To improve the trajectory tracking performance of piezoelectric actuators (PEAs), this study proposes a Kolmogorov-Arnold network feedforward model predictive control (MPC-KAN) based on a gated recurrent unit (GRU) neural network (NN) prediction model. Unlike neural network inverse model control, this method uses GRU-NN forward modeling and adjusts the model predictive control (MPC) output based on the model prediction results. First, this study selects the training input features of GRU-NN based on a linearized model and trains the network. Then, to improve optimization performance and shorten optimization time, the sparrow search algorithm (SSA) was used as the MPC optimizer, and a Kolmogorov-Arnold network (KAN) was established to replace the SSA optimization. The effectiveness of this method has been verified on the PEA platform. Compared with traditional methods, the control accuracy has been improved by approximately 30%.