A novel data-driven method for simulating non-Gaussian stochastic processes is proposed in this paper. The sample conversion model and power spectrum conversion model are established by using artificial neural network models respectively. A neural network model is constructed based on sample data to transform Gaussian samples into non-Gaussian samples. The distribution function of the samples is modeled using the shifted generalized lognormal distribution, and the latent Gaussian power spectrum is directly obtained through the backpropagation neural network model. The Gaussian stochastic process samples are generated using the spectral representation method, and then transformed into non-Gaussian process samples using the sample conversion neural network model. This method is capable of generating non-Gaussian stochastic process samples based on limited sample data, addressing the challenge of determining latent Gaussian power spectrum, and solving the problems such as poor accuracy and limited application range of the central moments-based transformation models. Through numerical simulations and validation in turbulent wind fields, the accuracy and effectiveness of the proposed method are further demonstrated.