  Deep neural networks that approximate nonlinear function-to-function
mappings, i.e., operators, which are called DeepONet, have been demonstrated in
recent articles to be capable of encoding entire PDE control methodologies,
such as backstepping, so that, for each new functional coefficient of a PDE
plant, the backstepping gains are obtained through a simple function
evaluation. These initial results have been limited to single PDEs from a given
class, approximating the solutions of only single-PDE operators for the gain
kernels. In this paper we expand this framework to the approximation of
multiple (cascaded) nonlinear operators. Multiple operators arise in the
control of PDE systems from distinct PDE classes, such as the system in this
paper: a reaction-diffusion plant, which is a parabolic PDE, with input delay,
which is a hyperbolic PDE. The DeepONet-approximated nonlinear operator is a
cascade/composition of the operators defined by one hyperbolic PDE of the
Goursat form and one parabolic PDE on a rectangle, both of which are bilinear
in their input functions and not explicitly solvable. For the delay-compensated
PDE backstepping controller, which employs the learned control operator,
namely, the approximated gain kernel, we guarantee exponential stability in the
$L^2$ norm of the plant state and the $H^1$ norm of the input delay state.
Simulations illustrate the contributed theory.
