Extremum Seeking for PDE Systems using Physics-Informed Neural Networks

TL;DR

This paper introduces a novel approach combining Physics-Informed Neural Networks with Extremum Seeking to automate motion planning in PDE systems, enabling real-time optimization without extensive analytical derivations.

Contribution

It presents an innovative integration of PINN and ES, automating perturbation signal design for PDE systems and improving efficiency in real-time optimization.

Findings

Successfully automates motion planning for PDEs

Enhances real-time PDE system optimization

Reduces need for case-specific analytical solutions

Abstract

Extremum Seeking (ES) is an effective real-time optimization method for PDE systems in cascade with nonlinear 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 (PINN) 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, nonlinear, 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.