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Extremum Seeking for PDE Systems using Physics-Informed Neural Networks

Haojin Guo, Zongyi Guo, Jianguo Guo, Tiago Roux Oliveira

TL;DR

This work tackles real-time extremum seeking for PDE-based systems by unifying trajectory generation with Physics-Informed Neural Networks (PINNs). By embedding physical laws into a single neural solver, the approach automates the design of probing signals for ES across parabolic and hyperbolic PDEs, eliminating case-by-case analytical derivations. A backstepping-based controller is combined with an averaging argument to prove exponential stability of the closed-loop infinite-dimensional system, with rigorous bounds on convergence to the optimum. Simulation results across distributed, advection-diffusion (RAD), wave, transport, and Stefan moving-boundary PDEs demonstrate that PINN-generated perturbations closely match analytical benchmarks and enable robust ES performance, suggesting significant practical impact for real-time PDE optimization in complex settings. The framework shows promise for extending ES to broader PDE families by jointly learning perturbations and leveraging physics-informed loss terms, enhancing computational efficiency and adaptability in control applications.

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.

Extremum Seeking for PDE Systems using Physics-Informed Neural Networks

TL;DR

This work tackles real-time extremum seeking for PDE-based systems by unifying trajectory generation with Physics-Informed Neural Networks (PINNs). By embedding physical laws into a single neural solver, the approach automates the design of probing signals for ES across parabolic and hyperbolic PDEs, eliminating case-by-case analytical derivations. A backstepping-based controller is combined with an averaging argument to prove exponential stability of the closed-loop infinite-dimensional system, with rigorous bounds on convergence to the optimum. Simulation results across distributed, advection-diffusion (RAD), wave, transport, and Stefan moving-boundary PDEs demonstrate that PINN-generated perturbations closely match analytical benchmarks and enable robust ES performance, suggesting significant practical impact for real-time PDE optimization in complex settings. The framework shows promise for extending ES to broader PDE families by jointly learning perturbations and leveraging physics-informed loss terms, enhancing computational efficiency and adaptability in control applications.

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.

Paper Structure

This paper contains 13 sections, 1 theorem, 57 equations, 16 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Actuation dynamics and demodulation signals
  4. Estimation Errors and PDE-Error Dynamics
  5. Methodology
  6. Physics-Informed Neural Networks (PINN)
  7. Control Design and Closed-Loop Error Dynamics
  8. Stability and convergence analysis
  9. Simulation results
  10. Comparison of PINN-Based and Numerical Solution Results
  11. Simulation Results of Probing Signals
  12. Simulation Results of Extremum Seeking Control
  13. Conclusions

Key Result

Theorem 1

For a sufficiently large $c>0$, there exists some $\bar{\omega}(c)>0$, such that $\forall\omega > \bar{\omega}$, the error-dynamics 42-45 with states $\vartheta(t)$, $u(x,t)$, has an unique exponentially stable periodic solution in $t$ of period $\Pi:=2\pi/\omega$, denoted by $\vartheta^\Pi(t),u^\Pi

Figures (16)

  • Figure 1: Schematic Diagram of the PINN Architecture
  • Figure 2: Schematic Diagram of the ES Architecture
  • Figure 3: Comparison of Numerical and PINN Solutions for PDEs
  • Figure 4: Comparison of Numerical and PINN Solutions for Stefan PDE
  • Figure 5: Distributed PDE perturbation signal evolution
  • ...and 11 more figures

Theorems & Definitions (3)

  • Remark 1
  • Theorem 1
  • Remark 2