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

A preprint version of the article is available at arXiv.


Achieving accurate and robust global situational awareness of a complex time-evolving field from a limited number of sensors has been a long-standing 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 could be in motion and could 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 naive use of machine learning becomes prohibitively expensive for global field reconstruction and is furthermore not adaptable to an arbitrary number of sensors. In this 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 our 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. Our technique opens a new pathway toward the practical use of neural networks for real-time global field estimation.

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Fig. 1: Voronoi tessellation-assisted global data recovery from discrete sensor locations for a two-dimensional cylinder wake.
Fig. 2: Voronoi tessellation-assisted spatial data recovery of a two-dimensional cylinder wake with nsensor = 8 and 16.
Fig. 3: Comparison of spatial data recovery for NOAA sea surface temperature.
Fig. 4: Voronoi-based spatial data recovery of NOAA sea surface temperature.
Fig. 5: Voronoi tessellation-assisted data recovery of turbulent channel flow.
Fig. 6: Robustness of our reconstruction technique for noisy input for the example of turbulent channel flow.

Data availability

Training data used in this study are available on the Open Science Framework (

Code availability

Sample codes for training our models are available on GitHub ( and


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R.M. and N.R. were supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, under contract DE-AC02-06CH11357. This research was funded in part and used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under contract DE-AC02-06CH11357. Koji.F. thanks support from the Japan Society for the Promotion of Science (grant nos. 18H03758, 21H05007). K.T. acknowledges support from the US Air Force Office of Scientific Research (grant nos. FA9550-16-1-0650 and FA9550-21-1-0178) and the US Army Research Office (grant no. W911NF-19-1-0032).

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Authors and Affiliations



Kai.F., R.M and N.R. designed research. Kai.F. performed research. Kai.F. analysed data. Kai.F. and K.T. wrote the paper. Koji.F. and K.T. supervised. All authors reviewed the manuscript.

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Correspondence to Kai Fukami.

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The authors declare no competing interests.

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Peer review information Nature Machine Intelligence thanks Mickaël Bourgoin and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Velocity RMS of turbulent channel flow.

Root mean squared value of streamwise velocity fluctuation in turbulent channel flow.

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Fukami, K., Maulik, R., Ramachandra, N. et al. Global field reconstruction from sparse sensors with Voronoi tessellation-assisted deep learning. Nat Mach Intell 3, 945–951 (2021).

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