Data-Driven or Model-Constrained? Comparing Airflow Estimation Methods under Sparse Sensing

Dominik Aigner; Thomas Wiedemann; Javier Monroy; Achim J. Lilienthal

cs.ROOct 29, 2026

Data-Driven or Model-Constrained? Comparing Airflow Estimation Methods under Sparse Sensing

Authors: Dominik Aigner, Thomas Wiedemann, Javier Monroy, Achim J. Lilienthal

Organizations: Technical University of Munich · German Aerospace Center · University of Málaga

Abstract

Airflow-driven advection is the primary mechanism for gas and aerosol dispersion in real-world scenarios, making accurate airflow estimation essential for ventilation monitoring, gas source localization, and contaminant spread analysis. Since obtaining airflow maps via high-resolution spatial sensing is usually infeasible in practice, flow field estimation methods must cope with sparse measurements by introducing statistical or physical priors to compensate for limited data. Existing approaches mainly differ in how they incorporate fluid-flow constraints and environmental geometry. However, due to the lack of systematic comparisons across this spectrum of methods, the potential benefits of physics-based priors for airflow estimation under sparse sensing remain underexplored. To address this question, we quantitatively evaluate four airflow estimators across three simulated 2D scenarios under steady-state conditions and varying sampling densities. The methods range from Gaussian Process Regression and graph-based Gaussian Markov Random Fields to two Navier-Stokes-based approaches, including a novel strong-form residual minimization variant. Our results show that physical priors provide the greatest benefit at low sampling densities by compensating for missing spatial information. Although data-driven Gaussian Process Regression becomes increasingly competitive as sensor coverage increases, model-based estimators retain an accuracy advantage across the evaluated scenarios.

Figures & tables

Fig. 1: Illustration of a dense, steady-state airflow field estimation (right) from sparse velocity samples (left) employing different priors.

Fig. 2: Combined estimation errors for increasing numbers of measurements. Solid lines denote magnitude error, dashed lines denote angular error. All results are averaged over 10 random sample sets and three different scenarios; uncertainty bands are omitted for readability.

$K$	GPR	GMRF	SFNS	WFNS
50	$1.9\pm 1.1$	$67.9\pm 37.7$	$341.6\pm 66.0$	$1030.9\pm 285.1$
100	$3.6\pm 2.2$	$60.4\pm 18.7$	$331.6\pm 83.5$	$975.0\pm 246.9$
200	$8.6\pm 5.1$	$51.8\pm 14.8$	$323.0\pm 98.9$	$903.9\pm 251.1$

TABLE I: Mean Estimator Runtimes in Milliseconds

Fig. 3: SFNS estimates for low sampling densities, with ground truth shown for comparison. Stars indicate random measurement locations. The example shows that even low sampling can recover the main flow-field structure.

Physics-Informed ML

Physics-informed machine learning combines data-driven models with physical laws to solve, predict, and infer scientific systems.

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May 15, 2026

FLUIDSPLAT: Reconstructing Physical Fields from Sparse Sensors via Gaussian Primitives

Reconstructing continuous flow fields from sparse surface-mounted sensors is central to aerodynamic design, flow control, and digital-twin instrumentation. Existing neural methods for this task typically encode sensor readings into implicit latent codes with little spatial interpretability and limited formal guidance on how representational capacity should scale with observation count. Inspired by 3D Gaussian Splatting, we introduce FLUIDSPLAT, a sensor-conditioned model that predicts K anisotropic Gaussian primitives forming a partition-of-unity scaffold, a spatially explicit and interpretable intermediate representation of the flow. For an idealized Gaussian primitive estimator, we prove an

O(K^{-s/d})

approximation rate for fields with Sobolev smoothness

s

; incorporating

N

noisy observations yields a squared-risk decomposition with bias

O(K^{-2s/d})

and variance

O(σ^{2}K/N)

.Balancing the two yields

K^{*}\!\sim\!(N/σ^{2})^{d/(2s+d)}

: primitive count cannot grow freely under sparse sensing, revealing a variance bottleneck that motivates complementing the scaffold with a state-conditioned residual decoder. Across four benchmarks spanning 2D and 3D, FLUIDSPLAT achieves 11-28% error reduction over several strong baselines on cylinder flow, AirfRANS, FlowBench LDC-3D, and PhySense-Car 3D benchmarks.

Huaxi Huang, Meng Li, Zhengqing Gao +3

Shanghai Artificial Intelligence Laboratory · The Hong Kong University of Science and Technology · Mohamed bin Zayed University of Artificial Intelligence +1

Neural Surrogate Modeling Gaussian Splatting Computational Fluid Dynamics

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Jun 22, 2026

Simulation-Free Estimation of Traffic Flows from Sparse Count Data

We propose a method for estimating time-varying traffic flow patterns from sparse aggregated vehicle counts. The method partitions the study area into spatial regions, constructs a set of feasible region-to-region routes, and solves a weighted least-squares optimization problem to determine the number of vehicles to allocate on each route. A weighted contribution matrix encodes sensor coverage, steering the optimizer toward flow configurations that are directly observable by sensors. Edge-level trajectories are then derived by scoring candidate routes against the temporal and volumetric profiles of aggregated regional sensor counts. The method is evaluated on the Brussels road network using real and synthetic traffic data. Results show that the proposed approach reproduces the daily traffic profile in the input data and outperforms the baseline methods at a fraction of the computational cost.

Davide Guastella, Gianluca Bontempi

Aix Marseille University, CNRS, LIS, Marseille, France · Université Libre de Bruxelles, Machine Learning Group, Brussels, Belgium · WEL Research Institute, Wavre, Belgium

Intelligent Transportation Systems Traffic Flow Estimation

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May 30, 2025

Diff-SPORT: Diffusion-based Sensor Placement Optimization and Reconstruction of Turbulent flows in urban environments

Rapid urbanization demands efficient monitoring of turbulent wind and pollutant dispersion, yet existing reconstruction and sensor placement strategies fail under realistic sparsity constraints. Here, we introduce Diff--SPORT, a diffusion-based framework that combines a generative diffusion prior with maximum a posteriori inference and Shapley-value attribution for high-fidelity flow reconstruction and optimal sensor placement. By training a diffusion prior model once over a domain, Diff--SPORT enables non-linear optimal sensor placement and near-real-time flow reconstruction from sparse measurements orders of magnitude faster than RANS or LES simulations, consistently outperforming state-of-the-art methods. The framework also extends, without algorithmic modification, to experimental passive scalar concentration dataset, a direct proxy for pollutant dispersion, measured in a 1:2400 scale water-flume model of the Beijing Haidian neighbourhood under realistic urban flow conditions. Shapley-guided sensor placement achieves up to 57% lower reconstruction error than randomly placed sensors at extreme sparsity, identifying compact and physically interpretable configurations. These results establish Diff--SPORT as a modular foundation offering a zero-shot alternative to retraining-intensive downstream strategies, supporting scalable urban flow monitoring for air quality management and resilient city design.

Abhijeet Vishwasrao, Sai Bharath Chandra Gutha, Andres Cremades +8

Department of Aerospace Engineering, University of Michigan, Ann Arbor, USA · FLOW, Engineering Mechanics, KTH Royal Institute of Technology, Stockholm, Sweden · Division of Robotics, Perception and Learning, School of EECS, KTH Royal Institute of Technology, Stockholm, Sweden +7

Diffusion-Based Inverse Problems Inverse Problems Computational Fluid Dynamics

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Abstract

Physics-Informed ML

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