DR-PDE-Net: A time-varying inverse multi-physics-informed neural network paradigm for solving dimension-reduced probability density evolution equation in noisy data regimes

The Dimension-Reduced Probability Density Evolution Equation (DR-PDEE) provides a promising tool for evaluating the evolution of probability density in high-dimensional stochastic dynamical systems. However, solving DR-PDEE relies heavily on accurately identifying unknown spatio-temporal-dependent intrinsic drift coefficients, which drive the evolution of probability density. Recognizing the potential to extract noisy information about Probability Density Functions (PDFs) from limited representative samples, a novel time-varying inverse Multi-Physics-Informed Neural Network (MPINN) framework, termed DR-PDE-Net, is proposed in this study to simultaneously predict the evolution of both intrinsic drift coefficients and response PDFs in noisy data regimes. DR-PDE-Net comprises two essential components: the use of normalizing flow techniques to obtain time-varying noisy response PDF data from limited samples, and a noisy-data-encoded MPINN featuring multiple outputs and parallel subnetworks dedicated to different target variables that integrates physics and data concurrently. By encoding noisy PDF data into the neural network architecture, domain knowledge and multiple physical constraints governing intrinsic drift coefficients and DR-PDEE can be seamlessly integrated to solve DR-PDEE from an inverse problem perspective. This new paradigm can leverage established relationships and principles to achieve more precise predictions, eliminating the need for separately estimating unknown spatio-temporal-dependent coefficients. Data encoding further empowers the network to efficiently explore the solution space and optimize more effectively. Illustrative examples and comparisons demonstrate the superior performance of DR-PDE-Net, showcasing accurate long-term predictions and improved estimates of intrinsic drift coefficients compared to numerical regression techniques. Additionally, DR-PDE-Net offers efficiency benefits for large-scale structures involving randomness both from structural parameters and excitations, providing comparable accuracy to path integral solutions (PIS) while requiring fewer sample points and potentially easing the constraints associated with the small-time step requirements inherent in PIS.