A physics informed machine learning framework for optimal sensor placement and parameter estimation

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 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. Additionally, comparison with traditional optimization approaches, shows competitive performance, particularly in scenarios involving expensive forward models or limited prior information.