Source-linked AI summary
Global field reconstruction from sparse sensors with Voronoi tessellation-assisted deep learning
Kai Fukami, Romit Maulik, Nesar Ramachandra, Koji Fukagata, Kunihiko Taira
TL;DR
Global field reconstruction from sparse, irregular, moving, and changing sensor networks remains difficult because conventional CNNs require structured inputs. The paper combines Voronoi tessellation with sensor-location masks to enable CNN-based reconstruction, demonstrating accurate and robust recovery across wake flow, geophysical data, and turbulent-flow examples. The framework handles varying sensor numbers and moving sensor positions with a single model.
Problem
Reconstructing a time-evolving global field from sparse sensors is challenging when sensor locations are irregular, moving, or changing over time.
Method
The method converts sensor measurements into Voronoi images and sensor-location masks, then feeds both structured representations into a convolutional neural network.
Results
The method demonstrated accurate and robust global field reconstruction across three flow and geophysical examples, including varying sensor numbers and placements.
Takeaways & Limitations
A single trained model can reconstruct global fields for arbitrary sensor counts and time-varying sensor positions.
Takeaways & Limitations
Supervised L2-minimization training can underestimate flow fluctuations, and broader physical constraints or turbulent-data generalization may require extensions such as constrained or unsupervised learning.
Abstract
from arXiv · showhide
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 naïve 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.
1 Introduction
The paper addresses global field reconstruction from sparse, irregular, and changing sensor data, which standard CNNs cannot directly process. It proposes Voronoi-based structured representations that allow CNN recovery for arbitrary sensor locations and sensor counts.
- 1 Introduction: Sparse global field reconstruction is difficult for high-dimensional physical systems, especially with unstructured or randomly placed sensors.Traditional methods face challenges, while CNNs assume structured, uniformly arranged training data.
- 1 Introduction: Figure 1 illustrates an eight-sensor cylinder-wake example in which Voronoi and mask images are jointly supplied to a CNN.The mask marks grid locations containing sensors with 1 and other grid locations with 0.
- 1 Introduction: The proposed method projects sparse sensor data onto a structured representation while retaining spatial sensor locations.It constructs a Voronoi tessellation and adds sensor-location information through a mask.
- 1 Introduction: Voronoi tessellation enables CNN-based reconstruction from arbitrary sensor locations and varying numbers of sensors.The framework is intended for applications that fuse information from discrete sensors, including buoy-based sensing and particle image velocimetry.
2 Problem setup and approach
The approach converts arbitrary sparse sensor measurements and locations into two structured-grid images for a single CNN model. Voronoi regions provide spatially assigned measurements, while a mask preserves sensor positions and supports changing sensor configurations.
- 2 Problem setup and approach: The objective is to reconstruct a high-resolution two-dimensional global field from local measurements at arbitrary locations and sensor counts.Sensors may move or change online status over time, so reconstruction should use one model without retraining.
- 2 Problem setup and approach: The CNN receives a Voronoi-projected measurement image and a mask image encoding sensor positions.The mask is 1 at sensor grid locations and 0 elsewhere.
- 2.1 Voronoi tessellation: Voronoi tessellation partitions the spatial domain into regions assigned according to distances from the sensors.For Euclidean domains, region boundaries are sensor bisectors, and each grid point receives its region’s representative sensor value.
- 2.1 Voronoi tessellation: The resulting structured-grid representation makes existing CNN architectures applicable to measurements from arbitrarily placed sensors.The tessellation need only be performed once for stationary sensors; sensor-count changes require updates near affected regions.
- 2.2 Convolutional neural network: The CNN uses convolutional layers to extract input features and produces the desired high-resolution global field.The implementation uses ReLU activations, ADAM optimization, early stopping, and three-fold cross-validation.
- 2.2 Convolutional neural network: Training learns CNN weights by minimizing the squared L2 reconstruction error between the target field and the model output.The learned model maps Voronoi measurements and mask information to the reconstructed field.
3 Applications
The Voronoi-assisted CNN reconstructs global fields from sparse, arbitrary sensor configurations across wake flow, sea-surface temperature, and turbulent channel-flow applications. Results show detailed recovery, generalization to unseen sensor placements and counts, and sensitivity to noise and training objectives.
- Two-dimensional cylinder wake: The method accurately reconstructs vortices and shear layers in a two-dimensional cylinder wake from sparse vorticity measurements.With 16 sensors, reconstruction error is reduced by half relative to 8 sensors.
- NOAA sea-surface temperature: For NOAA sea-surface temperature, the model recovers fine-grained global structures more accurately than linear and cubic interpolation.Unlike interpolation, it reconstructs the whole field beyond the sensors’ convex hull.
- NOAA sea-surface temperature: A single model successfully handles trained and unseen sensor placements, moving sensors, and sensor counts of 70 and 200 in sea-surface-temperature reconstruction.For unseen sensor counts up to 200, the L2 error remains below 0.1.
- Turbulent channel flow: In turbulent channel flow, finer velocity-fluctuation features are reconstructed from 200 sensors, representing 3.26% of the field grid points, with reasonable recovery at unseen counts of 150 and 250.Unseen placements have higher error than trained placements but show similar trends.
- Turbulent channel flow: The reconstructed turbulent channel-flow RMS profile captures the near-wall peak, although it underestimates the reference DNS curve.The authors attribute the underestimation to fluctuation inputs and L2-minimization training, which favors values near the training-data average.
- Turbulent channel flow: For noisy turbulent-flow inputs, noise added directly to sensor measurements has a larger influence than noise added after Voronoi tessellation preparation.The difference is attributed to the CNN learning the relationship between Voronoi images and high-resolution flow fields.
4 Discussion
The paper presents Voronoi tessellation combined with CNNs as a flexible framework for reconstructing global fields from arbitrary sensor collections. Demonstrations support accurate and robust recovery while motivating extensions for physical constraints, generalization, and adaptive sensing.
- 4 Discussion: The method combines Voronoi tessellation with CNNs to convert arbitrary sensor measurements and locations into a uniform-grid representation for global reconstruction.Mask images encode sensor locations, while Voronoi images represent measurements.
- 4 Discussion: Three global flow-reconstruction examples demonstrated the method’s accuracy and robustness from local sensor measurements.
- 4 Discussion: Adaptive Voronoi discretization changes only near altered sensor arrangements, providing computational savings and an opportunity for spatially adaptive techniques.
- 4 Discussion: The framework can be extended to enforce physical constraints, improve robustness across physical parameters, and support more generalizable turbulent-data models.Suggested directions include divergence-free constraints, dedicated training datasets, and unsupervised learning.
- 4 Discussion: Changing sensor numbers or positions can be accommodated at each time step, enabling extensions to satellite measurements, particle image velocimetry, intelligent placement, and turbulence statistics.