  We introduce a framework for eliminating the computation of controller gain
functions in PDE control. We learn the nonlinear operator from the plant
parameters to the control gains with a (deep) neural network. We provide
closed-loop stability guarantees (global exponential) under an NN-approximation
of the feedback gains. While, in the existing PDE backstepping, finding the
gain kernel requires (one offline) solution to an integral equation, the neural
operator (NO) approach we propose learns the mapping from the functional
coefficients of the plant PDE to the kernel function by employing a
sufficiently high number of offline numerical solutions to the kernel integral
equation, for a large enough number of the PDE model's different functional
coefficients. We prove the existence of a DeepONet approximation, with
arbitrarily high accuracy, of the exact nonlinear continuous operator mapping
PDE coefficient functions into gain functions. Once proven to exist, learning
of the NO is standard, completed "once and for all" (never online) and the
kernel integral equation doesn't need to be solved ever again, for any new
functional coefficient not exceeding the magnitude of the functional
coefficients used for training. We also present an extension from approximating
the gain kernel operator to approximating the full feedback law mapping, from
plant parameter functions and state measurement functions to the control input,
with semiglobal practical stability guarantees. Simulation illustrations are
provided and code is available on github. This framework, eliminating real-time
recomputation of gains, has the potential to be game changing for adaptive
control of PDEs and gain scheduling control of nonlinear PDEs.
  The paper requires no prior background in machine learning or neural
networks.
