Evolutionary probability density reconstruction of stochastic dynamic responses based on physics-aided deep learning

Probability density evolution is the vital probabilistic information for stochastic dynamic system. However, it may face big challenges when using numerical methods to solve the generalized probability density evolution equation (GDEE) of complicated real-world system under complex excitations. Recently, physics-informed deep learning has gained popularity in solving partial differential equations due to the superior approximation and generalization capabilities, which opens a promising intelligent way to solve the GDEE. In this work, a mesh-free learning model, i.e., Phy-EPDNN, is proposed for evolutionary probability density (EPD) reconstruction, where a normalized GDEE is derived and embedded in the developed network as the prior physical knowledge. The computational domain can be customized by users on demand, in which the GDEE is not required to be solved in the whole computational domain as that in numerical methods. There is no need of boundary and initial conditions for the proposed model. A data augmentation method is also proposed to obtain sufficient collection data for supervised learning. Three specially selected analytical models and two practical scenarios verify the potential applicability of the proposed general framework for EPD reconstruction. Parametric analysis is performed to discuss the influence of the major network parameters on reconstruction performance.