Abstract:  The belief that bringing ML into feedback stabilization problems means giving up on stability guarantees, while quite widespread, is unfounded. Adaptive control of complex systems, such as PDEs with unknown parameters, is an exemplary setting in which operator learning enables control implementation with stability guarantees. The exact ADAPTIVE backstepping designs, which require a solution of a PDE at each time step, for each new online-learned estimate of the system’s parameter function, are emulated with a thousandfold speedup using offline-learned DeepONet approximators. Stability is guaranteed using error estimates for the DeepONet approximations of the backstepping operators combined with Lyapunov theory for PDEs. Designs, theorems, and numerical tests for both hyperbolic and parabolic PDEs are presented. This is work with Bhan, Shi, and Lamarque.