This paper proposes a novel neural network (NN)-based vector control method for a three-phase induction motor. The proposed NN vector control utilizes the rotor flux-oriented reference frame, and the role of the NN controller is to substitute the two decoupled current-loop proportional-integral (PI) controllers in conventional vector control techniques. The objective of NN training is to approximate optimal control and the NN controller was trained by Levenberg-Marquardt (LM) algorithm. Forward Accumulation Through Time algorithm for induction motor was developed to calculate Jacobian matrix needed by the LM algorithm. The simulations showed that the NN vector control can provide better current tracking ability than the conventional vector control, such as less oscillations and low harmonics. Especially, the NN vector control can better overcome the problem of detuning effects than the conventional vector control. The hardware experiments further demonstrated the great advantage of the NN vector control. The NN vector control can succeed in driving the induction motor without audible noise using relatively lower switching frequency or lower sampling rate compared with the conventional vector control, and thus has the potential to improve efficiency and reduce size and cost of an induction motor drive system.