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

arXiv: https://arxiv.org/abs/2603.28073 · HTML: https://mlsanity.com/paper/2603.28073/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 78 references that are in the map.
- First posted on arXiv 30 Mar 2026.

## Builds on (most cited first, 30 of 36)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [SwinIR: Image Restoration Using Swin Transformer](https://mlsanity.com/paper/2108.10257/) (2021)
- [Enhanced Deep Residual Networks for Single Image Super-Resolution](https://mlsanity.com/paper/1707.02921/) (2017)
- [Image Super-Resolution via Iterative Refinement](https://mlsanity.com/paper/2104.07636/) (2021)
- [Image Super-Resolution Using Very Deep Residual Channel Attention Networks](https://mlsanity.com/paper/1807.02758/) (2018)
- [ESRGAN: Enhanced Super-Resolution Generative Adversarial Networks](https://mlsanity.com/paper/1809.00219/) (2018)
- [Image Super-Resolution Using Deep Convolutional Networks](https://mlsanity.com/paper/1501.00092/) (2014)
- [Real-Time Single Image and Video Super-Resolution Using an Efficient Sub-Pixel Convolutional Neural Network](https://mlsanity.com/paper/1609.05158/) (2016)
- [Accurate Image Super-Resolution Using Very Deep Convolutional Networks](https://mlsanity.com/paper/1511.04587/) (2015)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Residual Dense Network for Image Super-Resolution](https://mlsanity.com/paper/1802.08797/) (2018)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Deep Laplacian Pyramid Networks for Fast and Accurate Super-Resolution](https://mlsanity.com/paper/1704.03915/) (2017)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Accelerating the Super-Resolution Convolutional Neural Network](https://mlsanity.com/paper/1608.00367/) (2016)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Deeply-Recursive Convolutional Network for Image Super-Resolution](https://mlsanity.com/paper/1511.04491/) (2015)
- [Fast, Accurate, and Lightweight Super-Resolution with Cascading Residual Network](https://mlsanity.com/paper/1803.08664/) (2018)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [Deep Back-Projection Networks For Super-Resolution](https://mlsanity.com/paper/1803.02735/) (2018)
- [Enhancing Computational Fluid Dynamics with Machine Learning](https://mlsanity.com/paper/2110.02085/) (2021)

## Built on by (most cited first, 4 of 4)

- [WLNO: Wavelet-Laplace Neural Operator for Solving Partial Differential Equations](https://mlsanity.com/paper/2605.24658/) (2026)
- [The impact of observation density on Bayesian inversion of latent dynamics in shock-dominated flows](https://mlsanity.com/paper/2605.19076/) (2026)
- [Quantum SEDONet: Spectrally-Embedded Quantum Deep Operator Networks for Partial Differential Equations](https://mlsanity.com/paper/2608.27626/) (2026)
- [Extreme-Scale Super-Resolution of Homogeneous Turbulence from a Single Realization](https://mlsanity.com/paper/2610.20305/) (2026)

## Abstract (arXiv)

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 128×128 vorticity fields are reconstructed from extremely coarse 8×8 observations representing a 16× downsampling factor. Across 201 independent test realizations, SIMR-NO achieves a mean relative ℓ2 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.