Model Reduction for Nonlinear Dynamical Systems With Parametrized Boundary and Initial Conditions Using Support Vector Machine on a Manifold

This work introduces a model reduction approach for parametrized boundary and initial conditions using supervised learning techniques. The Grassmann distance is employed to measure differences between parameter values based on the angles of solution subspaces. The K-medoids clustering is then utilized to group the parameter values according to these Grassmann distances. The support vector machine (SVM) algorithm is performed to train and create an automated classifier that selects the most appropriate reduced basis sets corresponding to new parameter values of interest, which may not be in the training set. By using kernels, SVM algorithm can further handle nonlinear classification problems effectively. Model reduction techniques based on proper orthogonal decomposition and discrete empirical interpolation are finally applied to obtain the reduced systems. Numerical tests are performed on nonlinear differential equations, including the Burger’s and sine-Gordon equations with parametrized boundary and initial conditions, and compared with the standard global basis approach. The proposed method is shown to be accurate and efficient in these numerical tests, including when the solution behavior is sensitive to changes in the parameterized initial and boundary conditions over the training set. In addition, the generalization capability of the proposed framework is numerically investigated through unseen parameters beyond the training range and time intervals extended from training temporal domain. The reduced models remain stable and accurate for temporal extrapolation and maintain reliable performance under moderate parameter deviations.