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.

Deep Learning of Delay-Compensated Backstepping for Reaction-Diffusion PDEs

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.