Enhancing Reduced-Order Models in High-Dimensional Engineering Applications through Gradient Boosting Tree

Abstract

Addressing forward problems with high-fidelity models like Computational Fluid Dynamics (CFD) and Finite Element Analysis (FEA) presents significant challenges due to their computational demands from high-dimensional spatial discretization. These models are expensive, requiring powerful solvers and extensive computational resources to achieve precise results. Additionally, their sensitivity to parameter variations complicates the accurate development of Reduced-Order Models (ROMs). Projection-based methods such as Proper Orthogonal Decomposition (POD) are typically employed to extract optimal basis modes from snapshot data for ROMs. However, these methods struggle to robustly quantify uncertainty, affecting their reliability and accuracy in practical applications.My research explores a novel integration of XGBoost, known for its predictive accuracy and capability to handle complex relationships, with an innovative mapping technique between a multi-dimensional Euclidean space and the Grassmann Manifold, which represents low-dimensional subspaces. This mapping includes an injectivity constraint to enhance optimization, making XGBoost suitable with minor modifications. The approach involves mapping the subspaces from the Grassmann Manifold back to the Euclidean space to formulate a multivariate XGBoost regression. When introduced to a new parameter, this method projects the corresponding vector onto the Grassmann Manifold, facilitating the identification of the optimal subspace or POD basis. This process allows for a detailed understanding of how variations in settings affect the optimal representation in ROMs and quantifies the confidence in these insights, addressing limitations in traditional projection-based methods.