A time-varying inverse multi-physics-informed neural network paradigm for solving dimension-reduced probability density evolution equation

Large-scale structures, such as long-span bridges and high-rise buildings, often exhibit stochastic responses under excitations from winds and earthquakes. The dimension-reduced probability density evolution equation (DR-PDEE) provides a theoretically rigorous and computationally efficient framework for evaluating the transient probability density functions (PDFs) of such high-dimensional stochastic systems. However, effectively solving the DR-PDEE relies on accurately identifying intrinsic drift functions, which is challenging when only limited data are available. To address this issue, a time-varying inverse multi-physics-informed neural network (MPINN) for solving DR-PDEE from an inverse problem perspective is proposed in this study. The framework consists of two main components: a normalizing flow module that reconstructs noisy PDF data from limited deterministic analyses, and a noisy-data-encoded MPINN architecture with multiple parallel subnetworks tailored for distinct target variables that integrates physics and data concurrently. By extracting noisy PDF data from limited deterministic analyses, which is often neglected in conventional numerical methods, the proposed approach achieves the inverse identification of intrinsic drift functions and accurate prediction of response PDFs. Moreover, the data encoding enhances optimization efficiency by guiding the network to explore the solution space more effectively. Numerical results demonstrate that the proposed scheme achieves accurate long-term PDF predictions, improved identification of intrinsic drift functions, and high computational efficiency, requiring fewer representative deterministic analyses while maintaining accuracy comparable to path integral solutions (PIS) and potentially alleviating the small time-step limitations.