# Super-Resolution Neural Operator

arXiv: https://arxiv.org/abs/2303.02584 · HTML: https://mlsanity.com/paper/2303.02584/

## Key facts

- Cited by 49 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 43 references that are in the map.
- First posted on arXiv 5 Mar 2023.

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

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Swin Transformer: Hierarchical Vision Transformer using Shifted Windows](https://mlsanity.com/paper/2103.14030/) (2021)
- [Training data-efficient image transformers & distillation through attention](https://mlsanity.com/paper/2012.12877/) (2020)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (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 Using Very Deep Residual Channel Attention Networks](https://mlsanity.com/paper/1807.02758/) (2018)
- [Real-Time Single Image and Video Super-Resolution Using an Efficient Sub-Pixel Convolutional Neural Network](https://mlsanity.com/paper/1609.05158/) (2016)
- [Pre-Trained Image Processing Transformer](https://mlsanity.com/paper/2012.00364/) (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)
- [Learning Continuous Image Representation with Local Implicit Image Function](https://mlsanity.com/paper/2012.09161/) (2020)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [Meta-SR: A Magnification-Arbitrary Network for Super-Resolution](https://mlsanity.com/paper/1903.00875/) (2019)
- [How Much Position Information Do Convolutional Neural Networks Encode?](https://mlsanity.com/paper/2001.08248/) (2020)
- [Interpreting Super-Resolution Networks with Local Attribution Maps](https://mlsanity.com/paper/2011.11036/) (2020)
- [Local Texture Estimator for Implicit Representation Function](https://mlsanity.com/paper/2111.08918/) (2021)
- [MeshfreeFlowNet: A Physics-Constrained Deep Continuous Space-Time Super-Resolution Framework](https://mlsanity.com/paper/2005.01463/) (2020)
- [Learning A Single Network for Scale-Arbitrary Super-Resolution](https://mlsanity.com/paper/2004.03791/) (2020)
- [Implicit Transformer Network for Screen Content Image Continuous Super-Resolution](https://mlsanity.com/paper/2112.06174/) (2021)
- [Solving PDE-constrained Control Problems Using Operator Learning](https://mlsanity.com/paper/2111.04941/) (2021)
- [SRWarp: Generalized Image Super-Resolution under Arbitrary Transformation](https://mlsanity.com/paper/2104.10325/) (2021)

## Built on by (most cited first, 30 of 49)

- [Pixel to Gaussian: Ultra-Fast Continuous Super-Resolution with 2D Gaussian Modeling](https://mlsanity.com/paper/2503.06617/) (2025)
- [Mitigating Spectral Bias in Neural Operators via High-Frequency Scaling for Physical Systems](https://mlsanity.com/paper/2503.13695/) (2025)
- [Generalized and Efficient 2D Gaussian Splatting for Arbitrary-scale Super-Resolution](https://mlsanity.com/paper/2501.06838/) (2025)
- [Nonlocal Attention Operator: Materializing Hidden Knowledge Towards Interpretable Physics Discovery](https://mlsanity.com/paper/2408.07307/) (2024)
- [Basis-to-Basis Operator Learning Using Function Encoders](https://mlsanity.com/paper/2410.00171/) (2024)
- [HIIF: Hierarchical Encoding based Implicit Image Function for Continuous Super-resolution](https://mlsanity.com/paper/2412.03748/) (2024)
- [DiffFNO: Diffusion Fourier Neural Operator](https://mlsanity.com/paper/2411.09911/) (2024)
- [TULIP: Transformer for Upsampling of LiDAR Point Clouds](https://mlsanity.com/paper/2312.06733/) (2023)
- [S3Mamba: Arbitrary-Scale Super-Resolution via Scaleable State Space Model](https://mlsanity.com/paper/2411.11906/) (2024)
- [Hierarchical Neural Operator Transformer with Learnable Frequency-aware Loss Prior for Arbitrary-scale Super-resolution](https://mlsanity.com/paper/2405.12202/) (2024)
- [Exploring the Low-Pass Filtering Behavior in Image Super-Resolution](https://mlsanity.com/paper/2405.07919/) (2024)
- [Thera: Aliasing-Free Arbitrary-Scale Super-Resolution with Neural Heat Fields](https://mlsanity.com/paper/2311.17643/) (2023)
- [VQualA 2025 Challenge on Image Super-Resolution Generated Content Quality Assessment: Methods and Results](https://mlsanity.com/paper/2509.06413/) (2025)
- [Learning Dual-Level Deformable Implicit Representation for Real-World Scale Arbitrary Super-Resolution](https://mlsanity.com/paper/2403.10925/) (2024)
- [Space-Time Video Super-resolution with Neural Operator](https://mlsanity.com/paper/2404.06036/) (2024)
- [AnySR: Realizing Image Super-Resolution as Any-Scale, Any-Resource](https://mlsanity.com/paper/2407.04241/) (2024)
- [Solving Differential Equations with Constrained Learning](https://mlsanity.com/paper/2410.22796/) (2024)
- [NeurOp-Diff:Continuous Remote Sensing Image Super-Resolution via Neural Operator Diffusion](https://mlsanity.com/paper/2501.09054/) (2025)
- [MGHF: Multi-Granular High-Frequency Perceptual Loss for Image Super-Resolution](https://mlsanity.com/paper/2411.13548/) (2024)
- [Reconstruct Anything Model: a lightweight general model for computational imaging](https://mlsanity.com/paper/2503.08915/) (2025)
- [AnyTSR: Any-Scale Thermal Super-Resolution for UAV](https://mlsanity.com/paper/2504.13682/) (2025)
- [Localized FNO for Spatiotemporal Hemodynamic Upsampling in Aneurysm MRI](https://mlsanity.com/paper/2507.13789/) (2025)
- [Locally Adaptive Conformal Inference for Operator Models](https://mlsanity.com/paper/2507.20975/) (2025)
- [Adaptive Multi-modal Fusion of Spatially Variant Kernel Refinement with Diffusion Model for Blind Image Super-Resolution](https://mlsanity.com/paper/2403.05808/) (2024)
- [LeRF: Learning Resampling Function for Adaptive and Efficient Image Interpolation](https://mlsanity.com/paper/2407.09935/) (2024)
- [SetONet: A Set-Based Operator Network for Solving PDEs with Variable-Input Sampling](https://mlsanity.com/paper/2505.04738/) (2025)
- [Radiative-Structured Neural Operator for Continuous Spectral Super-Resolution](https://mlsanity.com/paper/2511.17895/) (2025)
- [Neuro-Symbolic Multitasking: A Unified Framework for Discovering Generalizable Solutions to PDE Families](https://mlsanity.com/paper/2602.11630/) (2026)
- [Beyond Visual Fidelity: Benchmarking Super-Resolution Models for Large-Scale Remote Sensing Imagery via Downstream Task Integration](https://mlsanity.com/paper/2605.00310/) (2026)
- [A Unified Framework to Super-Resolve Face Images of Varied Low Resolutions](https://mlsanity.com/paper/2306.03380/) (2023)

## Abstract (arXiv)

We propose Super-resolution Neural Operator (SRNO), a deep operator learning framework that can resolve high-resolution (HR) images at arbitrary scales from the low-resolution (LR) counterparts. Treating the LR-HR image pairs as continuous functions approximated with different grid sizes, SRNO learns the mapping between the corresponding function spaces. From the perspective of approximation theory, SRNO first embeds the LR input into a higher-dimensional latent representation space, trying to capture sufficient basis functions, and then iteratively approximates the implicit image function with a kernel integral mechanism, followed by a final dimensionality reduction step to generate the RGB representation at the target coordinates. The key characteristics distinguishing SRNO from prior continuous SR works are: 1) the kernel integral in each layer is efficiently implemented via the Galerkin-type attention, which possesses non-local properties in the spatial domain and therefore benefits the grid-free continuum; and 2) the multilayer attention architecture allows for the dynamic latent basis update, which is crucial for SR problems to "hallucinate" high-frequency information from the LR image. Experiments show that SRNO outperforms existing continuous SR methods in terms of both accuracy and running time. Our code is at https://github.com/2y7c3/Super-Resolution-Neural-Operator