# Global field reconstruction from sparse sensors with Voronoi tessellation-assisted deep learning

arXiv: https://arxiv.org/abs/2101.00554 · HTML: https://mlsanity.com/paper/2101.00554/

## Key facts

- Cited by 57 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 44 references that are in the map.
- First posted on arXiv 3 Jan 2021.

## Builds on (most cited first, 5 of 5)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Data-Driven Sparse Sensor Placement for Reconstruction](https://mlsanity.com/paper/1701.07569/) (2017)
- [Transfer Learning with Graph Neural Networks for Short-Term Highway Traffic Forecasting](https://mlsanity.com/paper/2004.08038/) (2020)

## Built on by (most cited first, 30 of 57)

- [Super-Resolution Analysis via Machine Learning: A Survey for Fluid Flows](https://mlsanity.com/paper/2301.10937/) (2023)
- [The transformative potential of machine learning for experiments in fluid mechanics](https://mlsanity.com/paper/2303.15832/) (2023)
- [RecFNO: a resolution-invariant flow and heat field reconstruction method from sparse observations via Fourier neural operator](https://mlsanity.com/paper/2302.09808/) (2023)
- [Grasping Extreme Aerodynamics on a Low-Dimensional Manifold](https://mlsanity.com/paper/2305.08024/) (2023)
- [SuperBench: A Super-Resolution Benchmark Dataset for Scientific Machine Learning](https://mlsanity.com/paper/2306.14070/) (2023)
- [Assessments of epistemic uncertainty using Gaussian stochastic weight averaging for fluid-flow regression](https://mlsanity.com/paper/2109.08248/) (2021)
- [Efficient deep data assimilation with sparse observations and time-varying sensors](https://mlsanity.com/paper/2310.16187/) (2023)
- [AgenticSciML: Collaborative Multi-Agent Systems for Emergent Discovery in Scientific Machine Learning](https://mlsanity.com/paper/2511.07262/) (2025)
- [Flow Completion Network: Inferring the Fluid Dynamics from Incomplete Flow Information using Graph Neural Networks](https://mlsanity.com/paper/2205.04739/) (2022)
- [Spatially-Aware Diffusion Models with Cross-Attention for Global Field Reconstruction with Sparse Observations](https://mlsanity.com/paper/2409.00230/) (2024)
- [Machine learning for modelling unstructured grid data in computational physics: a review](https://mlsanity.com/paper/2502.09346/) (2025)
- [Forecasting subcritical cylinder wakes with Fourier Neural Operators](https://mlsanity.com/paper/2301.08290/) (2023)
- [PhySense: Sensor Placement Optimization for Accurate Physics Sensing](https://mlsanity.com/paper/2505.18190/) (2025)
- [Guiding diffusion models to reconstruct flow fields from sparse data](https://mlsanity.com/paper/2510.19971/) (2025)
- [Multi-fidelity prediction of fluid flow and temperature field based on transfer learning using Fourier Neural Operator](https://mlsanity.com/paper/2304.06972/) (2023)
- [PiRD: Physics-informed Residual Diffusion for Flow Field Reconstruction](https://mlsanity.com/paper/2404.08412/) (2024)
- [A Machine Learning Surrogate Modeling Benchmark for Temperature Field Reconstruction of Heat-Source Systems](https://mlsanity.com/paper/2108.08298/) (2021)
- [Performance and accuracy assessments of an incompressible fluid solver coupled with a deep Convolutional Neural Network](https://mlsanity.com/paper/2109.09363/) (2021)
- [Multimodal Atmospheric Super-Resolution With Deep Generative Models](https://mlsanity.com/paper/2506.22780/) (2025)
- [Dynamical system prediction from sparse observations using deep neural networks with Voronoi tessellation and physics constraint](https://mlsanity.com/paper/2409.00458/) (2024)
- [Ensemble flow reconstruction in the atmospheric boundary layer from spatially limited measurements through latent diffusion models](https://mlsanity.com/paper/2303.00836/) (2023)
- [Constrained Sensing and Reliable State Estimation with Shallow Recurrent Decoders on a TRIGA Mark II Reactor](https://mlsanity.com/paper/2510.12368/) (2025)
- [Super-resolution GANs of randomly-seeded fields](https://mlsanity.com/paper/2202.11701/) (2022)
- [Uncertainty Guided Ensemble Self-Training for Semi-Supervised Global Field Reconstruction](https://mlsanity.com/paper/2302.11940/) (2023)
- [Spatiotemporal wall pressure forecast of a rectangular cylinder with physics-aware DeepU-Fourier neural network](https://mlsanity.com/paper/2508.03183/) (2025)
- [Applying Physics-Informed Enhanced Super-Resolution Generative Adversarial Networks to Turbulent Non-Premixed Combustion on Non-Uniform Meshes and Demonstration of an Accelerated Simulation Workflow](https://mlsanity.com/paper/2210.16248/) (2022)
- [Reconstruction of Fields from Sparse Sensing: Differentiable Sensor Placement Enhances Generalization](https://mlsanity.com/paper/2312.09176/) (2023)
- [Physics-aligned Schr"odinger bridge](https://mlsanity.com/paper/2409.17825/) (2024)
- [FR-Mamba: Time-Series Physical Field Reconstruction Based on State Space Model](https://mlsanity.com/paper/2505.16083/) (2025)
- [BLISSNet: Deep Operator Learning for Fast and Accurate Flow Reconstruction from Sparse Sensor Measurements](https://mlsanity.com/paper/2602.24228/) (2026)

## Abstract (arXiv)

Achieving accurate and robust global situational awareness of a complex time-evolving field from a limited number of sensors has been a longstanding challenge. This reconstruction problem is especially difficult when sensors are sparsely positioned in a seemingly random or unorganized manner, which is often encountered in a range of scientific and engineering problems. Moreover, these sensors can be in motion and can become online or offline over time. The key leverage in addressing this scientific issue is the wealth of data accumulated from the sensors. As a solution to this problem, we propose a data-driven spatial field recovery technique founded on a structured grid-based deep-learning approach for arbitrary positioned sensors of any numbers. It should be noted that the na"ive use of machine learning becomes prohibitively expensive for global field reconstruction and is furthermore not adaptable to an arbitrary number of sensors. In the present work, we consider the use of Voronoi tessellation to obtain a structured-grid representation from sensor locations enabling the computationally tractable use of convolutional neural networks. One of the central features of the present method is its compatibility with deep-learning based super-resolution reconstruction techniques for structured sensor data that are established for image processing. The proposed reconstruction technique is demonstrated for unsteady wake flow, geophysical data, and three-dimensional turbulence. The current framework is able to handle an arbitrary number of moving sensors, and thereby overcomes a major limitation with existing reconstruction methods. The presented technique opens a new pathway towards the practical use of neural networks for real-time global field estimation.