Model-Based Sensor Placement Criterion Overview

• Model-Based Sensor Placement Criterion is a mathematical objective that quantifies and optimizes sensor configurations for improved state inference and field reconstruction.
• It integrates physical, statistical, and information-theoretic models to enhance estimation accuracy while managing constraints from complex system dynamics.
• Modern algorithms, including gradient-based methods and reinforcement learning, enable efficient sensor layout optimization in both continuous and discrete settings.

A model-based sensor placement criterion is a mathematically formulated objective, derived from physical, statistical, or information-theoretic models, that quantifies the value of a sensor configuration for inference, estimation, or field reconstruction. The criterion is selected to directly impact estimation accuracy, uncertainty, or physical observability, and is optimized subject to constraints dictated by the underlying system dynamics, measurement modalities, and feasible deployment regions. This article surveys foundational principles, major subclasses of criteria (distance, CRB-based, information-theoretic, and uncertainty-minimization), and modern algorithmic realization across continuous, discrete, linear, nonlinear, and hybrid-modal settings.

1. Foundational Principles and Mathematical Formulation

The core principle of model-based sensor placement is to mathematically quantify, and then optimize, the informativeness, observability, or utility of a selected set of sensor locations with respect to a governing model. The most universal scenario considers estimating an unknown state $x$, field $u$, source location $s$, or parameter vector $\boldsymbol\lambda$ given sensor measurements subject to noise and possibly non-linear/heterogeneous dynamics. The criterion $\mathcal{C}(S)$ for a sensor subset $S$ typically takes the form of:

• Information-theoretic objective: expected information gain, reductions in Kullback–Leibler divergence, or entropy post-observation (Alexanderian et al., 31 Jan 2025, Waxman et al., 5 Dec 2025)
• Bayesian optimal design: A-, D-, E-, V-optimality, based on scalarizations of the expected or average posterior covariance matrix (Aarset, 22 Oct 2024, Aretz et al., 2023, Tang et al., 3 Apr 2025, Panwar et al., 2022, Zhang et al., 28 Oct 2024)
• Geometric or PDE-based metrics: minimum/mean/maximal distance-to-nearest-sensor (Ftouhi et al., 29 Aug 2025)
• Error-propagation/conditioning: condition number of operator mapping measurements to field or parameter (Liu et al., 27 Sep 2024)
• Uncertainty-minimization: reduction in epistemic (model) uncertainty in deep- or Gaussian process settings (Eksen et al., 27 Nov 2025)

The optimization is over feasible sensor locations $S\subseteq \mathcal{V}$, settings (discrete or continuous), or entire sensor trajectories $P(t)$ for moving sensors (Hodgson et al., 2019, Potter et al., 29 May 2024).

2. Classical Fisher Information and Cramér–Rao-Based Criteria

For parameter estimation or source localization with measurement model $y_i = h_i(\theta, s_i) + n_i$ ($n_i$ zero-mean Gaussian), the standard model-based placement criterion is the scalarization of the Cramér–Rao lower bound (CRB) matrix $J^{-1}$. The Fisher Information Matrix (FIM):

$$J(\theta; S) = \mathbb{E}[\nabla_\theta \ln p(y|\theta)\nabla_\theta \ln p(y|\theta)^T] = H^T\Sigma^{-1}H$$

where $H$ is the Jacobian with respect to $\theta$ evaluated at candidate sensor positions. The principal scalarizations are:

Criterion	Objective Function	Interpretation
A-optimal	$\mathrm{tr}(J^{-1})$	Minimize average MSE
D-optimal	$-\log\det J$ or $\log\det J$	Maximize volume of information
E-optimal	$\lambda_{\max}(J^{-1})$	Minimize worst-case variance

These are widely used in source localization—where the involved $J$ depend on sensor-source geometry (distance, angle, etc)—and inverse problems, and can be applied to hybrid measurement scenarios (TOA, TDOA, RSS, AOA) (Tang et al., 3 Apr 2025, Zhang et al., 28 Oct 2024, Aarset, 22 Oct 2024, Sahu et al., 2021, Panwar et al., 2022).

Optimality and global/necessary conditions for discrete or relaxed design variables are established via subgradient arguments, explicit orderings on gradient components (dominant/redundant sensor concept), or convex optimization (e.g., binary low-rank A-optimal design (Aarset, 22 Oct 2024)).

3. Information-Theoretic and Bayesian Experimental Design

Information-theoretic placement criteria are formulated by maximizing expected reductions in uncertainty, such as the expected Kullback–Leibler divergence from prior to posterior, or expected entropy reduction:

$$\mathbb{E}_{y\mid\text{prior}}\, \left[D_{\mathrm{KL}}(p(\theta|y)\,\|\,p(\theta))\right]$$

In linear-Gaussian models, this reduces to maximizing the log-determinant of the updated posterior covariance (D-optimality), while in weak-constraint 4D-Var data assimilation or dynamic models under uncertainty, it leads to matrix-based objectives involving the full observation, forecast, and model error statistics (Alexanderian et al., 31 Jan 2025). Recent work incorporates such objectives with randomized matrix sketching and Lanczos approximations to scale to large operator settings (Alexanderian et al., 31 Jan 2025, Waxman et al., 5 Dec 2025).

In nonparametric or spatiotemporal settings, model-based criteria are implemented as minimizing the information loss (e.g., KL divergence between full and sparse surrogate posteriors in a Gaussian process framework), leveraging physics-based simulation data and sparse variational inference (Waxman et al., 5 Dec 2025).

Deep learning-based surrogates, such as Physics-Informed Neural Networks (PINN), further allow efficient D-optimal sensor selection via direct evaluation of Fisher information via automatic differentiation (Venianakis et al., 19 Nov 2025). In data-driven regimes with neural process surrogates, epistemic uncertainty minimization (expected reduction in model uncertainty) is used as a placement criterion (Eksen et al., 27 Nov 2025).

4. Geometric and PDE-Based Criteria

When the objective is geometric coverage or direct field reconstruction, model-based sensor placement adopts geometric norms or distance-based functionals. This class is typified by:

• $L^p$-distance: minimize $\Vert d(\cdot, \cup_i B_i)\Vert_{L^p(\Omega)}$, where $d$ is the distance to nearest sensor ball $B_i$ (Ftouhi et al., 29 Aug 2025)
• Max-min formulations: minimum of the maximal or mean distance over the field
• Condition number minimization: minimize condition number $\kappa(\boldsymbol{A})$ of the interpolation/reconstruction operator as a proxy for robustness in least-squares or operator learning reconstruction (Liu et al., 27 Sep 2024)

In the geometric approach of (Ftouhi et al., 29 Aug 2025), combinatorial difficulty of the distance function is replaced by a PDE-based surrogate: using Varadhan’s asymptotic linking of the distance to the solution of an elliptic PDE, enabling gradient-based optimization via shape derivatives and adjoint states.

5. Optimization Algorithms, Numerical Realization, and Hybrid Formulations

A rich set of continuous and combinatorial optimization techniques is adapted to model-based sensor placement:

• Gradient-based optimization: using adjoints and shape derivatives for PDE-constrained problems (Ftouhi et al., 29 Aug 2025, Hodgson et al., 2019), or explicit gradients of SVD/low-rank decompositions in linear goals (Aarset, 22 Oct 2024)
• Alternating Direction Method of Multipliers (ADMM) combined with Majorization-Minimization (MM): to tackle nonconvex, constrained formulations under multiple scalarizations (Sahu et al., 2021, Panwar et al., 2022)
• Greedy orthogonal matching pursuit (OMP) for iterative maximization of observability or Fisher information (Aretz et al., 2023)
• Genetic algorithms and evolutionary strategies for combinatorial search, especially via fitness metrics such as log-condition number or risk-aware objectives (Liu et al., 27 Sep 2024, Candelieri et al., 2021, Waxman et al., 5 Dec 2025)
• Deep reinforcement learning: interpreting sensor placement under uncertainty as a Markov decision process, maximizing incremental information gain rewards (Jabini et al., 2023)
• Modular decomposition and answer set programming for diagnosis/actuation in complex networked or logical systems (Raju et al., 2022)

Hybrid and non-linear settings incorporate majorization-minimization routines with closed-form primal/dual updates for fusion of mixed modality measurements (TOA, TDOA, RSS, AOA) even in presence of correlated noise (Panwar et al., 2022, Tang et al., 3 Apr 2025), with geometric constraints on optimal layouts deduced explicitly for each measurement combination.

6. Analytical Insights, Constraints, and Theoretical Performance

Model-based sensor placement admits analytical geometric or algebraic constraints characterizing optimal sensor geometries. For instance, in hybrid source localization, achieving the A-optimal CRB bound for TOA/TDOA/AOA/RSS fusion requires that specific trigonometric and distance-weighted sums vanish (e.g., $\sum_i \sin \alpha_i = 0$, $\sum_i d_i^{-2} \cos 2\alpha_i=0$, etc.), which are rigorously classified and exploited for algorithmic initialization or validation (Tang et al., 3 Apr 2025).

Equivalence results show that moderate sensor location errors or uncertainties may not alter the optimality of classical layouts—e.g., for full-TDOA in the near field, the same uniform-angular configurations are optimal both with and without sensor location uncertainty (Zhang et al., 28 Oct 2024).

Probabilistic and robust estimation guarantees are derived for greedy and random-placement algorithms using submodularity of set functions related to distinguishability of nonlinear secant pairs (Otto et al., 2021), or via worst-case observability constants (Aretz et al., 2023).

7. Applications, Empirical Performance, and Impact

Model-based sensor placement criteria are applied widely:

• Source localization (acoustic, RF, radar, seismic): fusion of TDOA/TOA/AOA/RSS, hybrid geometries, consideration of sensor errors (Tang et al., 3 Apr 2025, Zhang et al., 28 Oct 2024, Aarset, 22 Oct 2024, Sahu et al., 2021, Panwar et al., 2022)
• Environmental and spatiotemporal field monitoring: temperature, climate, hydrology, oil spill mapping (Liu et al., 27 Sep 2024, Waxman et al., 5 Dec 2025, Hodgson et al., 2019)
• Structural health and distributed parameter inference, Bayesian inverse problems, active fault diagnosis (Aarset, 22 Oct 2024, Venianakis et al., 19 Nov 2025, Raju et al., 2022)
• Online/real-time adaptive and risk-aware monitoring in complex networked systems (Candelieri et al., 2021, Jabini et al., 2023)
• Deep learning and data-driven field reconstruction/forecasting (Semaan, 2016, Eksen et al., 27 Nov 2025, Waxman et al., 5 Dec 2025)

Numerical experiments consistently validate that model-based sensor placements delivered via these criteria outperform random, uniform, or heuristic-based alternatives in reducing estimation error, posterior variance, and prescribed risk metrics. The advantage is particularly pronounced for systems with spatially heterogeneous dynamics, hybrid measurement modalities, or when data acquisition costs impose a severe budget constraint.

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References

• "Optimal Sensor Placement Using Combinations of Hybrid Measurements for Source Localization" (Tang et al., 3 Apr 2025)
• "Sensor placement via large deviations in the Eikonal equation" (Ftouhi et al., 29 Aug 2025)
• "Global optimality conditions for sensor placement, with extensions to binary low-rank A-optimal designs" (Aarset, 22 Oct 2024)
• "A physics-driven sensor placement optimization methodology for temperature field reconstruction" (Liu et al., 27 Sep 2024)
• "Optimal sensor placement under model uncertainty in the weak-constraint 4D-Var framework" (Alexanderian et al., 31 Jan 2025)
• "A Physics Informed Machine Learning Framework for Optimal Sensor Placement and Parameter Estimation" (Venianakis et al., 19 Nov 2025)
• "Where to Measure: Epistemic Uncertainty-Based Sensor Placement with ConvCNPs" (Eksen et al., 27 Nov 2025)
• "Optimal sensor placement using machine learning" (Semaan, 2016)
• "A Greedy Sensor Selection Algorithm for Hyperparameterized Linear Bayesian Inverse Problems" (Aretz et al., 2023)
• "Risk Aware Optimization of Water Sensor Placement" (Candelieri et al., 2021)
• "Optimal Model-Based Sensor Placement & Adaptive Monitoring Of An Oil Spill" (Hodgson et al., 2019)
• "Optimal Sensor Placement for Source Localization: A Unified ADMM Approach" (Sahu et al., 2021)
• "Optimal Sensor Placement for TDOA-Based Source Localization with Sensor Location Errors" (Zhang et al., 28 Oct 2024)
• "Optimal Sensor Placement for Hybrid Source Localization Using Fused TOA-RSS-AOA Measurements" (Panwar et al., 2022)
• "Designing an Optimal Sensor Network via Minimizing Information Loss" (Waxman et al., 5 Dec 2025)
• "Inadequacy of Linear Methods for Minimal Sensor Placement and Feature Selection in Nonlinear Systems; a New Approach Using Secants" (Otto et al., 2021)
• "Measurement Optimization under Uncertainty using Deep Reinforcement Learning" (Jabini et al., 2023)