A new LBFNN algorithm to solve FPK equations for stochastic dynamical systems under Gaussian or Non-Gaussian excitation

In this paper, a new Logistic Basis Function Neural Network (LBFNN) algorithm is proposed and applied to solve Fokker–Planck–Kolmogorov (FPK) equations for stochastic dynamical systems under random excitation. Our attention is focused on the transient solution of the systems under Non-Gaussian excitation. Firstly, we built a neural network by using Logistic probability functions as the basis functions, in which the weighted coefficients are unknown and to be determined. After that, considering the constraint from FPK equation and the normalization condition from the weighted parameters, we construct the loss function to be comprised by these two parts. The innovation of our algorithm is that unknown weighted parameters in LBFNN can be obtained by solving a set of algebraic iteration formulas instead of testify by sample data. Four examples with different dimensions and different excitations are discussed to identify the availability of the LBFNN algorithm. All results show that LBFNN algorithm is not only capable to get transient solutions of the systems under Gaussian white-noise, but also enable us to get the transient solutions in the case of Non-Gaussian excitation. Other techniques including Exact expressions, Gaussian radial basis function neural network (RBFNN) and Monte-carlo simulation are utilized in order to examine the accuracy of the LBFNN algorithm. The good agreements in different comparisons demonstrate the validity and the strength of the LBFNN algorithm.