This paper proposes a deep operator learning framework for fast and accurate prediction of both primal and dual variables in alternating-current optimal power flow (AC-OPF). Unlike conventional neural-network surrogates that learn a fixed input--output mapping and often ignore dual information, the proposed method is designed to approximate the underlying operator that maps varying system operating conditions to optimal OPF solutions. The framework is built on a primal--dual DeepONet architecture, where branch and trunk networks jointly encode system conditions and query locations to predict generator dispatch, voltage states, and associated Lagrange multipliers.

To better capture optimization structure, a differentiable dual completion layer is incorporated to improve dual feasibility and complementary slackness, and a KKT-informed loss function is used to guide training toward physically consistent and optimization-aware solutions. The proposed model is evaluated on standard IEEE benchmark systems, where it achieves high prediction accuracy for both primal and dual variables, millisecond-level inference time, and significantly reduced KKT residuals compared with conventional learning-based OPF approximations. These results highlight the promise of deep operator learning as an efficient surrogate for real-time AC-OPF analysis and power-system operation.