Real-Time Sensing of Inaccessible Physical Fields via an Edge-Deployable Hardware-Portable Graph Neural Operator

William Howes; Jason Yoo; Kazuma Kobayashi; Subhankar Sarkar; Farid Ahmed; Souvik Chakraborty; Syed Bahauddin Alam

cs.LGApr 2, 2026

Real-Time Sensing of Inaccessible Physical Fields via an Edge-Deployable Hardware-Portable Graph Neural Operator

Authors: William Howes, Jason Yoo, Kazuma Kobayashi, Subhankar Sarkar, Farid Ahmed, Souvik Chakraborty, Syed Bahauddin Alam

Organizations: Grainger College of Engineering, Nuclear, Plasma & Radiological Engineering Department, University of Illinois Urbana-Champaign, Urbana, IL, USA · Yardi School of Artificial Intelligence, Indian Institute of Technology Delhi · Department of Applied Mechanics, Indian Institute of Technology Delhi, New Delhi, India · National Center for Supercomputing Applications, Urbana, IL, USA

Abstract

Real-time inference of inaccessible interior physical fields from sparse boundary observations is a fundamental but unresolved problem in scientific machine learning, with direct relevance to safety-critical monitoring across many engineering applications. Existing neural operators achieve high accuracy but leave deployment to embedded edge platforms unaddressed. Here we introduce VIRSO (Virtual Irregular Real-Time Sparse Operator), the first neural operator with a unique spatial-spectral architecture that explicitly addresses edge-deployment hardware. VIRSO learns a nonlinear mapping from sparse, geometrically disjoint boundary inputs to spatially continuous interior multiphysics fields on irregular unstructured meshes through a spectral-spatial decomposition explicitly aligned with hardware execution: a compute-bound graph spectral pathway and a memory-bandwidth-bound spatial-aggregation pathway, each independently characterized on datacenter and embedded accelerators. The design reduces the inference energy-delay product by 29 $\times$ relative to the vanilla graph-operator baseline (206 J $\cdot$ ms $\to$ 7.0 J $\cdot$ ms on an NVIDIA H200) and enables 17.0 samples/s embedded inference on an NVIDIA Jetson Orin Nano within 7.06 W board-level power, without modification. A mesh-density-adaptive graph construction strategy (V-KNN) simultaneously improves accuracy and reduces graph edge count by 34%. Across three benchmarks with reconstruction ratios from 47:1 to 156:1, VIRSO achieves mean relative $L_2$ errors below 1% with fewer parameters than operator baselines and delivers an inference speedup of $\approx 10^4$ times over the high-fidelity reference solver. To our knowledge, this is the first demonstration of a single-digit-watt neural operator, establishing hardware co-design as a missing ingredient in operator-based inference and a tractable path to real-time deployment.

Figures & tables

Figure 1: Multi-Physics Virtual Sensing Framework for VIRSO . (a) Training and validation for the VIRSO model originates from mesh-based numerical solvers. The boundary conditions at the geometry inlet and a discretization of the forcing profile (or signal for transient) are given as initial input to our framework along with the generated mesh which is converted to positional coordinates of each mesh cell throughout the domain of the desired multi-physics output. (b) The mesh cell coordinates become our graph nodes

\mathbb{V}

and utilizing graph creation methods such as K-Nearest Neighbors (KNN), Radius, or a custom Variable KNN (V-KNN), we can define crucial edge connections

\mathbb{E}

between node points. The input signal and values are mapped to a latent embedding space and then copied to each node and combined with node coordinates to define the initial node features

\mathbb{X}

. (c) The VIRSO model projects the input into a hidden function dimension and repeatedly extracts global and local features through spectral-spatial blocks. Eventually the operator downlifts to the final output space. (d) A close up into the VIRSO block architecture. The GFT allows for the spectral convolution of the first

m

modes, and the spatial aggregation GNN provides local feature extraction. The collaboration projection

f

prevents over-smoothing while skip connections assist training and also allow local information flow. (e) Defined on the mesh-node grid, we have multiple predicted field outputs that describe physics within our domain such as pressure, velocity, and turbulent kinetic energy (TKE).

Figure 2: Lid-Driven Cavity: Problem Formulation and Results. (a) Boundary conditions: no-slip at all walls except the driven lid at $y=1$ , where a time-varying forcing signal $V(t)$ prescribes the horizontal velocity. (b) Representative forcing signals $V(t)$ evaluated at 90 discrete time steps, spanning a broad range of transient dynamics. (c) Percentile error distributions from VIRSO for pressure, velocity magnitude, TKE, and overall mean error (blue, orange, green, red). All distributions are centered below $1\%$ with low spread, confirming consistent generalization across the test distribution. (d) 50th-percentile reconstruction: predicted fields, ground truth, and absolute error for pressure, velocity magnitude, and TKE. VIRSO accurately recovers the primary vortex, shear layer, and corner eddies. Elevated absolute error near boundaries and corners reflects the sensitivity of turbulent physics to regions underresolved by the uniform output grid.

Model	Parameters	Mean Relative $L_{2}$ Error (%)
Model	Parameters	$p(x,y)$	$\\|\textbf{v}(x,y)\\|$	$k(x,y)$	Mean
NOMAD	1.63 M	0.65	1.18	0.74	0.86
Geo-FNO	1.68 M	0.94	1.08	1.32	1.11
GNO	0.23 M	2.84	10.89	22.23	11.98
VIRSO	1.62 M	0.62	0.95	0.71	0.76

Table 1: Model Comparison: Lid-Driven Cavity

Figure 3: PWR Subchannel: Problem Formulation and Results. (a) Geometry: coolant flow between four reactor rods, analyzed at a fixed axial cross-section. No-slip boundary conditions are enforced on rod surfaces; inlet is at

z=0

. (b) Multimodal boundary inputs: axial heat source profile

q(z)

at 100 positions and two scalar inlet conditions. These inputs are defined on the axial direction, while outputs are defined on the transverse cross-section — a geometrically disjoint cross-domain configuration. (c) Percentile error distributions (orange: temperature, blue: velocity, green: TKE, red: mean). All distributions centered below

1\%

with narrow spread. (d) 50th-percentile reconstruction. VIRSO accurately captures wall shear layers, central velocity structure, and symmetric thermal gradient. Weak spatial correlation of error with boundary proximity for velocity and TKE; stochastic fluctuations for temperature.

Model	Parameters	Mean Relative $L_{2}$ Error (%)
Model	Parameters	$\\|\boldsymbol{v}(x,y)\\|$	$T(x,y)$	$k(x,y)$	Mean
NOMAD	0.42 M	0.39	0.27	0.96	0.54
Geo-FNO	2.70 M	1.55	1.06	2.88	1.83
GNO	0.27 M	4.99	0.50	3.47	2.99
VIRSO	0.34 M	0.37	0.28	0.88	0.51

Table 2: Model Comparison: PWR Subchannel

Figure 4: Heat Exchanger: Problem Formulation and Results. (a) Geometry: wavy tape insert inside a dimpled cylindrical channel. Analysis performed on the 2D axial cross-section at

x=0.789

. (b) Boundary inputs: axial heat profile and two scalar inlet conditions, geometrically disjoint from the output cross-section. (c) Percentile error distributions for the 10-layer VIRSO across all output fields. Distributions are centered below

1\%

with narrow spread. (d) 50th-percentile reconstruction. VIRSO accurately recovers the primary counter-rotating recirculation cells, secondary eddies, and pressure response to the insert gap. Elevated error is concentrated near the insert junction and channel wall, regions of maximum geometric complexity.

Model	Parameters	Mean Relative $L_{2}$ Error (%)
Model	Parameters	$p$	$u_{z}$	$u_{y}$	$u_{x}$	$\\|u\\|$	Mean
NOMAD	2.69 M	0.66	1.01	1.03	0.74	1.43	0.97
Geo-FNO	2.70 M	0.74	1.14	1.14	0.91	1.51	1.09
GNO	0.27 M	3.37	9.89	10.75	11.15	11.82	9.40
VIRSO ∗	1.66 M	0.47	0.87	0.86	0.71	1.24	0.83
VIRSO	2.31 M	0.40	0.71	0.66	0.52	1.18	0.70

Table 3: Model Comparison: Heat Exchanger

Figure 5: Graph Topology and Node Degree Distribution. (a) Mesh geometry of the heat exchanger, adapted to the VIRSO graph node configuration. Higher mesh density is visible near the insert junction and channel walls, corresponding to regions of steepest physical gradients. (b,c) Degree distribution for KNN with $k=30$ and $k=45$ : modest variation with slightly elevated connectivity near high-density regions, but insufficient targeting. (d) KNN with $k=93$ : high-density regions receive elevated degree, improving performance, but the enforced minimum of 93 throughout the interior introduces significant inefficiency. (e) Radius with $r=0.001$ : naturally concentrates high degree near geometrically complex regions; outperforms KNN with $k=30,45$ . (f) Radius with $r=0.0015$ : matches the maximum degree of KNN with $k=93$ near walls and junctions but maintains a minimum of only 24 in the interior, achieving superior performance with fewer edges. (g) Variable-KNN (V-KNN): explicitly assigns connectivity proportional to local mesh density. Achieves the same maximum degree as (d) and (f) near geometric complexity, maintains minimum degree 45 in the interior, and uses only 270K edges — the most efficient configuration achieving the best overall accuracy.

Model	Param.	FLOPs	GPU%	Mem%	Mem (MiB)	Pwr (W)	Lat. (ms)	Energy (J/it)
NOMAD	2.69 M	11.29 G	30.92	1.12	999.92	196.03	2.35	0.41
Geo-FNO	2.70 M	1.58 G	23.63	2.02	1059.39	139.39	4.94	0.59
GNO	0.27 M	429.49 G	86.78	36.15	1965.99	572.00	20.48	10.07
VIRSO	1.66 M	2.03 G	42.35	5.71	1039.98	193.35	7.77	1.30
VIRSO (Spatial)	0.11 M	0.99 G	32.77	6.95	1031.65	178.15	8.93	1.40
VIRSO (Spectral)	1.58 M	0.98 G	17.34	$<$ 1	937.93	124.41	8.18	0.86

Table 4: Inference Energy and Memory: Heat Exchanger

Model	Mean $L_{2}$ (%)	FLOPs	Energy (J/it)	Lat. (ms)	EDP (J $\cdot$ ms)
GNO	9.40	429.49 G	10.07	20.48	206.2
Geo-FNO (full, 2.70 M)	1.09	1.58 G	0.59	4.94	2.91
Geo-FNO (light, 0.26 M)	4.21	1.16 G	0.56	5.10	2.86
NOMAD (full, 2.69 M)	0.97	11.29 G	0.41	2.35	0.96
NOMAD (light, 0.26 M)	8.24	1.28 G	0.23	2.00	0.46
VIRSO (2-layer, 0.26 M)	1.95	0.61 G	0.54	4.29	2.32

Table 5: Accuracy–Efficiency Summary: Heat Exchanger (H200, device-level GPU measurements)

Model	Param.	FLOPs	Avg. Lat. (ms/it)	Avg. Pwr (W)	Energy (J/it)	Peak RAM (GB)	Mean L2 (%)
10-L (Full)	1.66 M	2.04 G	562.92	7.58	4.72	5.16	0.84
10-L (Spec.)	1.58 M	0.98 G	58.77	7.06	0.84	5.14	0.91
2-L	0.26 M	0.61 G	104.03	7.51	1.25	5.32	1.96

Table 6: Embedded inference performance of pretrained 10-layer (full and spectral-only) and the 2-layer VIRSO model.

\mathbf{v}_{t+1}^{spatial}(\mathcal{Y}_{n})=\mathbf{\Gamma}\odot\mathbf{A}\mathbf{v}_{t}\mathbf{W},\quad\mathbf{\Gamma}=[\gamma_{uv}],\mathbf{W}\in\mathbb{R}^{d_{v}\times d_{v}}

Algorithm 1 VIRSO Kernel Convolution Approximation

Model	Best	25th	50th	75th	95th	Worst
NOMAD	0.47%	0.56%	0.69%	0.94%	1.80%	7.78%
Geo-FNO	0.60%	0.89%	1.00%	1.15%	1.69%	6.86%
GNO	10.29%	10.82%	11.51%	12.57%	15.35%	25.05%
VIRSO	0.38%	0.48%	0.58%	0.84%	1.75%	6.31%

Table 7: Lid-Driven Cavity: Percentile Error Statistics (Mean Relative

L_{2}

Error)

Model	Best	25th	50th	75th	95th	Worst
NOMAD	0.28%	0.30%	0.31%	0.32%	0.40%	17.70%
Geo-FNO	1.17%	1.35%	1.53%	1.92%	2.56%	17.32%
GNO	2.76%	2.78%	2.80%	2.83%	2.89%	18.09%
VIRSO	0.25%	0.26%	0.28%	0.29%	0.34%	17.72%

Table 8: PWR Subchannel: Percentile Error Statistics

Model	Best	25th	50th	75th	95th	Worst
NOMAD	0.93%	0.93%	0.95%	0.99%	1.14%	1.20%
Geo-FNO	1.04%	1.05%	1.07%	1.11%	1.18%	1.20%
GNO	9.31%	9.33%	9.37%	9.45%	9.57%	9.61%
VIRSO ∗	0.80%	0.80%	0.81%	0.84%	0.92%	0.96%
VIRSO	0.66%	0.67%	0.69%	0.71%	0.77%	0.80%

Table 9: Heat Exchanger: Percentile Error Statistics

Configuration	$p$	$u_{z}$	$u_{y}$	$u_{x}$	$\\|u\\|$	Mean
No Embed, No Skip	0.98	5.83	5.84	4.58	3.57	4.16
Embed Only	0.70	1.01	0.94	0.65	1.34	0.93
Skip Only	0.58	1.41	1.57	1.45	1.60	1.32
Embed and Skip	0.47	0.87	0.86	0.71	1.24	0.83

Table 10: Component Ablation: Heat Exchanger (10-layer VIRSO)

Model	Parameters	Mean Relative $L_{2}$ Error (%)
Model	Parameters	$p$	$u_{z}$	$u_{y}$	$u_{x}$	$\\|u\\|$	Mean
Spatial only	0.11 M	3.62	18.71	20.54	20.71	17.70	16.27
Spectral only	1.58 M	0.55	1.00	0.93	0.72	1.29	0.90
Spectral only (no skip)	1.58 M	1.21	6.81	7.81	7.03	5.94	5.76
Spectral + Spatial	1.66 M	0.47	0.87	0.86	0.71	1.24	0.83

Table 11: Spectral-Spatial Component Analysis: Heat Exchanger

Graph	Edges	Min Degree	Max Degree	Mean Error
KNN, $k=30$	135K	30	53	0.83%
KNN, $k=45$	201K	45	75	0.79%
KNN, $k=93$	408K	93	134	0.67%
Radius, $r=0.001$	150K	7	77	0.71%
Radius, $r=0.0015$	292K	24	134	0.62%

Table 12: Graph Construction Comparison: Heat Exchanger (10-layer VIRSO)

Parameters	$p$	$u_{z}$	$u_{y}$	$u_{x}$	$\\|u\\|$	Mean
2.31 M	0.39	0.47	0.54	0.36	1.12	0.58
1.66 M	0.41	0.47	0.54	0.38	1.13	0.59

Table 13: V-KNN Graph Results: Heat Exchanger

Model	Parameters	Mean Relative $L_{2}$ Error (%)
Model	Parameters	$p$	$u_{z}$	$u_{y}$	$u_{x}$	$\\|u\\|$	Mean
Spatial only	0.11 M	10.12	31.58	24.02	25.82	29.26	24.16
Spectral only	1.58 M	0.50	0.66	0.71	0.50	1.20	0.71
Spectral only (no skip)	1.58 M	1.46	8.04	8.78	8.54	7.34	6.83
Spectral + Spatial	1.66 M	0.41	0.47	0.54	0.38	1.13	0.59

Table 14: Spectral-Spatial Analysis with V-KNN Graph: Heat Exchanger

Model	Param.	Mean Relative $L_{2}$ Error (%)
Model	Param.	$p$	$u_{z}$	$u_{y}$	$u_{x}$	$\\|u\\|$	Mean
NOMAD	0.26 M	4.61	11.47	9.72	6.78	8.60	8.24
Geo-FNO	0.26 M	2.58	5.54	5.46	3.21	4.26	4.21
GNO	0.27 M	3.37	9.89	10.75	11.15	11.82	9.40
VIRSO (2L)	0.26 M	1.36	2.00	2.14	1.73	2.53	1.95
VIRSO (14L, small) ∗	0.18 M	1.39	2.09	1.94	1.33	2.14	1.78

Table 15: Lightweight Model Comparison: Heat Exchanger

Model	Param.	GPU%	Mem%	Mem (MiB)	Pwr (W)	Lat. (ms)	Energy (J/it)
NOMAD	0.26 M	15.78	$<$ 1	949.95	131.74	2.00	0.23
Geo-FNO	0.26 M	19.63	1.21	1017.31	129.27	5.10	0.56
GNO	0.27 M	86.78	36.15	1965.99	572.00	20.48	10.07
VIRSO (2L)	0.26 M	26.01	2.49	1033.34	146.35	4.29	0.54
VIRSO (14L, small)	0.18 M	29.92	2.69	999.99	156.31	13.74	1.81

Table 16: Lightweight Model Inference Energy: Heat Exchanger

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Apr 17, 2026J/K move · Enter open · S save

Neuroscience Inspired Graph Operators Towards Edge-Deployable Virtual Sensing for Irregular Geometries

William Howes, Farid Ahmed, Kazuma Kobayashi, Souvik Chakraborty+1

Grainger College of Engineering, Nuclear, Plasma & Radiological Engineering Department, University of Illinois Urbana-Champaign, Urbana, IL, USA · Department of Applied Mechanics, Indian Institute of Technology Delhi, New Delhi, India · Yardi School of Artificial Intelligence, Indian Institute of Technology Delhi+1

Edge Computing Graph Neural Operators Spiking Neural Networks Neural Operators Virtual Sensing

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Predicting full-field physics through the real-time virtual sensing of engineering systems can enhance limited physical sensors but often requires sparse-to-dense reconstruction, complex multiphysics, and highly irregular geometries as well as strict latency and energy constraints for edge-deployability. Neural operators have been presented as a potential candidate for such applications but few architectures exist that explicitly address power consumption. Spiking neuron integration can provide a potential solution when integrated on neuromorphic hardware but the current existing neuron models result in severe performance degradation towards regression-based virtual sensing. To address the performance concerns and edge-constraints, we present the Variable Spiking Graph Neural Operator (VS-GNO) which integrates a sophisticated spectral-spatial convolutional analysis and a previously developed Variable Spiking Neuron (VSN) and energy-error balance loss function. With a non-spiking

L_2

error baseline of

0.4\%

, VS-GNO can provide a reconstruction error of

0.71\%

with

15\%

average spiking in its spectral-only form and

1.04\%

with

24.5\%

spiking in its entire form. These results position VS-GNO as a promising step towards energy-efficient, edge-deployable neural operators for real-time sparse-to-dense virtual sensing in complex, highly irregular engineering environments.

Abstract

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