  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.
