A Comparative Study of Subspace Measures for Parametric Model Reduction Using K-Medoids Clustering and Neural Networks: A Case Study for Burgers’ Equation

Abstract

This work investigates the performance of three distance measures including Grassmann, Binet-Cauchy and Chordal distances in the complexity reduction of parameterized nonlinear dynamical systems using K-medoids clustering and neural network. The Grassmann distance provides a geometric perspective by measuring the subspace angles, while the Binet-Cauchy distance utilizes determinants to measure volume distortion between subspaces. The Chordal distance, on the other hand, considers a straightforward metric that uses the Euclidean distance between points on a unit sphere. Each of these distances is employed in the K-medoids clustering process to construct a dictionary of projection basis sets used in reduced-order modeling. The neural network is then trained to automatically select the most suitable basis sets in the dictionary for a given parameter vector. The numerical experiments are demonstrated using the parameterized Burgers’ equation, where the governing partial differential equation, initial conditions and boundary conditions are all parameter-dependent. The Grassmann distance is shown to give the most accurate result across different parameter settings when compared with the Binet-Cauchy and Chordal distances.