A decoupled approach for determination of the joint probability density function of a high-dimensional nonlinear stochastic dynamical system via the probability density evolution method

Abstract

The joint probability density function (PDF) of multiple response processes of a system is a crucial topic in the fields of science and engineering. It is adopted to describe the dependent uncertainty propagation in complex stochastic dynamical systems, including the displacement and velocity of one degree of freedom (DOF), and the nonlinear probabilistic dependencies between multiple variables. However, the partial differential equations (PDEs) governing the time evolution of joint PDFs for multiple stochastic processes are often high-dimensional and coupled, making them analytically or numerically intractable for systems with high dimensionality. The probability density evolution method (PDEM) has provided a new perspective from the standpoint of random events description of the principle of preservation of probability. For an n-dimensional system, if merely the joint PDF of m processes is needed, where m is not larger than n, then it is sufficient to solve the generalized density evolution equation (GDEE), i.e., the Li–Chen equation, in m dimensions. This approach is particularly convenient for cases where n is large while m is small. However, for cases where m is not less than two, it is still challenging to avoid solving high-dimensional PDEs. This paper proposes a new method for the determination of joint PDF of multiple responses based on the principle of preservation of probability. To capture the joint PDF of m processes, the m-dimensional PDE can be transformed into m one-dimensional PDEs, significantly reducing the computational complexity of numerical implementation. The proposed method gives an enhanced version to the PDEM, enabling it to not only determine the individual probabilistic response, but also efficiently determine the dependent probabilistic distribution of multiple responses of high-dimensional stochastic dynamical systems without solving the coupling PDE. It reduces the computational cost from exponential growth of m powers to linear growth. Some numerical examples are presented to verify the effectiveness of the proposed method. Finally, problems for future investigations are discussed.