Projection-based model reduction is among the most widely adopted approaches for constructing parametric Reduced-Order Models (ROMs). Utilizing snapshot data from solving full-order governing equations, the Proper Orthogonal Decomposition (POD) computes the optimal basis modes that span a low-dimensional subspace where the ROM resides. Challenges arise when one would like to investigate how systems behave differently over the parameter space (in design, diagnosis, control, uncertainty quantification and real-time operations). In this case, the optimal basis needs to be efficiently updated to adapt ROM that accurately captures the variation of a system's behavior over its parameter space. In this talk, we introduce a Projected Gaussian Process (pGP) model and formulate the problem of adapting POD basis as a supervised statistical subspace learning problem, for which the goal is to learn a mapping
(injective) from the parameter space to the Grassmann Manifold that contains the optimal vector subspaces. To establish such a relationship, a mapping is 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 GP regression, and such a distribution is then projected to the Grassmann Manifold that enables us to find the optimal subspace, i.e., POD basis, for the new parameter. Compared with existing interpolation method, the proposed statistical learning approach allows us to optimally estimate (or tune) 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.

Dr. Xiao Liu is the David M. McKenney Family Associate Professor at the H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology. His research focuses on data-driven approaches for scientific and engineering applications, and his work have been published on both Industrial Engineering and Statistics journals; e.g., JASA, Technometrics, IISE Transactions, AOAS, etc. He served as the President of the Data Analytics & Information Systems division of IISE, and the Program co-Chair for the 2025 IISE Annual Conference & Expo. Before joining GT, he held positions at the National University of Singapore, IBM Thomas J. Watson Research Center, and University of Arkansas.