Dynamic Mode Decomposition with Multiple Initial Conditions for Linearizing Nonlinear Partial Differential Equations

Dynamic Mode Decomposition (DMD) has emerged as a widely used data-driven tool for analyzing complex dynamical systems. While numerous extensions of DMD have been proposed to improve its applicability to nonlinear partial differential equations (PDEs), the standard formulation of DMD has not always been explored to its full potential. In this work, we demonstrate that DMD via ensembling can recover the spectra of the linearized dynamics, corresponding to the Koopman operator, restricted to linear observables of nonlinear PDEs in the neighborhood of their equilibria with good accuracy. By stacking trajectories from multiple initial conditions, the ensemble approach yields a more comprehensive representation of the underlying dynamics and captures richer spatiotemporal structures. In the local regime, the resulting model captures the essential dynamics without the need for nonlinear observable lifting, kernel-based formulations, or deep learning architectures. We investigated the role of the sampling distribution by relating sampling strategies to convergence rates associated with corresponding quadrature rules. We evaluated the ensemble formulation on several systems, including the convection-diffusion equation, Burgers’ equation, Fisher’s equation, the Kuramoto–Sivashinsky equation, as well as the Navier–Stokes dynamics corresponding to flow past a circular cylinder, and compared the learned spectra with their analytical counterparts. The learned spectra show excellent agreement with analytical counterparts, demonstrating that DMD via ensembling can reliably uncover the underlying dynamics of complex dynamical systems.