Nonlinear Model Order Reduction via Dynamic Mode Decomposition

Abstract

Abstract. We propose a new technique for obtaining reduced order models for nonlinear dynamical systems. Specifically, we advocate the use of the recently developed Dynamic Mode Decomposition (DMD), an equation-free method, to approximate the nonlinear term. DMD is a spatio-temporal matrix decomposition of a data matrix that correlates spatial features while simultaneously associating the activity with periodic temporal behavior. With this decomposition, one can obtain a fully reduced dimensional surrogate model and avoid the evaluation of the nonlinear term in the online stage. This allows for an impressive speed up of the computational cost, and, at the same time, accurate approximations of the problem. We present a suite of numerical tests to illustrate our approach and to show the effectiveness of the method in comparison to existing approaches.Key words. nonlinear dynamical systems, proper orthogonal decomposition, dynamic mode decomposition, data-driven modeling, reduced-order modeling, dimensionality reduction AMS subject classifications. 65L02, 65M02, 37M05, 62H251. Introduction. Reduced-order models (ROMs) are of growing importance in scientific computing as they provide a principled approach to approximating highdimensional PDEs with low-dimensional models. Indeed, the dimensionality reduction provided by ROMs help reduce the computational complexity and time needed to solve large-scale, engineering systems [24,3], enabling simulation based scientific studies not possible even a decade ago. One of the primary challenges in producing the low-rank dynamical system is efficiently projecting the nonlinearity of the governing PDEs (inner products) [2,6] on to the proper orthogonal decomposition (POD) [18,10,30] basis. This fact was recognized early on in the ROM community, and methods such as gappy POD [8,31,32] where proposed to more efficiently enable the task. More recently, the empirical interpolation method (EIM) [2], and the simplified discrete empirical interpolation method (DEIM) [6] for the proper orthogonal decomposition (POD) [18,10,30], have provided a computationally efficient method for discretely (sparsely) sampling and evaluating the nonlinearity. These broadly used and highly-successful methods ensure that the computational complexity of ROMs scale favorably with the rank of the approximation, even for complex nonlinearities. As an alternative to the EIM/DEIM architecture, we propose using the recently developed Dynamic Mode Decomposition (DMD) for producing low-rank approximations of the PDE nonlinearities. DMD provides a decomposition of data into spatio-temporal modes that correlates the data across spatial features (like POD), but also associates the correlated data to unique temporal Fourier modes, allowing for a computationally efficient regression of the nonlinear terms to a least-square fit linear dynamics approximation. We demonstrate that the POD-DMD method produces a viable ROM architecture, scaling favorable in computational efficiency relative to

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