A new polynomial fitting model based on a neural network is presented to characterize the hysteresis in piezoelectric actuators. As hysteresis is multi-valued mapping, and traditional neural networks can only solve one-to-one mapping, a hysteresis mathematical model is proposed to expand the input of the neural network by converting the multi-valued into one-to-one mapping. Experiments were performed under designed excitation with different driven voltage amplitudes to obtain the parameters of the model using the polynomial fitting method. The simulation results were in good accordance with the measured data and demonstrate the precision with which the model can predict the hysteresis. Based on the proposed model, a single-neuron adaptive proportional–integral–derivative controller combined with a feedforward loop is designed to correct the errors induced by the hysteresis in the piezoelectric actuator. The results demonstrate superior tracking performance, which validates the practicability and effectiveness of the presented approach.