讲座摘要 This talk will present a novel non-linear model reduction method: Probabilistic Manifold Decomposition (PMD), which provides a powerful framework for constructing non-intrusive reduced-order models (ROMs) by embedding a high-dimensional system into a low-dimensional probabilistic manifold and predicting the dynamics. Through explicit mappings, PMD captures both linearity and non-linearity of the system. A key strength of PMD lies in its predictive capabilities, allowing it to generate stable dynamic states based on embedded representations.
The method also offers a mathematically rigorous approach to analyze the convergence of linear feature matrices and low-dimensional probabilistic manifolds, ensuring that sample-based approximations converge to the true data distributions as sample sizes increase. These properties, combined with its computational efficiency, make PMD a versatile tool for applications requiring high accuracy and scalability, such as fluid dynamics simulations and other engineering problems. By preserving the geometric and probabilistic structures of the high-dimensional system, PMD achieves a balance between computational speed, accuracy, and predictive capabilities, positioning itself as a robust alternative to the traditional model reduction methods such as DMD and POD.

Xiao Dunhui is a professor at the School of Mathematical Sciences, Tongji University. He also serves as the Deputy Director (on secondment) of the Information Office at Tongji University, the Director of the Computational Mathematics Department, a standing member of the 11th Council of the Computational Mathematics Branch of the Chinese Mathematical Society, a standing committee member of the first AI Application Committee of the Chinese Society for Rock Mechanics and Engineering, and a member of the Shanghai CSIAM. He previously worked at the Department of Earth Science and the Data Science Institute at Imperial College London, and the Zienkiewicz Centre for Computational Engineering at Swansea University, one of the birthplaces of the finite element method.