The evolution of the probability density function of a stochastic dynamical system over time can be described by a Fokker–Planck–Kolmogorov(FPK) equation, the solution of which determines the distribution of macrosc...The evolution of the probability density function of a stochastic dynamical system over time can be described by a Fokker–Planck–Kolmogorov(FPK) equation, the solution of which determines the distribution of macroscopic variables in the stochastic dynamic system. Traditional methods for solving these equations often struggle with computational efficiency and scalability, particularly in high-dimensional contexts. To address these challenges, this paper proposes a novel deep learning method based on prior knowledge with dual training to solve the stationary FPK equations. Initially, the neural network is pre-trained through the prior knowledge obtained by Monte Carlo simulation(MCS). Subsequently, the second training phase incorporates the FPK differential operator into the loss function, while a supervisory term consisting of local maximum points is specifically included to mitigate the generation of zero solutions. This dual-training strategy not only expedites convergence but also enhances computational efficiency, making the method well-suited for high-dimensional systems. Numerical examples, including two different two-dimensional(2D), six-dimensional(6D), and eight-dimensional(8D) systems, are conducted to assess the efficacy of the proposed method. The results demonstrate robust performance in terms of both computational speed and accuracy for solving FPK equations in the first three systems. While the method is also applicable to high-dimensional systems, such as 8D, it should be noted that computational efficiency may be marginally compromised due to data volume constraints.展开更多
Stochastic dynamic analysis of the nonlinear system is an open research question which has drawn many scholars'attention for its importance and challenge.Fokker–Planck–Kolmogorov(FPK)equation is of great signifi...Stochastic dynamic analysis of the nonlinear system is an open research question which has drawn many scholars'attention for its importance and challenge.Fokker–Planck–Kolmogorov(FPK)equation is of great significance because of its theoretical strictness and computational accuracy.However,practical difficulties with the FPK method appear when the analysis of multi-degree-offreedom(MDOF)with more general nonlinearity is required.In the present paper,by invoking the idea of equivalence of probability flux,the general high-dimensional FPK equation related to MDOF system is reduced to one-dimensional FPK equation.Then a cell renormalized method(CRM)which is based on the numerical reconstruction of the derived moments of FPK equation is introduced by coarsening the continuous state space into a discretized region of cells.Then the cell renormalized FPK(CR-FPK)equation is solved by difference method.Three numerical examples are illustrated and the effectiveness of proposed method is assessed and verified.展开更多
In this paper, the method proposed recently by the author for the solution of probability density function (PDF) of nonlinear stochastic systems is presented in detail and extended for more general problems of stochas...In this paper, the method proposed recently by the author for the solution of probability density function (PDF) of nonlinear stochastic systems is presented in detail and extended for more general problems of stochastic differential equations (SDE), therefore the Fokker Planck Kolmogorov (FPK) equation is expressed in general form with no limitation on the degree of nonlinearity of the SDE, the type of δ correlated excitations, the existence of multiplicative excitations, and the dimension of SDE or FPK equation. Examples are given and numerical results are provided for comparing with known exact solution to show the effectiveness of the method.展开更多
基金Project supported by the National Natural Science Foundation of China (Grant No.12172226)。
文摘The evolution of the probability density function of a stochastic dynamical system over time can be described by a Fokker–Planck–Kolmogorov(FPK) equation, the solution of which determines the distribution of macroscopic variables in the stochastic dynamic system. Traditional methods for solving these equations often struggle with computational efficiency and scalability, particularly in high-dimensional contexts. To address these challenges, this paper proposes a novel deep learning method based on prior knowledge with dual training to solve the stationary FPK equations. Initially, the neural network is pre-trained through the prior knowledge obtained by Monte Carlo simulation(MCS). Subsequently, the second training phase incorporates the FPK differential operator into the loss function, while a supervisory term consisting of local maximum points is specifically included to mitigate the generation of zero solutions. This dual-training strategy not only expedites convergence but also enhances computational efficiency, making the method well-suited for high-dimensional systems. Numerical examples, including two different two-dimensional(2D), six-dimensional(6D), and eight-dimensional(8D) systems, are conducted to assess the efficacy of the proposed method. The results demonstrate robust performance in terms of both computational speed and accuracy for solving FPK equations in the first three systems. While the method is also applicable to high-dimensional systems, such as 8D, it should be noted that computational efficiency may be marginally compromised due to data volume constraints.
文摘Stochastic dynamic analysis of the nonlinear system is an open research question which has drawn many scholars'attention for its importance and challenge.Fokker–Planck–Kolmogorov(FPK)equation is of great significance because of its theoretical strictness and computational accuracy.However,practical difficulties with the FPK method appear when the analysis of multi-degree-offreedom(MDOF)with more general nonlinearity is required.In the present paper,by invoking the idea of equivalence of probability flux,the general high-dimensional FPK equation related to MDOF system is reduced to one-dimensional FPK equation.Then a cell renormalized method(CRM)which is based on the numerical reconstruction of the derived moments of FPK equation is introduced by coarsening the continuous state space into a discretized region of cells.Then the cell renormalized FPK(CR-FPK)equation is solved by difference method.Three numerical examples are illustrated and the effectiveness of proposed method is assessed and verified.
文摘In this paper, the method proposed recently by the author for the solution of probability density function (PDF) of nonlinear stochastic systems is presented in detail and extended for more general problems of stochastic differential equations (SDE), therefore the Fokker Planck Kolmogorov (FPK) equation is expressed in general form with no limitation on the degree of nonlinearity of the SDE, the type of δ correlated excitations, the existence of multiplicative excitations, and the dimension of SDE or FPK equation. Examples are given and numerical results are provided for comparing with known exact solution to show the effectiveness of the method.
