Output Regulation of an ODE-Parabolic PIDE Cascade System by Neural Operators

In this article, we consider the neural operator (NO)-based output regulation for the ordinary differential equation (ODE)-parabolic partial integro-differential equation (PIDE) cascade systems with spatially varying in-domain coefficients, in which both the body equations and the uncontrolled end are subject to disturbances. The disturbances are generated by an exosystem. A boundary feedback regulator is designed using the backstepping method, while NOs are employed to efficiently generate the controller. Specifically, DeepONet, a representative NO capable of learning nonlinear operators, is utilized to approximate the backstepping kernel functions, thereby accelerating their computation. Our approach demonstrates that DeepONet generates the kernel functions with a loss on the order of (Formula presented.), nearly two orders of magnitude faster than conventional PDE solvers. Moreover, by integrating DeepONet-approximated kernels into the feedback regulator, Lyapunov analysis rigorously ensures that the closed-loop system achieves exponential tracking of the reference trajectory.