Radial basis function neural networks solution for stationary probability density function of nonlinear stochastic systems

A radial basis function neural networks (RBF-NN) solution of the reduced Fokker-Planck-Kolmogorov (FPK) equation is proposed in this paper. The activation functions consist of normalized Gaussian probability density functions (PDFs). The use of normalized Gaussian PDFs leads to a simple constraint on the coefficients for normalization of the RBF-NN solution, which as a constraint is imposed with the help of the method of Lagrange multiplier. The relationship between the proposed RBF-NN PDF solution and the generalized cell mapping with short-time Gaussian approximation is discussed, which provides a justification for Gaussian PDFs with varying means and variances in the state space. The optimal number of neurons or activation functions, which leads to the smallest error, is investigated. Four examples are presented to show the effectiveness of the proposed solution method. The results indicate that the proposed solution method is a very efficient and accurate way to compute the stationary PDF of nonlinear stochastic systems. It is also found that the distribution of the optimal coefficients as a function of the mean of Gaussian activation functions is similar to the steady-state PDF solution. Finally, we should point out that an important advantage of the RBF-NN method over methods such as finite element and finite difference is its ability to obtain solutions of the FPK equation for multi-degree-of-freedom stochastic systems.