# RecFNO: a resolution-invariant flow and heat field reconstruction method from sparse observations via Fourier neural operator

arXiv: https://arxiv.org/abs/2302.09808 · HTML: https://mlsanity.com/paper/2302.09808/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 45 references that are in the map.
- First posted on arXiv 20 Feb 2023.

## Builds on (most cited first, 15 of 15)

- [Fully Convolutional Networks for Semantic Segmentation](https://mlsanity.com/paper/1411.4038/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Generative Adversarial Networks: An Overview](https://mlsanity.com/paper/1710.07035/) (2017)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Dynamic mode decomposition with control](https://mlsanity.com/paper/1409.6358/) (2014)
- [The Random Feature Model for Input-Output Maps between Banach Spaces](https://mlsanity.com/paper/2005.10224/) (2020)
- [Global field reconstruction from sparse sensors with Voronoi tessellation-assisted deep learning](https://mlsanity.com/paper/2101.00554/) (2021)
- [Multi-level Convolutional Autoencoder Networks for Parametric Prediction of Spatio-temporal Dynamics](https://mlsanity.com/paper/1912.11114/) (2019)
- [A Comparison of Neural Network Architectures for Data-Driven Reduced-Order Modeling](https://mlsanity.com/paper/2110.03442/) (2021)
- [A Deep Neural Network Surrogate Modeling Benchmark for Temperature Field Prediction of Heat Source Layout](https://mlsanity.com/paper/2103.11177/) (2021)
- [A Machine Learning Surrogate Modeling Benchmark for Temperature Field Reconstruction of Heat-Source Systems](https://mlsanity.com/paper/2108.08298/) (2021)
- [State estimation with limited sensors -- A deep learning based approach](https://mlsanity.com/paper/2101.11513/) (2021)
- [Parameter-Conditioned Sequential Generative Modeling of Fluid Flows](https://mlsanity.com/paper/1912.06752/) (2019)
- [Observable dictionary learning for high-dimensional statistical inference](https://mlsanity.com/paper/1702.05289/) (2017)

## Built on by (most cited first, 19 of 19)

- [DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Training](https://mlsanity.com/paper/2403.03542/) (2024)
- [A Mathematical Guide to Operator Learning](https://mlsanity.com/paper/2312.14688/) (2023)
- [Mixture-of-Experts Operator Transformer for Large-Scale PDE Pre-Training](https://mlsanity.com/paper/2510.25803/) (2025)
- [AgenticSciML: Collaborative Multi-Agent Systems for Emergent Discovery in Scientific Machine Learning](https://mlsanity.com/paper/2511.07262/) (2025)
- [Time Series Analysis in Frequency Domain: A Survey of Open Challenges, Opportunities and Benchmarks](https://mlsanity.com/paper/2504.07099/) (2025)
- [Deep Operator Learning for High-Fidelity Fluid Flow Field Reconstruction from Sparse Sensor Measurements](https://mlsanity.com/paper/2412.08009/) (2024)
- [Generating Full-field Evolution of Physical Dynamics from Irregular Sparse Observations](https://mlsanity.com/paper/2505.09284/) (2025)
- [RAMS: Residual-based adversarial-gradient moving sample method for scientific machine learning in solving partial differential equations](https://mlsanity.com/paper/2509.01234/) (2025)
- [Spatiotemporal wall pressure forecast of a rectangular cylinder with physics-aware DeepU-Fourier neural network](https://mlsanity.com/paper/2508.03183/) (2025)
- [FieldFormer: Locality-Aware Transformers for Spatio-Temporal Modeling on Sparse Sensor Networks](https://mlsanity.com/paper/2510.03589/) (2025)
- [FLUIDSPLAT: Reconstructing Physical Fields from Sparse Sensors via Gaussian Primitives](https://mlsanity.com/paper/2605.18866/) (2026)
- [A novel Fourier neural operator framework for classification of multi-sized images: Application to three dimensional digital porous media](https://mlsanity.com/paper/2402.11568/) (2024)
- [Multiscale autonomous forecasting of plasma systems' dynamics using neural networks](https://mlsanity.com/paper/2502.11203/) (2025)
- [CS-SHRED: Enhancing SHRED for Robust Recovery of Spatiotemporal Dynamics](https://mlsanity.com/paper/2507.22303/) (2025)
- [Learning to Reconstruct Temperature Field from Sparse Observations with Implicit Physics Priors](https://mlsanity.com/paper/2512.01196/) (2025)
- [GLU: Global-Local-Uncertainty Fusion for Scalable Spatiotemporal Reconstruction and Forecasting](https://mlsanity.com/paper/2603.26023/) (2026)
- [LAtent Phase Inference from Short time sequences using SHallow REcurrent Decoders (LAPIS-SHRED)](https://mlsanity.com/paper/2604.01216/) (2026)
- [Inferring physical fields in coupled systems with unknown parameters from incomplete observations using physics-constrained attentive neural operators](https://mlsanity.com/paper/2610.05723/) (2026)
- [Learning Incompressible 3D Flows from Sparse Samples: A Set Transformer Approach to Divergence-Free Kernel Fitting](https://mlsanity.com/paper/2610.16645/) (2026)

## Abstract (arXiv)

Perception of the full state is an essential technology to support the monitoring, analysis, and design of physical systems, one of whose challenges is to recover global field from sparse observations. Well-known for brilliant approximation ability, deep neural networks have been attractive to data-driven flow and heat field reconstruction studies. However, limited by network structure, existing researches mostly learn the reconstruction mapping in finite-dimensional space and has poor transferability to variable resolution of outputs. In this paper, we extend the new paradigm of neural operator and propose an end-to-end physical field reconstruction method with both excellent performance and mesh transferability named RecFNO. The proposed method aims to learn the mapping from sparse observations to flow and heat field in infinite-dimensional space, contributing to a more powerful nonlinear fitting capacity and resolution-invariant characteristic. Firstly, according to different usage scenarios, we develop three types of embeddings to model the sparse observation inputs: MLP, mask, and Voronoi embedding. The MLP embedding is propitious to more sparse input, while the others benefit from spatial information preservation and perform better with the increase of observation data. Then, we adopt stacked Fourier layers to reconstruct physical field in Fourier space that regularizes the overall recovered field by Fourier modes superposition. Benefiting from the operator in infinite-dimensional space, the proposed method obtains remarkable accuracy and better resolution transferability among meshes. The experiments conducted on fluid mechanics and thermology problems show that the proposed method outperforms existing POD-based and CNN-based methods in most cases and has the capacity to achieve zero-shot super-resolution.