Adapting Projection-Based Reduced-Order Models using Projected Gaussian Process

Projection-based model reduction is among the most widely adopted methods for constructing parametric Reduced-Order Models (ROM). Utilizing the snapshot data from solving full-order governing equations, the Proper Orthogonal Decomposition (POD) computes the optimal basis modes that represent the data, and a ROM can be constructed in the low-dimensional vector subspace spanned by the POD basis. For parametric governing equations, a potential challenge arises when there is a need to update the POD basis to adapt ROM that accurately capture the variation of a system's behavior over its parameter space (in design, control, uncertainty quantification, digital twins applications, etc.). In this paper, we propose a Projected Gaussian Process (pGP) and formulate the problem of adapting the POD basis as a supervised statistical learning problem, for which the goal is to learn a mapping from the parameter space to the Grassmann manifold that contains the optimal subspaces. A mapping is firstly established between the Euclidean space and the horizontal space of an orthogonal matrix that spans a reference subspace in the Grassmann manifold. A second mapping from the horizontal space to the Grassmann manifold is established through the Exponential/Logarithm maps between the manifold and its tangent space. Finally, given a new parameter, the conditional distribution of a vector can be found in the Euclidean space using the Gaussian Process (GP) regression, and such a distribution is then projected to the Grassmann manifold that enables us to predict the optimal subspace for the new parameter. As a statistical learning approach, the proposed pGP allows us to optimally estimate (or tune) the model parameters from data and quantify the statistical uncertainty associated with the prediction. The advantages of the proposed pGP are demonstrated by numerical experiments.