An Online Manifold Learning Approach for Model Reduction of Dynamical Systems

Abstract

This article discusses a newly developed online manifold learning method, subspace iteration using reduced models (SIRM), for the dimensionality reduction of dynamical systems. This method may be viewed as subspace iteration combined with a model reduction procedure. Specifically, starting with a test solution, the method solves a reduced model to obtain a more precise solution, and it repeats this process until sufficient accuracy is achieved. The reduced model is obtained by projecting the full model onto a subspace that is spanned by the dominant modes of an extended data ensemble. The extended data ensemble in this article contains not only the state vectors of some snapshots of the approximate solution from the previous iteration but also the associated tangent vectors. Therefore, the proposed manifold learning method takes advantage of the information of the original dynamical system to reduce the dynamics. Moreover, the learning procedure is computed in the online stage, as opposed to being computed offline, which is used in many projection-based model reduction techniques that require prior calculations or experiments. After providing an error bound of the classical POD-Galerkin method in terms of the projection error and the initial condition error, we prove that the sequence of approximate solutions converge to the actual solution of the original system as long as the vector field of the full model is locally Lipschitz on an open set that contains the solution trajectory. Good accuracy of the proposed method has been demonstrated in two numerical examples, from a linear advection-diffusion equation to a nonlinear Burgers equation. In order to save computational cost, the SIRM method is extended to a local model reduction approach by partitioning the entire time domain into several subintervals and obtaining a series of local reduced models of much lower dimensionality. The accuracy and efficiency of the local SIRM are shown through the numerical simulation of the Navier--Stokes equation in a lid-driven cavity flow problem.