Physics Informed Deep Learning Method To Solve The FPK Equation

Fokker-Planck-Kolmogorov(FPK)equation is a set of partial differential equations that describes the transfer probability density function of the response of nonlinear system under random input,such as the response of transmission tower system under random wind load.Generally,only the numerical solution can be obtained for this equation because only the stable analytical solution in a few cases can be obtained under strict constraints.In this paper,we use the deep learning neural network based on physical information to solve several typical FPK equations,and verify the effectiveness of the method by analyzing the actual model.First of all,this paper summarizes the current research overview of various commonly used numerical solutions of FPK equation and deep learning based on physical information to solve partial differential equations.This paper briefly introduces the principle of deep learning neural network based on physical information for solving partial differential equations and the selection of parameters of deep learning model to establish a deep learning neural network for solving high dimensional stationary FPK equation.The algorithm is applied to the random response analysis of wind-induced vibration of tower,and is to solve the response of tower structure under Gaussian white noise parameter excitation.The effectiveness of the method is verified by comparing with Monte Carlo simulation results.Secondly,the method of solving the stable FPK equation based on the physical information deep learning neural network is extended to the multi-dimensional state space.With the increase of dimension of state space,it is not suitable to adopt grid type selection rule.In order to improve the efficiency and accuracy of calculation,the selection method of data points is optimized.The number theory method is used to select points,and the number theory selection method in unit hypercube is selected according to the spherical symmetry or approximate radiation attenuation of probability density,which is that the probability space selection in the super sphere is reserved,and the number of point selection is greatly reduced.By solving the probability density function of the stationary response of coupled Duffin oscillator excited by Gaussian white noise,the influence of the number of number theory points,hidden layers and number of neurons on the calculation accuracy is discussed in detail.Therefore,the parameter selection of depth learning algorithm for solving multi-dimensional stationary FPK equation is proposed.The results of the depth learning solution are compared with the results of the exact solution or the finite difference method,which verifies the efficiency and accuracy of the method used in this paper to solve the multi-dimensional FPK equation.Finally,by combining the probability density evolution theory developed in recent years with the FPK equation,the reduced dimension equivalent flux probability density evolution equation is derived,and then the reduced dimension FPK type equation is obtained by introducing the equivalent drift coefficient.In this paper,a conditional mean function method for constructing equivalent drift coefficients is selected.The accuracy and efficiency of the proposed method are discussed by solving the stationary velocity response probability density function of a single degree of freedom Duffing oscillator and a nine-story shear type linear frame structure,and the effectiveness of the proposed method is verified.