A Statistical Machine Learning Approach for Adapting Reduced-Order Models using Projected Gaussian Process

The Proper Orthogonal Decomposition (POD) computes the optimal basis modes that span a low-dimensional subspace where the Reduced-Order Models (ROMs) reside. Because a governing equation is often parameterized by a set of parameters, challenges immediately arise when one would like to investigate how systems behave differently over the parameter space (in design, control, uncertainty quantification and real-time operations). In this case, the POD basis needs to be updated so as to adapt ROM that accurately captures the variation of a system's behavior over its parameter space. This paper proposes a Projected Gaussian Process (pGP) and formulate the problem of adapting 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 vector subspaces. A mapping is firstly found between the Euclidean space and the horizontal space of an orthogonal matrix that spans a reference subspace in the Grassmann Manifold. Then, 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 projected to the Grassmann Manifold that yields the optimal subspace for the new parameter. The proposed statistical learning approach allows us to optimally estimate model parameters given data (i.e., the prediction/interpolation becomes problem-specific), and quantify the uncertainty associated with the prediction. Numerical examples are presented to demonstrate the advantages of the proposed pGP for adapting POD basis against parameter changes.