In the seismic analysis of structural systems, dynamic response prediction is an essential problem and is significant in every stage during the structural life cycle. Conventionally, response analysis is carried out by numerical analysis. However, when the structural parameter is unknown, the establishment of a numerical model will be difficult. Enlightened by the Runge-Kutta (RK) numerical algorithm, this paper proposes a novel recurrent neural network named Runge-Kutta recurrent neural network (RKRNN) to realize the seismic response prediction. A partition training strategy is formulated to train the proposed neural network and to improve the efficiency of training. The proposed model can be trained by using a limited number of samples. Three numerical examples are utilized to validate the feasibility of RKRNN model including a linear three degrees of freedom (DOFs) system, a nonlinear single DOF system with Bouc-Wen hysteresis, and a numerical reinforced concrete bridge model. Additionally, the site monitoring data from a real-world bridge is utilized to further validate the proposed network. The results show that the proposed RKRNN model can effectively and efficiently predict the structural response under seismic load and exhibits robustness to noise, with good potential for applications in engineering practice.