Gaussian Processes for Sensor Repositioning in PDE-Driven Systems

Abstract

Field estimation in spatio-temporally evolving environments remains challenging, particularly when limited sensor resources must capture dynamic features while contending with modeling errors and measurement noise e.g., in environmental monitoring using aerial vehicles, where system dynamics interact with practical sensing limitations. In this work, we consider a scenario where a network of mobile sensor nodes measure an advection-diffusion field, where the sensor locations can be dynamically optimized based on PDE residuals e.g., sensors on-board drones. Our novel two-stage framework strategically integrates Gaussian Process regression with PDE constraints. An initial inference stage estimates key parameters (e.g., advection velocity, diffusion coefficient) through stationary sensor measurements and finite-difference derivative approximations, while a subsequent mobility stage employs forward-Euler time-stepping to dynamically relocate the sensors toward regions of high PDE residual. Simulations based on a 2D advection-diffusion field experiment reveals upto an order magnitude improvement in field reconstruction error, as compared to information theoretic deployments. We conclude with future directions of extending our work and suggest applications.