  Deep neural network approximation of nonlinear operators, commonly referred
to as DeepONet, has proven capable of approximating PDE backstepping designs in
which a single Goursat-form PDE governs a single feedback gain function. In
boundary control of coupled PDEs, coupled Goursat-form PDEs govern two or more
gain kernels-a PDE structure unaddressed thus far with DeepONet. In this paper,
we explore the subject of approximating systems of gain kernel PDEs for
hyperbolic PDE plants by considering a simple counter-convecting $2\times 2$
coupled system in whose control a $2\times 2$ kernel PDE system in Goursat form
arises. Engineering applications include oil drilling, the Saint-Venant model
of shallow water waves, and the Aw-Rascle-Zhang model of stop-and-go
instability in congested traffic flow. We establish the continuity of the
mapping from a total of five plant PDE functional coefficients to the kernel
PDE solutions, prove the existence of an arbitrarily close DeepONet
approximation to the kernel PDEs, and ensure that the DeepONet-approximated
gains guarantee stabilization when replacing the exact backstepping gain
kernels. Taking into account anti-collocated boundary actuation and sensing,
our $L^2$-Globally-exponentially stabilizing (GES) approximate gain
kernel-based output feedback design implies the deep learning of both the
controller's and the observer's gains. Moreover, the encoding of the
output-feedback law into DeepONet ensures semi-global practical exponential
stability (SG-PES). The DeepONet operator speeds up the computation of the
controller gains by multiple orders of magnitude. Its theoretically proven
stabilizing capability is demonstrated through simulations.
