Extremum seeking boundary control for PDE systems using PINNs

Extremum seeking (ES) is an effective real-time optimization method for partial differential equation (PDE) systems in cascade with non-linear quadratic maps. To address PDEs in the feedback loop, a boundary control law and a re-design of the additive probing signal are mandatory. The latter, commonly called ‘trajectory generation’ or ‘motion planning’, involves designing perturbation signals that anticipate their propagation through PDEs. Specifically, this requires solving motion planning problems for systems governed by parabolic and hyperbolic PDEs. Physics-informed neural networks (PINNs) is a powerful tool for solving PDEs by embedding physical laws as constraints in the neural network’s loss function, enabling efficient solutions for high-dimensional, non-linear and complex problems. This paper proposes a novel construction integrating PINN and ES, automating the motion planning process for specific PDE systems and eliminating the need for case-by-case analytical derivations. The proposed strategy efficiently extracts perturbation signals, optimizing the PDE system.