Abstract
Data-driven mathematical methods are increasingly important for characterizing complex systems across the physical, engineering, and biological sciences. These methods aim to discover and exploit a relatively small subset of the full highdimensional state space where low-dimensional models can be used to describe the evolution of the system. Emerging dimensionality reduction methods, such as the dynamic mode decomposition (DMD) and its Koopman generalization, have garnered attention due to the fact that they can (i) discover low-rank spatio-temporal patterns of activity, (ii) embed the dynamics in the subspace in an equation-free manner (i. e., the governing equations are unknown), unlike Galerkin projection onto proper orthogonal decomposition modes, and (iii) provide approximations in terms of linear dynamical systems, which are amenable to simple analysis techniques. The selection of observables (features) for the DMD/Koopman architecture can yield accurate low-dimensional embeddings for nonlinear partial differential equations (PDEs) while limiting computational costs. Indeed, a good choice of observables, including time delay embeddings, can often linearize the nonlinear manifold by making the spatiotemporal dynamics weakly nonlinear. In addition to DMD/Koopman decompositions, coarse-grained models for spatio-temporal systems can also be discovered using the sparse identification of nonlinear dynamics (SINDy) algorithm which allows one to construct reduced-order models in low-dimensional embeddings. These methods can be used in a nonintrusive, equation-free manner for improved computational performance on parametric PDE systems.
