Data-driven linear analysis of dynamical systems via nonlinearity-subtracted dynamic mode decomposition

Abstract

The Dynamic Mode Decomposition (DMD) has been consolidated as a basic tool for data-driven analysis of dynamical systems, allowing simultaneous identification of coherent structures and their dynamics from time-resolved measurements. However, with a linear regression at its core, DMD is unable to produce accurate models from recordings of dynamics that are inherently nonlinear, such as the response to large perturbations and the evolution on chaotic attractors. Recent approaches attempt to simultaneously fit the linear and nonlinear contributions to the dynamics by performing a regression onto a physically motivated model structure. However, although the resulting nonlinear models can produce accurate short-term predictions, their linearization does not necessarily agree with that of the original system. In this work, we introduce a novel data-driven method --- nonlinearity-subtracted DMD (NSDMD) --- that focuses on producing an accurate linearization of a system when the nonlinear contribution to its dynamics are available while the linear part is not. This scenario is encountered, for example, when the nonlinear terms in the governing equations are known, while the linear operator contains uncertain material properties or it accounts for the closure of unresolved dynamics. This also arises when the data is generated by a black-box simulation code that is able to output the nonlinearity, but not the action of the linear operator on the snapshots. NSDMD leverages data snapshots of the nonlinearity to explicitly account for the purely nonlinear contributions to the dynamics and formulate a regression problem that finds a low-rank approximation of the underlying linear operator. We demonstrate the approach on several numerical examples, showcasing its improved capabilities for data-driven linear analysis of chaotic, partially observed, advection-dominated, and high-dimensional dynamics.