This paper studies the dynamic response of wind turbine systems (WTS) under non-Gaussian stochastic excitation. Firstly, considering the limitations of traditional Gaussian noise in representing the actual wind speed and system uncertainty, the α-stable Lévy noise with heavy tail and pulse characteristics is introduced to establish a more practical WTS stochastic dynamical model. Secondly, based on the theory of stochastic differential, the fractional Fokker-Planck-Kolmogorov (FPK) equation corresponding to WTS under the excitation of α-stable Lévy noise is derived, which precisely describes the evolution law of the transient probability density function (PDF) for the system state. Finally, to effectively solve the fractional partial differential equation, a physics-informed neural networks (PINNs) framework is proposed, which takes the physical control equation as the constrained embedding loss function, and can directly learn the space-time continuous PDF solution without grid discretization. Numerical experiments show that the PINNs solution is highly consistent with the Monte Carlo simulation results, which verifies the accuracy of this method in solving fractional FPK equations. Meanwhile, PINNs shows much higher computational efficiency than traditional Monte Carlo methods.