Multi-output multi-physics-informed neural network for learning dimension-reduced probability density evolution equation with unknown spatio-temporal-dependent coefficients.

• A novel framework called MO-MPINN is proposed to solve the DR-PDEE with unknown spatio-temporal-dependent coefficients. • The proposed MO-MPINN can simultaneously predict the evolution of time-varying coefficients and PDF of QoI. • A network architecture with multiple parallel subnetworks with each one corresponding to a different output is employed to reduce the training complexity. • Multiple physical laws are embedded in the loss function to ensure an accurate representation of the underlying principles. • The presented method can be applied to high-dimensional linear and nonlinear systems involving double randomness in structural parameters and excitations. The Dimension-Reduced Probability Density Evolution Equation (DR-PDEE) offers a promising approach for evaluating probability density evolution in stochastic dynamical systems. Physics-Informed Neural Networks (PINNs) are well-suited for solving DR-PDEE due to their ability to encode physical laws into the learning process. However, challenges arise from the spatio-temporal-dependence of unknown intrinsic drift and diffusion coefficients, which drive DR-PDEE, along with their derivatives. To address these challenges, a novel framework called Multi-Output Multi-Physics-Informed Neural Network (MO-MPINN) is proposed to predict the evolution of time-varying coefficients and response probability density simultaneously. MO-MPINN features multiple output neurons, eliminating the necessity for distinct identification of unknown spatio-temporal-dependent coefficients separately. It uses parallel subnetworks to reduce training complexity and embeds multiple physical laws in the loss function to ensure an accurate representation of the underlying principles. Leveraging automatic differentiation, MO-MPINN efficiently computes derivatives of coefficients without resorting to numerical differentiation. The framework is applicable to high-dimensional stochastic nonlinear systems with double randomness in structural parameters and excitations. Several structures are presented to validate the performance of the MO-MPINN. This study introduces a new paradigm for solving partial differential equations involving differentiation of spatio-temporal-dependent coefficients. [ABSTRACT FROM AUTHOR]