Neural operator for observer gains and output-feedback control of coupled PDE–ODE systems

This paper presents a novel data-driven framework for designing observers and controllers in coupled ODE-PDE systems of reaction-diffusion type. Leveraging the DeepONet architecture as a neural operator, the method directly approximates nonlinear mappings between function spaces, eliminating the need for analytical solutions of kernel equations. The observer is first constructed to estimate the system states, followed by the design of the controller based on the estimated states. Simulation results, validated against exact solutions of Goursat equations and evaluated through metrics such as convergence rate, estimation error, and control effort, demonstrate the high accuracy and computational efficiency of the proposed approach. Finally, an ablation study was conducted as well to evaluate the building blocks of the proposed method.