Stationary response probability density of nonlinear random vibrating systems: a data-driven method

A data-driven method is established to derive the (approximately) analytical expression of the stationary response probability density of nonlinear random vibrating system, which explicitly includes system features and intensity of excitation. The stationary response probability density is first assumed as an exponential form by using the principle of maximum entropy. Through the rule of dimensional consistency, the power of exponential function is expressed as a linear combination of a set of nondimensional parameter clusters which are constituted by system features, intensity of excitation, and state variables. By comparing the power of exponential function with the approximate logarithm probability density evaluated from simulated data statistically, the determination of unknown coefficients comes down to the solution of (overdetermined) simultaneous linear algebraic equations. The data-driven method rediscovers the exact stationary response probability density of random-excited Duffing oscillator and derives an approximately analytical expression of stationary response probability density of van der Pol system from the simulated data of six cases with different values of system features and intensity of excitation. This data-driven method is a unique method which can explicitly include the information of system and excitation in the analytical expression of stationary response probability density. It avoids the solution of simultaneous nonlinear algebraic equations encountered in the maximum entropy method and closure methods and, in the meanwhile, avoids the sophisticated selection of weighting functions in closure methods.