This work addresses the full-information output regulation problem for nonlinear systems, assuming the states of both the plant and the exosystem are known. In this setting, perfect tracking or rejection is achieved by constructing a zero-regulation-error manifold $π(w)$ and a feedforward input $c(w)$ that render such manifold invariant. The pair $(π(w), c(w))$ is characterized by the regulator equations, i.e., a system of PDEs with an algebraic constraint. We focus on accurately solving the regulator equations introducing a physics-informed neural network (PINN) approach that directly approximates $π(w)$ and $c(w)$ by minimizing the residuals under boundary and feasibility conditions, without requiring precomputed trajectories or labeled data. The learned operator maps exosystem states to steady state plant states and inputs, enables real-time inference and, critically, generalizes across families of the exosystem with varying initial conditions and parameters. The framework is validated on a regulation task that synchronizes a helicopter's vertical dynamics with a harmonically oscillating platform. The resulting PINN-based solver reconstructs the zero-error manifold with high fidelity and sustains regulation performance under exosystem variations, highlighting the potential of learning-enabled solvers for nonlinear output regulation. The proposed approach is broadly applicable to nonlinear systems that admit a solution to the output regulation problem.

This paper investigates the robust output regulation problem of second-order nonlinear uncertain systems with an unknown exosystem. Instead of the adaptive control approach, this paper resorts to a robust control methodology to solve the problem and thus avoid the bursting phenomenon. In particular, this paper constructs generic internal models for the steady-state state and input variables of the system. By introducing a coordinate transformation, this paper converts the robust output regulation problem into a nonadaptive stabilization problem of an augmented system composed of the second-order nonlinear uncertain system and the generic internal models. Then, we design the stabilization control law and construct a strict Lyapunov function that guarantees the robustness with respect to unmodeled disturbances. The analysis shows that the output zeroing manifold of the augmented system can be made attractive by the proposed nonadaptive control law, which solves the robust output regulation problem. Finally, we demonstrate the effectiveness of the proposed nonadaptive internal model approach by its application to the control of the Duffing system.