Neural Operator Approximations of Backstepping Kernels for $2\times 2$ Hyperbolic PDEs

Deep neural network approximation of nonlinear operators, commonly referred to as DeepONet, has so far 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 note we open 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 $2\times 2$ Goursat form kernel PDE system arises. Such a coupled kernel PDE problem arises in several canonical $2\times 2$ hyperbolic PDE problems: oil drilling, Saint-Venant model of shallow water waves, and Aw-Rascle model of stop-and-go instability in congested traffic flow. In this paper, 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 establish that the DeepONet-approximated gains guarantee stabilization when replacing the exact backstepping gain kernels. The DeepONet operator speeds the computation of the controller gains by multiple orders of magnitude and its theoretically proven stabilizing capability is illustrated by simulations.