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 , 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.