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.

Regression Trees on Grassmann Manifold for Adapting Reduced-Order Models

Low dimensional and computationally less expensive Reduced-Order Models (ROMs) have been widely used to capture the dominant behaviors of high-dimensional systems. A ROM can be obtained, using the well-known Proper Orthogonal Decomposition (POD), by projecting the full-order model to a subspace spanned by modal basis modes which are learned from experimental, simulated or observational data, i.e., training data. However, the optimal basis can change with the parameter settings. When a ROM, constructed using the POD basis obtained from training data, is applied to new parameter settings, the model often lacks robustness against the change of parameters in design, control, and other real-time operation problems. This paper proposes to use regression trees on Grassmann Manifold to learn the mapping between parameters and POD bases that span the low-dimensional subspaces onto which full-order models are projected. Motivated by the fact that a subspace spanned by a POD basis can be viewed as a point in the Grassmann manifold, we propose to grow a tree by repeatedly splitting the tree node to maximize the Riemannian distance between the two subspaces spanned by the predicted POD bases on the left and right daughter nodes. Five numerical examples are presented to comprehensively demonstrate the performance of the proposed method, and compare the proposed tree-based method to the existing interpolation method for POD basis and the use of global POD basis. The results show that the proposed tree-based method is capable of establishing the mapping between parameters and POD bases, and thus adapt ROMs for new parameters.