<p>Solving the Fokker-Planck-Kolmogorov (FPK) equation is a key problem for obtaining the transient response of stochastic dynamical systems. For shape memory alloy (SMA) oscillator systems, factors such as temperature and damping coefficients can affect the safety and normal operation of the system. Therefore, this paper studies the nonlinear dynamic response of SMA oscillator under periodic and Gaussian colored noise excitation. First, the stochastic Itô differential equations and the FPK equation for stochastic SMA system under non-resonant and resonant conditions are derived using the stochastic averaging method. Subsequently, Logistic Basis Function Neural Network (LBFNN) is proposed to solve the FPK equation. In the LBFNN algorithm, a three-layer neural network is used to approximate the solution of the FPK equation. The characteristic of this method lies in transforming the process of solving the FPK equation into solving a system of algebraic equations. The stationary and transient probability density functions of the SMA oscillator under periodic and Gaussian colored noise excitation are obtained. The influence of different parameter values on the SMA oscillator is analyzed, and the correctness of the approximate analytical solutions calculated by the LBFNN is verified using the Radial Basis Function Neural Network(RBFNN). The consistency of the comparison results demonstrates the effectiveness and superiority of the LBFNN algorithm in studying SMA oscillator. The study finds that temperature, damping coefficient and noise intensity can affect the performance of SMA oscillator. This paper shows that using neural networks to study the practical application of SMA materials has good potential.</p>