An Online Manifold Learning Approach for Model Reduction of Dynamical Systems

Abstract

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.

A Certified Trust Region Reduced Basis Approach to PDE-Constrained Optimization

“…The bound (24) is standard; see, e.g., [27]. We follow an analogous procedure to show (25). We first note from (11) and (22) that the dual error satisfies (26) a(v, e du (µ); µ) = r du (v; µ) + 2d(e pr (µ), v).…”

“…Thus, approaches that tailor the surrogate model-in our case a projection-based reduced model-to the optimization problem are of particular interest. While a number of adaptation approaches have been proposed for projection-based reduced models (see, e.g., [10,20,23,25]), the challenge in the optimization setting is that regions of interest are not known a priori. Iterative approaches that adapt the reduced model as the optimization progresses have been considered in [5,24].…”

Nonlinear model reduction via a locally weighted POD method

“…On one hand, e k provides a lower bound for the reduced system, as ke k k 6 kek is always satisfied. On the other hand, for both elliptic PDEs [41] and parabolic PDEs [16,42] with a fixed time domain, if the Galerkin method is used to produce the reduced equation, then there respectively exists a constant C such that kek 6 C ke k k. Therefore, an upper bound of e is also related to e k . The first component e r of e k is directly related to sampling input parameters during the offline stage.…”

“…To improve computational efficiency with a fixed subspace dimension, the precomputed snapshots can be clustered, either through time domain partitions , space domain partitions , or parameter domain partitions . Either of these functionalities can be realized through local POD.…”

“…To obtain an accurate model, a subspace with a relatively high dimension should be used to construct the reduced system, especially when many solution modes exist for the whole domain in interest. Thus, global POD inevitably keeps redundant dimensions, which can lead to long online simulation times.To improve computational efficiency with a fixed subspace dimension, the precomputed snapshots can be clustered, either through time domain partitions [15,16], space domain partitions [17][18][19], or parameter domain partitions [20][21][22][23]. Either of these functionalities can be realized through local POD.…”