The backstepping method has been widely used in boundary control problems of PDE systems, but solving the backstepping kernel function can be time-consuming. To address this, a neural operator (NO) learning scheme is leveraged for accelerating the control design of cascaded parabolic PDEs. We establish the continuity and boundedness of the kernels, and demonstrate the existence of arbitrarily close DeepONet approximations to the kernel PDEs. Furthermore, we demonstrate that the DeepONet approximation gain kernels ensure stability when replacing the exact backstepping gain kernels. Notably, DeepONet operator exhibits computation speeds two orders of magnitude faster than PDE solvers for such gain functions.
