  To stabilize PDEs, feedback controllers require gain kernel functions, which
are themselves governed by PDEs. Furthermore, these gain-kernel PDEs depend on
the PDE plants' functional coefficients. The functional coefficients in PDE
plants are often unknown. This requires an adaptive approach to PDE control,
i.e., an estimation of the plant coefficients conducted concurrently with
control, where a separate PDE for the gain kernel must be solved at each
timestep upon the update in the plant coefficient function estimate. Solving a
PDE at each timestep is computationally expensive and a barrier to the
implementation of real-time adaptive control of PDEs. Recently, results in
neural operator (NO) approximations of functional mappings have been introduced
into PDE control, for replacing the computation of the gain kernel with a
neural network that is trained, once offline, and reused in real-time for rapid
solution of the PDEs. In this paper, we present the first result on applying
NOs in adaptive PDE control, presented for a benchmark 1-D hyperbolic PDE with
recirculation. We establish global stabilization via Lyapunov analysis, in the
plant and parameter error states, and also present an alternative approach, via
passive identifiers, which avoids the strong assumptions on kernel
differentiability. We then present numerical simulations demonstrating
stability and observe speedups up to three orders of magnitude, highlighting
the real-time efficacy of neural operators in adaptive control. Our code
(Github) is made publicly available for future researchers.
