Stochastic Subspace Via Probabilistic Principal Component Analysis for Characterizing Model Error

Abstract

This paper proposes a probabilistic model of subspaces based on the probabilistic principal component analysis (PCA). Given a sample of vectors in the embedding space—commonly known as a snapshot matrix—this method uses quantities derived from the probabilistic PCA to construct distributions of the sample matrix, as well as the principal subspaces. It is applicable to projection-based reduced-order modeling methods, such as proper orthogonal decomposition and related model reduction methods. The stochastic subspace thus constructed can be used, for example, to characterize model-form uncertainty in computational mechanics. The proposed method has multiple desirable properties: (1) it is naturally justified by the probabilistic PCA and has analytic forms for the induced random matrix models; (2) it satisfies linear constraints, such as boundary conditions of all kinds, by default; (3) it has only one hyperparameter, which significantly simplifies training; and (4) its algorithm is very easy to implement. We compare the proposed method with existing approaches in a low-dimensional visualization example and a parametric static problem, and demonstrate its performance in a dynamics model of a space structure.

Keywords: Model error, Model-form uncertainty, Stochastic reduced-order modeling, Probabilistic principal component analysis