A real-time nonlinear model predictive controller is presented for point-absorbing wave energy converter (PA-WEC) to maintain the optimal wave energy extraction by tracking the reference PA-WEC velocity. In order to reduce computational cost, the predictive controller is designed using wave condition preview and the Taylor series expansion based nonlinear optimization to determine the q-axis current control signals while the constraints on both the device velocity and q-axis current are respected. The dynamics of the PA-WEC is modeled and details of the nonlinear model predictive controller are presented. A pragmatic method is proposed to obtain the optimal reference PA-WEC velocity by observing the time instants when the excitation force passes a threshold. A long short-term memory recurrent neural network (LSTM-RNN) identifier is designed to identify the wave condition term for implementing the controller. The overall stability of the closed control system including the predictive control and the LSTM-RNN identifier is proved. The proposed nonlinear predictive controller is validated under realistic wave conditions. The results indicate that the proposed control allows higher PA-WEC power production than the conventional control under representative irregular wave conditions.