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

TL;DR

This paper extends neural operator-based adaptive control to reaction-diffusion PDEs, proving stability and regulation while achieving significant computational speedups over traditional methods.

Contribution

It introduces a neural operator approach for adaptive control of parabolic PDEs, with stability proofs and practical speedup demonstrations.

Findings

Proved global stability and asymptotic regulation for the PDE control system.

Achieved up to 45x speedup in simulations compared to finite difference methods.

Extended neural operator methodology from hyperbolic to parabolic PDEs.

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.