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This paper applies a constrained variant of gradient boosting to predict reduced-order model subspaces across parameter changes, a practical problem in computational engineering. It reports lower reconstruction errors than Gaussian-process and Grassmann-interpolation baselines on four test problems. For ML readers, the main points are the Grassmann-to-Euclidean mapping and the norm constraint that preserves injectivity, but the evidence rests on a small set of benchmark PDE problems, so generalisation to other settings remains to be shown.
Constrained Extreme Gradient Boosting for Parametric Proper Orthogonal Decomposition Subspace Prediction
arXiv:2605.04130v3 Announce Type: replace Abstract: High-fidelity simulations such as computational fluid dynamics and finite element analysis are essential for modelling complex engineering systems but remain prohibitively expensive for tasks including parametric studies, optimisation and real-time control. Projection-based reduced-order models (ROMs) alleviate this cost by projecting full-order dynamics onto low-dimensional subspaces. On the other hand, projection-based ROMs can become less robust under parameter variation, and one approach is to adapt the projection basis (that is, the subspace) as the parameters change. We propose constrained extreme gradient boosting (cXGBoost) for predicting the proper orthogonal decomposition (POD) basis at given parameter settings. In particular, given the observed parameter-subspace pairs, we map the subspaces from the Grassmann manifold to a Euclidean space. Then, cXGBoost is trained in this Euclidean space with a constraint that the norm of the predicted vectors is bounded to ensure the injectivity of the mapping between the Euclidean space and the Grassmann manifold. On four parametric problems, a cylinder wake, Kolmogorov flow, the Kuramoto-Sivashinsky equation and a forced beam, cXGBoost has lower mean reconstruction errors than existing approaches, namely the projected Gaussian process and Grassmann interpolation.
Abstract:High-fidelity simulations such as computational fluid dynamics and finite element analysis are essential for modelling complex engineering systems but remain prohibitively expensive for tasks including parametric studies, optimisation and real-time control. Projection-based reduced-order models (ROMs) alleviate this cost by projecting full-order dynamics onto low-dimensional subspaces. On the other hand, projection-based ROMs can become less robust under parameter variation, and one approach is to adapt the projection basis (that is, the subspace) as the parameters change. We propose constrained extreme gradient boosting (cXGBoost) for predicting the proper orthogonal decomposition (POD) basis at given parameter settings. In particular, given the observed parameter-subspace pairs, we map the subspaces from the Grassmann manifold to a Euclidean space. Then, cXGBoost is trained in this Euclidean space with a constraint that the norm of the predicted vectors is bounded to ensure the injectivity of the mapping between the Euclidean space and the Grassmann manifold. On four parametric problems, a cylinder wake, Kolmogorov flow, the Kuramoto-Sivashinsky equation and a forced beam, cXGBoost has lower mean reconstruction errors than existing approaches, namely the projected Gaussian process and Grassmann interpolation.
Machine Learning (cs.LG)
arXiv:2605.04130 [cs.LG]
(or arXiv:2605.04130v3 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2605.04130 Focus to learn more arXiv-issued DOI via DataCite
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