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Adaptive control of reaction-diffusion PDEs via neural operator-approximated gain kernels

Luke Bhan, Yuanyuan Shi, Miroslav Krstic

TL;DR

This work extends neural-operator gain-kernel backstepping to adaptive control of parabolic reaction-diffusion PDEs with unknown spatial coefficients by learning the full 2D gain kernel mapping from online coefficient estimates to the kernel itself. It establishes a Lyapunov-based stability framework that accounts for neural-operator approximation errors and parameter-estimation dynamics, proving asymptotic regulation of the plant state under suitable update gains. The paper also formalizes a universal nonlocal neural-operator theorem and demonstrates that DeepONet/FNO variants can implement the needed kernel mappings with provable accuracy, enabling real-time online kernel recomputation. Empirical results show the method stabilizes the plant and delivers substantial speedups (up to tens of times) compared to conventional finite-difference kernel solvers, highlighting practical potential for fast adaptive PDE control in chemical diffusion and related applications.

Abstract

Neural operator approximations of the gain kernels in PDE backstepping has emerged as a viable method for implementing controllers in real time. With such an approach, one approximates the gain kernel, which maps the plant coefficient into the solution of a PDE, with a neural operator. It is in adaptive control that the benefit of the neural operator is realized, as the kernel PDE solution needs to be computed online, for every updated estimate of the plant coefficient. We extend the neural operator methodology from adaptive control of a hyperbolic PDE to adaptive control of a benchmark parabolic PDE (a reaction-diffusion equation with a spatially-varying and unknown reaction coefficient). We prove global stability and asymptotic regulation of the plant state for a Lyapunov design of parameter adaptation. The key technical challenge of the result is handling the 2D nature of the gain kernels and proving that the target system with two distinct sources of perturbation terms, due to the parameter estimation error and due to the neural approximation error, is Lyapunov stable. To verify our theoretical result, we present simulations achieving calculation speedups up to 45x relative to the traditional finite difference solvers for every timestep in the simulation trajectory.

Adaptive control of reaction-diffusion PDEs via neural operator-approximated gain kernels

TL;DR

This work extends neural-operator gain-kernel backstepping to adaptive control of parabolic reaction-diffusion PDEs with unknown spatial coefficients by learning the full 2D gain kernel mapping from online coefficient estimates to the kernel itself. It establishes a Lyapunov-based stability framework that accounts for neural-operator approximation errors and parameter-estimation dynamics, proving asymptotic regulation of the plant state under suitable update gains. The paper also formalizes a universal nonlocal neural-operator theorem and demonstrates that DeepONet/FNO variants can implement the needed kernel mappings with provable accuracy, enabling real-time online kernel recomputation. Empirical results show the method stabilizes the plant and delivers substantial speedups (up to tens of times) compared to conventional finite-difference kernel solvers, highlighting practical potential for fast adaptive PDE control in chemical diffusion and related applications.

Abstract

Neural operator approximations of the gain kernels in PDE backstepping has emerged as a viable method for implementing controllers in real time. With such an approach, one approximates the gain kernel, which maps the plant coefficient into the solution of a PDE, with a neural operator. It is in adaptive control that the benefit of the neural operator is realized, as the kernel PDE solution needs to be computed online, for every updated estimate of the plant coefficient. We extend the neural operator methodology from adaptive control of a hyperbolic PDE to adaptive control of a benchmark parabolic PDE (a reaction-diffusion equation with a spatially-varying and unknown reaction coefficient). We prove global stability and asymptotic regulation of the plant state for a Lyapunov design of parameter adaptation. The key technical challenge of the result is handling the 2D nature of the gain kernels and proving that the target system with two distinct sources of perturbation terms, due to the parameter estimation error and due to the neural approximation error, is Lyapunov stable. To verify our theoretical result, we present simulations achieving calculation speedups up to 45x relative to the traditional finite difference solvers for every timestep in the simulation trajectory.
Paper Structure (16 sections, 10 theorems, 82 equations, 4 figures, 1 table)

This paper contains 16 sections, 10 theorems, 82 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. PDE backstepping for adaptive control
  3. Kernel Implementations in PDE Control
  4. Paper organization
  5. Notation
  6. Nominal controller for PDE backstepping
  7. Properties of gain kernel PDE
  8. Neural operator approximation of the gain kernel
  9. Stability under NO Approximated Gain Kernel
  10. Experimental Simulations
  11. Conclusion
  12. Wendroff Inequality
  13. Nonlocal Neural Operator
  14. DeepONet form of NNO
  15. FNO form of NNO
  16. ...and 1 more sections

Key Result

Theorem 1

(smyshlyaevBook Stabilization under exact adaptive control scheme). There exists a $\gamma^*$ such that for $\gamma \in (0, \gamma^*)$, for any initial estimate $\hat{\lambda}(x, 0) \in C^1([0, 1])$ with $\|\hat{\lambda}(x, 0)\|_\infty \leq \bar{\lambda}$ and for any initial condition $u_0 \in H^2(0

Figures (4)

  • Figure 1: Simulation of the plant \ref{['eq:parabolicMain1']}, \ref{['eq:parabolicMain2']}, \ref{['eq:parabolicMain3']} with openloop controller $U(t) = 0$. We note that the plant is openloop unstable.
  • Figure 2: The $\hat{\lambda}$ estimates according to the feedback loop in Figure \ref{['fig:plant']} (top). The blue lines indicate the true $\lambda$ and red line the final $\hat{\lambda}$ estimate. The bottom figure shows the final estimates, initial guess, and true $\lambda$ for the simulation.
  • Figure 3: Simulation of the plant \ref{['eq:parabolicMain1']}, \ref{['eq:parabolicMain2']}, \ref{['eq:parabolicMain3']} with the update law \ref{['eq:mainResultUpdateLaw1']}, \ref{['eq:mainResultUpdateLaw2']}, \ref{['eq:mainResultUpdateLaw3']}, and the controller \ref{['eq:finalController']} where $\hat{k}$ is calculated using a neural operator.
  • Figure 4: Neural operator approximated kernel $\hat{k}(1, y, t)$ and kernel error corresponding to the plant in Figure \ref{['fig:plant']}.

Theorems & Definitions (15)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 2
  • Corollary 1
  • Lemma 4
  • proof
  • ...and 5 more