A Physics Informed Machine Learning Framework for Optimal Sensor Placement and Parameter Estimation

TL;DR

This work tackles parameter estimation in distributed-parameter PDEs when data are costly or sparse by coupling sensor placement with physics-informed neural networks. It introduces a two-PINN framework: a Sensitivity PINN that computes parameter derivatives via automatic differentiation to build the Fisher Information Matrix, and an Inference PINN that estimates the unknown parameters using data from optimally placed sensors chosen by D-optimality (maximizing $\det(F)$ or $\operatorname{tr}(F)$ for steady cases). The method is demonstrated on 1D steady-state and 2D transient reaction–diffusion–advection problems, showing that optimally placed sensors lead to substantially more accurate estimates of $Pe$ and $Da$ than intuitive placements, even under noise. The approach offers a mesh-free, adaptable workflow for sensor placement and parameter inference in complex, large-scale systems, with transfer learning enabling rapid reconfiguration for new conditions or geometries.

Abstract

Parameter estimation remains a challenging task across many areas of engineering. Because data acquisition can often be costly, limited, or prone to inaccuracies (noise, uncertainty) it is crucial to identify sensor configurations that provide the maximum amount of information about the unknown parameters, in particular for the case of distributed-parameter systems, where spatial variations are important. Physics-Informed Neural Networks (PINNs) have recently emerged as a powerful machine-learning (ML) tool for parameter estimation, particularly in cases with sparse or noisy measurements, overcoming some of the limitations of traditional optimization-based and Bayesian approaches. Despite the widespread use of PINNs for solving inverse problems, relatively little attention has been given to how their performance depends on sensor placement. This study addresses this gap by introducing a comprehensive PINN-based framework that simultaneously tackles optimal sensor placement and parameter estimation. Our approach involves training a PINN model in which the parameters of interest are included as additional inputs. This enables the efficient computation of sensitivity functions through automatic differentiation, which are then used to determine optimal sensor locations exploiting the D-optimality criterion. The framework is validated on two illustrative distributed-parameter reaction-diffusion-advection problems of increasing complexity. The results demonstrate that our PINNs-based methodology consistently achieves higher accuracy compared to parameter values estimated from intuitively or randomly selected sensor positions.

Figures (13)

• Figure 1: Schematic representation of the methodology for computing sensitivity derivatives using PINNs. The neural network admits as input the spatio-temporal coordinates $(\mathbf{x},t)$ along with the parameters of interest, $\lambda$. The loss function $\mathcal{L}$ to be minimized is constructed from the system residuals (PDE and boundary/initial conditions) and their derivatives with respect to the parameters, $\lambda$.
• Figure 2: Spatial profile of the FIM trace computed by PINNs and numerically for $Pe = 0.1$ ( a priori estimate) and $Da = 1.0$.
• Figure 3: Inference of $Pe$ vs. PINN training epochs with sensor placed at the FIM trace maximum ($x^* = 1.81$ -blue curve) versus at the outlet ($x^* = 10.0$ -red curve)
• Figure 4: FIM trace for the a priori estimate $Pe=0.1$ and true value $Pe=1.0$.
• Figure 5: Computational domain: the fluid enters the channel with a parabolic velocity profile at the horizontal direction carrying a reactive species, flows past a fixed obstacle, and undergoes transport and reaction.