Non-Intrusive Parametric Reduced Order Models with High-Dimensional Inputs via Gradient-Free Active Subspace
Dushhyanth Rajaram, Raphael H. Gautier, Christian Perron, Olivia J. Pinon-Fischer, Dimitri N. Mavris
Abstract
This work presents the development of a method for the construction of parametric, interpolation-based non-intrusive Reduced Order Models (ROMs) for predicting field outputs when the input parameter space is high-dimensional. The Proper Orthogonal Decomposition (POD) is used to decrease the dimensionality of the field outputs. Instead of directly approximating the map between the input parameters and the low-dimensional POD subspace, the proposed method trains Gaussian processes to simultaneously discover a projection of the inputs onto a low-dimensional subspace and learn a map between the low-dimensional input subspace and the POD subspace. The proposed technique is gradient-free i.e., it exclusively relies on input-output pairs. Additionally, this work proposes a novel method to train the Gaussian processes on product manifolds to enforce orthogonality in the estimated basis of the low-dimensional input subspace. Experiments on several scalar canonical functions and engineering problems show that the Gaussian process regression successfully discovers the low-dimensional subspace and yields good predictive performance. Construction of a ROM to emulate fields for a canonical elliptic PDE problem using the developed method shows satisfactory predictive performance in addition to successful discovery of low-dimensional input subspaces in the POD subspace.