# Deep Operator Learning for High-Fidelity Fluid Flow Field Reconstruction from Sparse Sensor Measurements

arXiv: https://arxiv.org/abs/2412.08009 · HTML: https://mlsanity.com/paper/2412.08009/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 30 references that are in the map.
- First posted on arXiv 11 Dec 2024.

## Builds on (most cited first, 12 of 12)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [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)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [CFDBench: A Large-Scale Benchmark for Machine Learning Methods in Fluid Dynamics](https://mlsanity.com/paper/2310.05963/) (2023)
- [Continuous Field Reconstruction from Sparse Observations with Implicit Neural Networks](https://mlsanity.com/paper/2401.11611/) (2024)
- [RecFNO: a resolution-invariant flow and heat field reconstruction method from sparse observations via Fourier neural operator](https://mlsanity.com/paper/2302.09808/) (2023)
- [Efficient deep data assimilation with sparse observations and time-varying sensors](https://mlsanity.com/paper/2310.16187/) (2023)
- [Challenges and opportunities for machine learning in multiscale computational modeling](https://mlsanity.com/paper/2303.12261/) (2023)

## Built on by (most cited first, 9 of 9)

- [Constrained Sensing and Reliable State Estimation with Shallow Recurrent Decoders on a TRIGA Mark II Reactor](https://mlsanity.com/paper/2510.12368/) (2025)
- [PAINT: Parallel-in-time Neural Twins for Dynamical System Reconstruction](https://mlsanity.com/paper/2510.16004/) (2025)
- [Noise-adapted Neural Operator for Robust Non-Line-of-Sight Imaging](https://mlsanity.com/paper/2508.09655/) (2025)
- [Adapting Automotive Aerodynamics Surrogates to New Vehicle Families via Transfer Learning](https://mlsanity.com/paper/2605.27968/) (2026)
- [A Physics-Augmented GraphGPS Framework for the Reconstruction of 3D Riemann Problems from Sparse Data](https://mlsanity.com/paper/2505.21421/) (2025)
- [Equation-Free Digital Twins for Nonlinear Structural Dynamics](https://mlsanity.com/paper/2605.00950/) (2026)
- [Christoffel-DPS: Optimal sensor placement in diffusion posterior sampling for arbitrary distributions](https://mlsanity.com/paper/2605.06861/) (2026)
- [SimCast-S2S: A Computationally Efficient Diffusion Model for Subseasonal Precipitation Forecasting](https://mlsanity.com/paper/2608.26594/) (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)

Reconstructing high-fidelity fluid flow fields from sparse sensor measurements is vital for many science and engineering applications but remains challenging because of dimensional disparities between state and observational spaces. Due to such dimensional differences, the measurement operator becomes ill-conditioned and non-invertible, making the reconstruction of flow fields from sensor measurements extremely difficult. Although sparse optimization and machine learning address the above problems to some extent, questions about their generalization and efficiency remain, particularly regarding the discretization dependence of these models. In this context, deep operator learning offers a better solution as this approach models mappings between infinite-dimensional functional spaces, enabling superior generalization and discretization-independent reconstruction. We introduce FLRONet, a deep operator learning framework that is trained to reconstruct fluid flow fields from sparse sensor measurements. FLRONet employs a branch-trunk network architecture to represent the inverse measurement operator that maps sensor observations to the original flow field, a continuous function of both space and time. Validation performed on the CFDBench dataset has demonstrated that FLRONet consistently achieves high levels of reconstruction accuracy and robustness, even in scenarios where sensor measurements are inaccurate or missing. Furthermore, the operator learning approach endows FLRONet with the capability to perform zero-shot super-resolution in both spatial and temporal domains, offering a solution for rapid reconstruction of high-fidelity flow fields.