Random perturbations are common in nature and engineering, and most of them exhibit inherent non-Gaussian properties. Thus, it may lead to huge errors if Gaussian excitation is used for modeling. As a typical and important non-Gaussian excitation model, Poisson white noise has attracted extensive attention. At present, the dynamic characteristic analysis of the system subjected to Poisson white noise is mainly focused on the study of the stationary response, while the solution of the transient response is still difficult and needs further development. In this paper, an efficient semi-analytical method based on radial basis function neural networks (RBF-NN) are proposed for transient response prediction of single-degree-of-freedom strong nonlinear systems under Poisson white noise excitation. Firstly, the transient solution of the generalized Fokker-Plank-Kolmogorov (FPK) equation is expressed as a set of Gaussian RBF-NN with unknown time-varying weight coefficients. Then, the finite difference method is applied to discretize and approximate the time derivative term, and the loss function with time recurrence is constructed by the random sampling technique. Finally, the time-varying optimal weight coefficients can be determined by minimizing the loss function through the Lagrange multiplier method. As examples, two classical strong nonlinear systems are investigated, and the solutions are validated by the Monte Carlo simulation (MCS) method. The results show that the transient probability density functions (PDFs) obtained by the proposed scheme agree well with the MCS data, and the algorithm has high computational efficiency. In the whole evolution process of the system response, the proposed scheme can effectively capture the complex nonlinear characteristics of the system response at each moment. Furthermore, the high precision semi-analytical transient solution obtained by the proposed scheme can not only be used as a benchmark to test the accuracy of other nonlinear random vibration analysis methods, but also has great potential application value for the structural optimum design.