Constrained Extreme Gradient Boosting for Adapting Reduced-Order Models

TL;DR

This paper introduces cXGBoost, a geometric constrained gradient boosting method for predicting parameter-dependent reduced bases, improving adaptive ROM performance in complex simulations.

Contribution

It proposes a novel constrained ensemble learning framework that leverages Grassmann manifold geometry for accurate, robust basis prediction in reduced-order modeling.

Findings

cXGBoost accurately predicts POD bases across parameter variations.

The method maintains geometric validity of subspaces through norm constraints.

Numerical examples demonstrate robustness in nonlinear regimes.

Abstract

High-fidelity simulations, such as computational fluid dynamics and finite element analysis, are essential for modeling complex engineering systems but are often prohibitively expensive for tasks including parametric studies, optimization, and real-time control. Projection-based reduced-order models (ROMs) alleviate this cost by projecting the governing dynamics onto low-dimensional subspaces. However, their performance can deteriorate under parameter variation, motivating the need for adaptive basis construction. In this work, we propose a constrained ensemble learning framework, termed Constrained Extreme Gradient Boosting (cXGBoost), for predicting Proper Orthogonal Decomposition (POD) bases as functions of system parameters. The approach leverages a geometric representation of subspaces on the Grassmann manifold, which are mapped to a Euclidean space to enable efficient regression using gradient boosting trees. A norm constraint is imposed during training to ensure the validity of the inverse mapping and preserve the geometric structure of the predicted subspaces. The proposed method is evaluated on four numerical examples, including fluid dynamics and wave propagation problems, demonstrating its ability to accurately predict parameter-dependent bases while maintaining robustness across nonlinear regimes. These results highlight the potential of combining geometric learning with constrained ensemble methods for scalable and reliable reduced-order modeling of high-dimensional parametric systems.