Generative Refinement Learning for Temporal Interpolation in Scientific Machine Learning

Pu Ren; Michael W. Mahoney; N. Benjamin Erichson

cs.LGOct 27, 2026

Generative Refinement Learning for Temporal Interpolation in Scientific Machine Learning

Authors: Pu Ren, Michael W. Mahoney, N. Benjamin Erichson

Organizations: Lawrence Berkeley National Lab · International Computer Science Institute · University of California, Berkeley

Abstract

Computationally expensive simulations and real-world observations often provide data at a lower temporal resolution than downstream tasks require. Averaging observations at fixed grid points can introduce errors, and motion estimates alone do not determine how the physical field values change. In this work, we study whether learning to correct a simple numerical interpolation improves the reconstruction of intermediate scientific fields. Our model, interpolative FLow EXpert (iFLEX), first estimates the missing field by taking a weighted average of the two observations at each grid point, giving more weight to the observation closer to the requested time. During training, subtracting this estimate from an observed intermediate field gives the residual that a conditional diffusion model learns to generate. Both observations condition the model, together with the target time, the interval between observations, and available physical parameters. To reconstruct an unobserved intermediate field, the generated residual is added to the weighted average. We evaluate iFLEX on Navier-Stokes Kraichnan turbulence, Shanghai radar precipitation, and sea surface temperature. Across these datasets, iFLEX achieves lower reconstruction error than the evaluated baselines, including optical-flow and diffusion models developed for video frame interpolation in computer vision. In a controlled turbulence comparison at a Reynolds number of 36,000, learning the residual reduces relative Frobenius norm error from about 13.7 for direct field generation to 10.5, with the diffusion backbone and training setup held fixed. Code is available at https://github.com/paulpuren/iflex.

Figures & tables

Figure 1: Interpolation using two observed states achieves lower reconstruction error than the tested one-endpoint predictor on NSKT. (a) Observations are separated by a time interval $\Delta t$ , and intermediate fields are reconstructed at a finer temporal spacing $\delta t$ . Unlike prediction from an initial state, interpolation uses observations from both ends of the interval. (b) At $Re=36000$ , iFLEX achieves lower relative Frobenius norm error (RFNE) than the one-endpoint predictor (Model A) across the tested intermediate times. Over the full evaluation set, the RFNE is $10.5\%$ for iFLEX and $20.4\%$ for Model A. Reconstruction errors are generally larger near the midpoint, where the requested state is farthest in time from both observations.

Figure 2: In these examples, iFLEX more closely reproduces the vortex structures in the ground-truth NSKT fields than Super SloMo. The panels compare the ground truth with reconstructions from Super SloMo and iFLEX at $t=0.005\,\mathrm{s}$ and $t=0.02\,\mathrm{s}$ , using the same two endpoint observations. Red boxes highlight differences in vortex shape and fine-scale structures. Each intermediate field is reconstructed independently.

Panel of Figure 3 — Figure 3: Overview of iFLEX for temporal interpolation between two observed physical fields. Linear interpolation provides an initial estimate of the intermediate state, while a conditional diffusion model generates the correction needed to reconstruct the missing field. The two endpoint observations are processed by a shared spatial encoder, and their features are combined and supplied to the denoising network through learned gates. The requested interpolation time, observation interval, available physical parameters, and diffusion noise level are incorporated through feature-wise linear modulation (FiLM). After the final denoising step, the generated residual is added to the linear estimate to obtain the reconstructed intermediate field.

			$Re=12000$			$Re=24000$			$Re=36000$
Method	#Params [M]	IT [ms]	RFNE $\downarrow$	PC $\uparrow$	SSIM $\uparrow$	RFNE $\downarrow$	PC $\uparrow$	SSIM $\uparrow$	RFNE $\downarrow$	PC $\uparrow$	SSIM $\uparrow$
Linear Interp.	–	0.37	32.12%	0.93	0.43	39.55%	0.90	0.35	43.96%	0.88	0.30
U-Net	260.33	29.41	39.54%	0.92	0.41	50.97%	0.88	0.27	59.94%	0.83	0.18
EDEN	194.00	9.53	13.15%	0.99	0.82	20.14%	0.98	0.71	25.52%	0.96	0.63
Super SloMo	39.59	3.54	13.20%	0.99	0.83	18.38%	0.98	0.74	22.87%	0.97	0.67
iFLEX	110.4	35.39	4.11%	1.00	0.96	7.10%	1.00	0.92	10.48%	0.99	0.87

Table 1: iFLEX achieves the lowest reconstruction error at all three unseen Reynolds numbers. All methods are trained using

Re\in\{2000,4000,8000,16000,32000\}

and evaluated with 20 intermediate states between the endpoint observations. The values

Re=12000

and

24000

are unseen but lie within the training range, while

Re=36000

tests extrapolation beyond that range. At

Re=36000

, iFLEX achieves

10.48\%

RFNE, compared with

22.87\%

for Super SloMo and

25.52\%

for EDEN. RFNE denotes relative Frobenius norm error, PC denotes Pearson correlation, and SSIM denotes structural similarity. Inference time (IT) is reported in milliseconds per generated intermediate field. Bold values indicate the best reconstruction scores.

		Shanghai Radar Data				Sea Surface Temperature
Method	#Params [M]	IT [ms]	RFNE $\downarrow$	PC $\uparrow$	SSIM $\uparrow$	IT [ms]	RFNE $\downarrow$	PC $\uparrow$	SSIM $\uparrow$
Linear Interp.	–	0.09	47.78%	0.84	0.79	0.12	1.94%	0.99	0.83
U-Net	260.33	13.25	48.51%	0.85	0.82	13.27	2.29%	0.996	0.80
EDEN	194.00	2.32	56.54%	0.77	0.77	2.45	2.48%	0.996	0.77
Super SloMo	39.59	1.45	36.77%	0.91	0.87	1.45	2.28%	0.996	0.79
iFLEX	110.4	10.53	33.52%	0.92	0.87	9.35	1.81%	0.998	0.87

Table 2: iFLEX achieves the lowest reconstruction error on Shanghai radar precipitation and sea surface temperature. The evaluations use eight intermediate states for radar and ten for sea surface temperature. Radar precipitation shows substantially larger reconstruction errors, while linear interpolation is already accurate for the more slowly varying temperature fields. On SST, iFLEX reduces RFNE from

1.94\%

for linear interpolation to

1.81\%

. Inference time (IT) is reported in milliseconds per generated intermediate field. Bold values indicate the best reconstruction scores.

Figure 4: iFLEX achieves lower reconstruction error across the observation interval on NSKT, Shanghai radar, and SST. The panels report RFNE at each requested intermediate step, using 20 intermediate states for NSKT at

Re=12000

, eight for radar, and ten for SST. Errors are generally largest near the middle of the interval, where the requested field is farthest in time from both observations. The curves display RFNE as a fraction rather than a percentage.

Panel of Figure 5 — Figure 5: Reconstruction accuracy depends on both spatial structure and computational cost. (a) Comparison of the ground-truth and reconstructed spatial spectra for NSKT at $Re=12000$ . The intended diagnostic is the windowed vorticity power spectrum; the underlying figure export requires the verification noted in the text. (b) RFNE versus inference time for the evaluated methods at the same Reynolds number. iFLEX achieves $4.11\%$ RFNE but requires $35.39\,\mathrm{ms}$ per generated field, compared with $13.20\%$ and $3.54\,\mathrm{ms}$ for Super SloMo. Bubble size represents model parameter count.

Figure 6: iFLEX achieves lower reconstruction error than Super SloMo and EDEN across three temporal gaps on NSKT at

Re=36000

. The panels show RFNE at each intermediate step for

T=12

16

, and

24

. During training, iFLEX samples

T\in\{5,\ldots,20\}

; therefore, the first two gaps lie within its training range, while

T=24

exceeds it. Errors generally increase toward the midpoint of the observation interval, particularly for the longer gap. RFNE is plotted as a fraction.

Panel of Figure 7 — Figure 7: Learning the interpolation residual gives the lowest reconstruction error in the NSKT ablation study at $Re=36000$ . (a) The full model achieves $10.48\%$ RFNE, compared with $20.44\%$ for the one-endpoint predictor (Model A), $12.71\%$ without interval duration conditioning (Model B), $10.75\%$ with fixed-gap training (Model C), and $13.73\%$ for direct generation of the full intermediate field (Model D). (b) Model A uses only the first observation and has larger errors toward the end of the interval, whereas iFLEX uses both endpoints. (c) The full model is compared with Model C, which is trained at a fixed gap of $T=20$ . The curves show the error at each intermediate time.

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Evaluation $T$	Resolution	(Training, Validation, Test)	Observed field	Evaluated $Re$
NSKT	20	$256\times 256$	$(16000,4000,2000)$	Vorticity	$600$ – $36000$
Shanghai Radar	8	$128\times 128$	$(1545,850,850)$	Precipitation	–
Sea Surface Temperature	10	$128\times 128$	$(172765,18278,66222)$	Temperature	–

Appendix

Table 3: Dataset characteristics and evaluation configurations. NSKT contains simulated vorticity fields, while Shanghai radar and SST contain observed precipitation and temperature fields. The reported

T

values specify the number of intermediate states in the main evaluation. The split counts are those supplied for each dataset.

Dataset	Epochs	Batch size	Learning rate	Optimizer
NSKT	150	48	$1\times 10^{-4}$	Adam
Shanghai	400	48	$1\times 10^{-4}$	Adam
SST	50	64	$1\times 10^{-4}$	Adam

Appendix

Table 4: Training settings for the supervised U-Net. All three datasets use Adam with a learning rate of

10^{-4}

; training duration and batch size vary by dataset.

Dataset	Epochs	Batch size	Learning rate	Optimizer
NSKT	200	256	$6\times 10^{-5}$	Adam
Shanghai	400	128	$1\times 10^{-4}$	Adam
SST	200	128	$1\times 10^{-4}$	Adam

Appendix

Table 5: Training settings for Super SloMo. All datasets use Adam. NSKT uses a learning rate of

6\times 10^{-5}

, while Shanghai radar and SST use

10^{-4}

Dataset	VAE			DiT			Optimizer
	Epochs	Batch	LR	Epochs	Batch	LR
NSKT	300	48	$1.25\times 10^{-5}$	300	48	$1\times 10^{-4}$	AdamW
Shanghai	300	256	$5\times 10^{-5}$	400	256	$5\times 10^{-5}$	AdamW
SST	100	256	$5\times 10^{-5}$	50	256	$5\times 10^{-4}$	AdamW

Appendix

Table 6: Training settings for the two stages of EDEN. The VAE learns the field representation, and the DiT learns to generate its latent tokens. Both stages use AdamW with dataset-specific schedules.

Dataset	Epochs	Batch size	Learning rate	Optimizer
NSKT	150	48	$1\times 10^{-4}$	Adam
Shanghai	300	48	$1\times 10^{-4}$	Adam
SST	50	128	$1\times 10^{-4}$	Adam

Appendix

Table 7: Training settings for iFLEX . All three datasets use Adam with a learning rate of

10^{-4}

. The denoising architecture is shared, while training duration and batch size vary by dataset.

Figure 8: iFLEX more closely reproduces the vortex structures in the illustrated NSKT fields at $Re=36000$ . The rows show the ground truth and reconstructions from U-Net, EDEN, Super SloMo, and iFLEX , evaluated at intermediate indices $k=5,10,15,20$ . The panel labels $t$ indicate target indices, rather than physical time in seconds. Differences are visible in the location and shape of vortices and in smaller vorticity structures. Over the corresponding test set, iFLEX achieves $10.48\%$ RFNE compared with $22.87\%$ for Super SloMo (Table 1 ); these aggregate values are not errors for the individual displayed fields.

Figure 9: The illustrated radar reconstructions show differences in the location and intensity of localized precipitation. The rows compare ground-truth fields with reconstructions from U-Net, EDEN, Super SloMo, and iFLEX at intermediate indices $k=2,4,6,8$ . The panel labels $t$ denote these indices. In this example, iFLEX more closely retains localized precipitation structures at the middle target times than EDEN or Super SloMo. Aggregate RFNE is $32.69\%$ for iFLEX , $56.54\%$ for EDEN, and $36.77\%$ for Super SloMo (Table 2 ).

Figure 10: The SST example shows similar broad temperature patterns across the methods, with differences in local gradients and finer spatial variations. The rows compare the ground truth with U-Net, EDEN, Super SloMo, and iFLEX at intermediate indices $k=2,4,6,10$ . The panel labels $t$ denote target indices. On the held-out South Pacific test set, iFLEX achieves $1.81\%$ RFNE and $0.87$ SSIM (Table 2 ). These scores summarize the full evaluation set rather than the displayed examples.

Intermediate states $T$	Model	RFNE $\downarrow$	SSIM $\uparrow$	PC $\uparrow$
4	Linear interpolation	36.85%	0.85	0.91
4	iFLEX	21.08%	0.93	0.97
8	Linear interpolation	47.78%	0.79	0.84
8	iFLEX	33.52%	0.87	0.92
12	Linear interpolation	55.02%	0.76	0.79
12	iFLEX	40.42%	0.84	0.89

Appendix

Table 8: iFLEX reduces reconstruction error relative to linear interpolation at all three tested radar observation gaps. Both methods become less accurate as the gap increases. At

T=12

, residual refinement reduces RFNE from

55.02\%

40.42\%

. Bold indicates the better result within each gap.

$Re$	Model	RFNE (%) $\downarrow$	SSIM $\uparrow$	PC $\uparrow$
12000	PIT-UNet	7.81	0.88	1.00
12000	iFLEX	4.11	0.96	1.00
24000	PIT-UNet	12.23	0.80	0.99
24000	iFLEX	7.10	0.92	1.00
36000	PIT-UNet	15.94	0.74	0.98
36000	iFLEX	10.48	0.87	0.99

Appendix

Table 9: iFLEX achieves lower RFNE and higher SSIM than the implemented PIT-UNet at three unseen Reynolds numbers. At

Re=36000

, RFNE decreases from

15.94\%

10.48\%

. Both models are evaluated with 20 intermediate states per interval. Bold denotes the best value at the reported precision.

Reynolds number	Method	RFNE (%) $\downarrow$	PC $\uparrow$	SSIM $\uparrow$
$600$	iFLEX	0.50	1.00	1.00
$600$	Endpoint-only	0.55	1.00	0.99
$1000$	iFLEX	0.73	1.00	0.99
$1000$	Endpoint-only	0.79	1.00	0.99
$12000$	iFLEX	4.11	1.00	0.96
$12000$	Endpoint-only	3.99	1.00	0.96

Appendix

Table 10: Explicit mean and difference features improve reconstruction at the three evaluated Reynolds numbers outside the training range, while endpoint-only fusion performs comparably at the two unseen values within it. Both models are trained at

Re\in\{2000,4000,8000,16000,32000\}

and evaluated with

T=20

. At

Re=36000

, complete fusion achieves

10.48\%

RFNE, compared with

11.26\%

for endpoint-only fusion. Bold indicates the best value at the reported precision.

$Re$	Method	RFNE (%) $\downarrow$	SSIM $\uparrow$	PC $\uparrow$
2000	Linear interpolation	10.97	0.805	0.991
	U-Net	20.85	0.725	0.983
	EDEN	3.11	0.948	0.999
	Super SloMo	4.36	0.956	0.999
	PIT-UNet	2.76	0.964	1.000
	iFLEX	1.21	0.989	1.000

Appendix

Table 11: iFLEX achieves the lowest RFNE and highest SSIM at every Reynolds number used during training, when evaluated on independent held-out trajectories. RFNE ranges from

1.21\%

Re=2000

9.35\%

Re=32000

. All methods use

256\times 256

crops and

T=20

. Bold denotes the best value at the reported precision.

$Re$	Method	RFNE (%) $\downarrow$	SSIM $\uparrow$	PC $\uparrow$
12,000	Linear interpolation	32.12	0.433	0.932
	U-Net	39.54	0.414	0.921
	EDEN	13.15	0.815	0.989
	Super SloMo	13.20	0.826	0.989
	PIT-UNet	7.81	0.880	0.996
	iFLEX	4.11	0.957	0.999

Appendix

Table 12: iFLEX achieves the lowest reconstruction error at both unseen Reynolds numbers within the training range. At

Re=12000

and

24000

, its RFNE is

4.11\%

and

7.10\%

, compared with

7.81\%

and

12.23\%

for PIT-UNet. All methods use

256\times 256

crops and

T=20

. Bold denotes the best value at the reported precision.

$Re$	Direction	Method	RFNE (%) $\downarrow$	SSIM $\uparrow$	PC $\uparrow$
600	Lower	Linear interpolation	2.18	0.977	1.000
		U-Net	43.19	0.622	0.990
		EDEN	1.55	0.970	1.000
		Super SloMo	1.80	0.986	1.000
		PIT-UNet	2.35	0.972	1.000
		iFLEX	0.50	0.995	1.000

Appendix

Table 13: iFLEX achieves the lowest RFNE for all three evaluated Reynolds numbers outside the training range. Its errors are

0.50\%

Re=600

0.73\%

Re=1000

, and

10.48\%

Re=36000

. The larger error in the upper-range case reflects the greater reconstruction difficulty of the tested flow, rather than parameter extrapolation distance alone. All methods use

256\times 256

crops and

T=20

$Re$	iFLEX	EDEN	EDEN+ $Re$	Super SloMo	Super SloMo+ $Re$
600	0.50%	1.55%	1.27%	1.80%	1.95%
1,000	0.73%	1.81%	1.46%	2.54%	2.70%
12,000	4.11%	13.15%	11.67%	13.20%	14.22%
24,000	7.10%	20.14%	18.43%	18.38%	19.85%
36,000	10.48%	25.52%	23.67%	22.87%	24.26%

Appendix

Table 14: iFLEX retains the lowest RFNE when EDEN and Super SloMo are additionally conditioned on Reynolds number. Conditioning improves EDEN under the tested implementation, while the constant-channel condition increases Super SloMo’s error. At

Re=36000

, the conditioned baselines achieve

23.67\%

and

24.26\%

RFNE, respectively, compared with

10.48\%

for iFLEX . Bold indicates the lowest error at each Reynolds number.

$Re$	Test seeds	RFNE (%) $\downarrow$
8,000	3407, 1000	$2.93\pm 0.35$
12,000	3407, 1000	$4.29\pm 0.74$
32,000	3407, 987	$6.14\pm 3.84$

Appendix

Table 15: Reconstruction error varies more between the two evaluated trajectories at

Re=32000

than at

Re=8000

12000

. Values report the mean and sample standard deviation of trajectory-level RFNE for a fixed trained checkpoint. Each trajectory contributes 256 spatiotemporal samples with 20 intermediate targets. The two realizations per Reynolds number provide a limited estimate of sensitivity to initial conditions.

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