Flow Field Reconstruction via Voronoi-Enhanced Physics-Informed Neural Networks with End-to-End Sensor Placement Optimization

Renjie Xiao; Bingteng Sun; Yiling Chen; Lin Lu; Qiang Du; Junqiang Zhu

physics.flu-dynMar 10, 2026

Flow Field Reconstruction via Voronoi-Enhanced Physics-Informed Neural Networks with End-to-End Sensor Placement Optimization

Authors: Renjie Xiao, Bingteng Sun, Yiling Chen, Lin Lu, Qiang Du, Junqiang Zhu

Organizations: Advanced Gas Turbine Laboratory, Institute of Engineering Thermophysics, Chinese Academy of Sciences, Beijing, 100190, China · National Key Laboratory of Science and Technology on Advanced Light-duty Gas-turbine, Beijing, 100190, China · School of Computer Science and Technology Shandong University Qingdao Shandong 266237 China · University of Chinese Academy of Sciences, Beijing, 100190, China ·

Abstract

(short version abstract, full in article)High-fidelity flow field reconstruction is important in fluid dynamics, but it is challenged by sparse and spatiotemporally incomplete sensor measurements, as well as failures of pre-deployed measurement points that can invalidate pre-trained reconstruction models. Physics-informed neural networks (PINNs) alleviate dependence on large labeled datasets by incorporating governing physics, yet sensor placement optimization, a key factor in reconstruction accuracy and robustness, remains underexplored. In this study, we propose a PINN with Voronoi-enhanced Sensor Optimization (VSOPINN). VSOPINN enables differentiable soft Voronoi construction for sparse sensor data rasterization, end-to-end fusion of centroidal Voronoi tessellation (CVT) with PINNs for adaptive sensor placement, and unified layout optimization for multi-condition flow reconstruction through a shared encoder-multi-decoder architecture. We validate VSOPINN on three representative problems: lid-driven cavity flow, vascular flow, and annular rotating flow. Results show that VSOPINN significantly improves reconstruction accuracy across different Reynolds numbers, adaptively learns effective sensor layouts, and remains robust under partial sensor failure. The study clarifies the intrinsic relationship between sensor placement and reconstruction precision in PINN-based flow field reconstruction.

Figures & tables

Figure 1: Diagram of the VSOPINN model. (a) Encoder/Decoder modules, (b) Attention-enhanced convolutional decoder. (c) Finite difference convolution kernel.

Figure 2: Single-encoder/multi-decoder architecture of the multi-VSOPINN framework.

Problems	Depth	Width	Kernel Size ( $K$ )
Section 3.1 (Cavity)	5	32, 64	5
Section 3.2 (Vascular)	8	16, 32, 64	5
Section 3.3 (Annulus)	8	16, 32, 64	5
Section 3.4 (Multi-cond)	5	32, 64	5

Table 1: The network architecture configurations for the proposed VSOPINN framework in each case study, including depth, channel width, and kernel size.

Figure 3: Reconstructed velocity magnitude and pressure fields for the lid-driven cavity flow ( $Re=100$ ). Compared with the CFD reference, the CVT-optimized sensor locations in VSOPINN yield more accurate resolutions of the primary vortex structures and corner singularities.

Method Configuration	Rel. $L^{2}$ Error
Method Configuration	$\bm{u}$	$\bm{v}$	$\|\mathbf{v}\|$
PhyGeoNet (Origin)	4.426e-01	8.602e-01	4.885e-01
PhyGeoNet + Data	2.988e-01	6.665e-01	3.660e-01
PhyGeoNet + Data + Voronoi	2.140e-01	4.884e-01	2.663e-01
VSOPINN (Sensor Opt)	1.128e-01	2.745e-01	1.366e-01
VSOPINN (with Attention)	9.826e-02	2.500e-01	1.252e-01

Table 2: Ablation study of relative

L^{2}

errors for the lid-driven cavity flow (

Re=100

) with

N_{s}=4

sensors. The comparison includes velocity components (

u,v

) and velocity magnitude (

|\mathbf{v}|

). The integration of CVT optimization yields the most significant performance gain.

Figure 4: Soft-Voronoi images of the velocity magnitude $|\mathbf{v}|$ for the lid-driven cavity flow ( $Re=100$ ), constructed using the best-performing sensor subsets at different numbers of active sensors: $k=4$ ( $\{S_{1},S_{2},S_{3},S_{4}\}$ ), $k=3$ ( $\{S_{1},S_{2},S_{4}\}$ ), $k=2$ ( $\{S_{2},S_{4}\}$ ), and $k=1$ ( $\{S_{4}\}$ ), as highlighted in Table 3 . The rasterization follows the differentiable soft assignment in Eq. ( 8 ), providing an image-like representation of sparse pointwise measurements on the reference grid.

$k$	Active sensors	$\epsilon_{u}$	$\epsilon_{v}$	$\epsilon_{\|\mathbf{v}\|}$
4	$\{S_{1},S_{2},S_{3},S_{4}\}$	4.707e-02	6.828e-02	4.214e-02
3	$\{S_{1},S_{2},S_{4}\}$	6.843e-02	1.154e-01	5.526e-02
3	$\{S_{2},S_{3},S_{4}\}$	1.028e-01	8.850e-02	8.032e-02
3	$\{S_{1},S_{2},S_{3}\}$	1.004e-01	1.218e-01	8.661e-02
3	$\{S_{1},S_{3},S_{4}\}$	1.011e-01	2.382e-01	1.481e-01
2	$\{S_{2},S_{4}\}$	1.006e-01	1.341e-01	8.141e-02

Table 3: Robustness test for the lid-driven cavity flow (

Re=100

) under sensor failures. The VSOPINN(with CVT) model is fixed after training with the CVT-optimized four-sensor set

\{S_{1},S_{2},S_{3},S_{4}\}

, and then evaluated using all sensor subsets via a single forward pass.

Method	Rel. $L^{2}$ Error ( $\|\mathbf{v}\|$ )
PhyGeoNet (Origin)	1.192e-01
PhyGeoNet + Data	1.188e-01
PhyGeoNet + Data + Voronoi	1.099e-01
VSOPINN (Sensor Opt)	1.031e-01
VSOPINN (with CVT)	9.678e-02

Table 4: Quantitative comparison of the relative

L^{2}

error of velocity magnitude (

|\mathbf{v}|

) for vascular flow reconstruction at

Re=450

(

N_{s}=4

Figure 5: Flow field and absolute error distributions for the vascular flow ( $Re=450$ ). VSOPINN exhibits superior accuracy over the baseline reference, particularly in capturing flow features near the bifurcation and unstructured wall boundaries.

Method Configuration	Rel. $L^{2}$ Error
Method Configuration	$\bm{u}$	$\bm{v}$	$\|\mathbf{v}\|$
PhyGeoNet (Origin)	6.698e-01	7.664e-01	6.396e-01
PhyGeoNet + Data	2.048e-01	2.202e-01	2.089e-01
PhyGeoNet + Data + Voronoi	1.255e-01	1.509e-01	1.319e-01
VSOPINN (Sensor Opt)	9.921e-02	1.151e-01	1.017e-01
VSOPINN (with CVT)	6.808e-02	7.869e-02	6.885e-02

Table 5: Relative

L^{2}

errors for the annulus driving flow under high-speed rotation (

\Omega=14

) with optimized sensor placement.

Figure 6: Reconstructed flow fields for the annulus problem ( $\Omega=14$ ). Comparisons among the five network models and the ground truth highlight their respective capabilities in resolving nonlinear velocity distributions and steep shear stress gradients near the high-speed rotating inner boundary.

Figure 7: Radial velocity magnitude profiles at $\theta=90^{\circ}$ , demonstrating the varying accuracy of the models in capturing the steep velocity gradients near the boundaries.

Figure 8: Visualization of the sensor placement strategy. The background displays the velocity magnitude field at $Re=600$ (interpolation case). Cyan dots represent the 40 candidate sensor positions generated from 10 independent training sessions, which tend to cluster in high-velocity regions. Red stars indicate the final Optimal Layout derived via k-means clustering, partitioned by the Voronoi tessellation (white dashed lines).

Sensor Layout	Interpolation ( $Re=600$ )				Extrapolation ( $Re=1000$ )
Sensor Layout	$\epsilon_{u}$	$\epsilon_{v}$	$\epsilon_{\|\mathbf{v}\|}$	$\epsilon_{p}$	$\epsilon_{u}$	$\epsilon_{v}$	$\epsilon_{\|\mathbf{v}\|}$	$\epsilon_{p}$
Random (Fixed)	1.729e-01	2.030e-01	1.650e-01	3.221e-01	2.417e-01	3.331e-01	2.621e-01	3.433e-01
Optimal (k-means)	1.514e-01	1.434e-01	1.313e-01	2.842e-01	2.226e-01	2.186e-01	2.034e-01	2.880e-01

Table 6: Quantitative comparison of relative

L^{2}

errors for flow reconstruction under interpolation (

Re=600

) and extrapolation (

Re=1000

) conditions. The comparison encompasses velocity components (

u,v

), velocity magnitude (

|\mathbf{v}|

), and pressure (

p

Figure 9: Comparative visualization of flow field reconstruction across interpolation ( $Re=600$ ) and extrapolation ( $Re=1000$ ) regimes. Rows (a) and (d) show the Ground Truth. The Optimal Layout (Rows c, f) significantly mitigates local errors compared to the Random Layout (Rows b, e), particularly in the extrapolation regime. Subplots (g) and (h) show the centerline velocity profiles ( $u$ at $x=0.5$ ), confirming the superior alignment of the proposed method with the reference solution.

Computational Fluid Dynamics

Computational fluid dynamics numerically models fluid motion, turbulence, heat and mass transport, and fluid–structure interactions.

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Observation-Aligned Two-Stage Domain Decomposition for Physics-Informed Traffic State Estimation with Sparse Fixed Sensors

Eunhan Ka, Ludovic Leclercq, Satish V. Ukkusuri

Lyles School of Civil and Construction Engineering, Purdue University, 610 Purdue Mall, West Lafayette, 47907, IN, United States · Univ Gustave Eiffel, ENTPE, LICIT-ECO7, Lyon, F-69675, France

Traffic Flow Estimation Domain Decomposition Methods Physics-Informed ML

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Traffic state estimation from sparse fixed sensors is challenging because physics-informed neural networks (PINNs) tend to over-smooth sharp transitions admitted by the Lighthill-Whitham--Richards (LWR) model. This study proposes Two-Stage Domain Decomposition Physics-Informed Neural Networks (TSDD-PINN), an observation-aligned framework for LWR-based offline speed-field reconstruction. The framework supports spatial, temporal, and space--time refinement. Matched direction analysis shows that spatial refinement has the lowest mean error and less than half the training time of space--time refinement in the tested setting, while temporal refinement is faster. A global parent PINN is first trained. In the controlled spatial implementation, its residual profile guides a deterministic partition for warm-started child networks. An optional operational safeguard retains Stage~1 when the prespecified screen does not activate. The primary I-24 MOTION evaluation spans five days, five sensor configurations, and ten seeds per configuration, yielding 1{,}500 runs. Controlled TSDD-PINN attains the lowest relative

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Abstract

Computational Fluid Dynamics

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