A data-driven, physics-informed framework for forecasting the spatiotemporal evolution of chaotic dynamics with nonlinearities modeled as exogenous forcings

Abstract We introduce a data-driven, linear method for the spatiotemporal prediction of high-dimensional and chaotic dynamical systems. In this method, the observables are vector-valued and delay-embedded, and the nonlinearities are treated as external forcings. The proper representation of the nonlinear terms is found in a physics-informed way or a purely data-driven fashion, depending on whether any knowledge of governing dynamics is available or not. The unknown matrices appearing in the method are found using the dynamic mode decomposition with control technique. The distinctive features of the method enable it to accurately capture the system's dynamically important unstable modes while suppressing their unbounded growth via the included nonlinearities. The predictive capabilities of the method are demonstrated for well-known prototypes of chaotic dynamics such as the Kuramoto-Sivashinsky equation and Lorenz-96 system, for which the method predictions are accurate for several Lyapunov timescales. Similar performance is shown for two-dimensional lid-driven cavity flows at high Reynolds numbers.