages.

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