SIMR-NO: A Spectrally-Informed Multi-Resolution Neural Operator for Turbulent Flow Super-Resolution

Muhammad Abid; Omer San

cs.LGMar 30, 2026

SIMR-NO: A Spectrally-Informed Multi-Resolution Neural Operator for Turbulent Flow Super-Resolution

Authors: Muhammad Abid, Omer San

Organizations: Department of Mechanical and Aerospace Engineering, University of Tennessee, Knoxville, TN 37996, USA.

Abstract

Reconstructing high-resolution turbulent flow fields from severely under-resolved observations is a fundamental inverse problem in computational fluid dynamics and scientific machine learning. Classical interpolation methods fail to recover missing fine-scale structures, while existing deep learning approaches rely on convolutional architectures that lack the spectral and multiscale inductive biases necessary for physically faithful reconstruction at large upscaling factors. We introduce the Spectrally-Informed Multi-Resolution Neural Operator (SIMR-NO), a hierarchical operator learning framework that factorizes the ill-posed inverse mapping across intermediate spatial resolutions, combines deterministic interpolation priors with spectrally gated Fourier residual corrections at each stage, and incorporates local refinement modules to recover fine-scale spatial features beyond the truncated Fourier basis. The proposed method is evaluated on Kolmogorov-forced two-dimensional turbulence, where $128\times128$ vorticity fields are reconstructed from extremely coarse $8\times8$ observations representing a $16\times$ downsampling factor. Across 201 independent test realizations, SIMR-NO achieves a mean relative $\ell_2$ error of $26.04\%$ with the lowest error variance among all methods, reducing reconstruction error by $31.7\%$ over FNO, $26.0\%$ over EDSR, and $9.3\%$ over LapSRN. Beyond pointwise accuracy, SIMR-NO is the only method that faithfully reproduces the ground-truth energy and enstrophy spectra across the full resolved wavenumber range, demonstrating physically consistent super-resolution of turbulent flow fields.

Figures & tables

Figure 1: Overview of the SIMR-NO architecture as a composition of two hierarchical stage blocks corresponding to ( 25 )–( 26 ). Each stage combines a deterministic bicubic upsampling prior with a learned spectrally-informed residual correction scaled by learnable parameters $\alpha_{1}$ and $\alpha_{2}$ . Stage 1 resolves large-scale coherent structures across the $32\rightarrow 64$ resolution transition, while Stage 2 refines fine-scale turbulent features across the $64\rightarrow 128$ transition. Both FNO and SIMR-NO receive the same pseudo high-resolution conditioning field $\widetilde{\omega}^{128}$ ; SIMR-NO internally constructs $a^{32}=\mathcal{U}_{128\rightarrow 32}(\widetilde{\omega}^{128})$ before applying the two-stage reconstruction defined in ( 22 ).

Figure 2: Mean relative $\ell_{2}$ error of 201 held-out test samples by the standard deviation. SIMR-NO with the lowest mean error and standard deviation shows superiority in accuracy and consistency compared to all other baseline methods.

Model	Mean RelL2 $\downarrow$	Std RelL2 $\downarrow$
FNO	0.3813	0.1129
EDSR	0.3517	0.1051
LapSRN	0.2879	0.1074
SIMR-NO	0.2604	0.0980

Table 1: Mean relative

\ell_{2}

error with standard deviation over 201 held-out test samples. Best results in bold .

Figure 3: The box plots display the distribution of relative $\ell_{2}$ error throughout all 201 held-out test samples. SIMR-NO demonstrates superior performance because it has the lowest median value $0.2383$ together with its most compact interquartile range, which measures between $0.1990$ and $0.3010$ and its smallest number of high-error outliers.

Figure 4: The reconstruction results show the best results from the test sample. The first row shows input LR ( $8\times 8$ ) and pseudo HR ( $\text{LR8}\rightarrow 128$ ) and ground truth HR ( $128\times 128$ ) and model predictions. The second row displays absolute pointwise error maps. SIMR-NO achieves the lowest and most spatially uniform error while accurately reconstructing fine-scale vortical structures that all baseline reconstructions failed to detect.

Figure 5: The best-case sample shows enstrophy spectrum $\mathcal{E}(k)$ and energy spectrum $E(k)$ and cumulative POD energy results. The SIMR-NO model tracks ground-truth spectral decay across all wavenumbers, while all baseline methods fail to accurately measure high-wavenumber energy content.

Metric	FNO	EDSR	LapSRN	SIMR-NO
MSE	0.2320	0.1686	0.1276	0.0604
SSIM	0.7691	0.7991	0.8644	0.9025
PSNR	25.07	26.46	27.67	30.92
RelL2 (%)	24.49	20.88	18.16	12.49

Table 2: The best-case example demonstrates its performance through per-sample metrics. SIMR-NO exceeds all baseline systems in every performance metric by a significant distance.

Figure 6: The reconstruction results show test sample results, which serve as the representative test sample. The baseline models show increasing oversmoothing, which results in their complete loss of intermediate-scale vortical structures. SIMR-NO maintains the correct vortical topology and fine-scale filamentary features with substantially lower reconstruction error.

Figure 7: The enstrophy spectrum $\mathcal{E}(k)$ and energy spectrum $E(k)$ and cumulative POD energy for the representative sample provide the three main components of the research study. The SIMR-NO method exhibits better spectral stability than other methods, especially when analyzing high wavenumbers that connect to detailed vortex patterns.

Metric	FNO	EDSR	LapSRN	SIMR-NO
MSE	0.6890	0.6718	0.3637	0.2176
SSIM	0.6367	0.6717	0.7981	0.8396
PSNR	21.62	21.73	24.40	26.63
RelL2 (%)	36.15	35.69	26.26	20.32

Table 3: The representative case study shows its metrics through per-sample evaluation. The SIMR-NO system reaches its highest performance across all metrics when tested under standard operational conditions.

Figure 8: The most difficult test sample shows reconstruction results that represent the most complex testing scenario. FNO exhibits its worst-case performance, which surpasses all other baseline systems through its higher MSE results. SIMR-NO maintains correct large-scale vortical topology and produces the most accurate reconstruction under these worst-case conditions.

Figure 9: The challenging sample shows three different measurements through its enstrophy spectrum, which is described by $\mathcal{E}(k)$ and its energy spectrum, which is represented by $E(k)$ and its cumulative POD energy. Baseline methods show their first departure from ground truth between low wavenumber values. The spectral behavior of SIMR-NO maintains its physical consistency throughout all resolved frequency intervals.

Metric	FNO	EDSR	LapSRN	SIMR-NO
MSE	1.1558	0.9917	0.7805	0.4805
SSIM	0.5474	0.5863	0.6624	0.7453
PSNR	19.16	19.82	20.86	22.97
RelL2 (%)	45.05	41.73	37.02	29.05

Table 4: Per-sample metrics for the challenging case. The SIMR-NO achieves its highest performance improvements when it operates under its most difficult testing conditions, which demonstrates its architectural strength that other systems lack.

Neural Operators

Neural operators learn mappings between functions, enabling fast surrogates for PDEs and other continuous physical systems.

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High-fidelity three-dimensional turbulent flow fields are essential for scientific and engineering analyses, yet obtaining them through direct numerical simulation or high-resolution measurements remains expensive. Super-resolution reconstruction from coarse-grained data therefore offers a promising alternative, but remains fundamentally challenging because fine-scale structures are intrinsically three-dimensional and dynamically coupled across scales. Existing data-driven approaches still struggle to recover multiscale volumetric structures and their coherent temporal evolution. To address these limitations, we propose a turbulence-aware framework for time-resolved three-dimensional flow reconstruction. The framework couples a multiscale neural operator, designed to recover volumetric structures, with a state space model to promote temporal consistency in the reconstructed dynamics. In addition, Navier-Stokes residuals are incorporated as soft constraints to regularize the reconstruction toward physically plausible dynamics. A mixture-of-experts mechanism is also introduced to support unsupervised adaptation to unknown coarse-graining. Across evaluations on canonical turbulent systems and atmospheric reanalysis data, the proposed framework reduces reconstruction errors relative to learning-based and classical baselines, while better preserving multiscale spatial structure and coherent temporal evolution. These results validate the effectiveness of the proposed framework in reconstructing high-fidelity three-dimensional flow fields from coarse-grained data.

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