Koopman Operator Inspired Nonlinear System Identification

Koopman analysis provides a general theoretical underpinning for widely used dynamic mode de-composition algorithms which approximate the behavior of a nonlinear dynamical system using a linear operator. While such methods have proven to be remarkably useful in the analysis of time -series data, the resulting linear models must generally be of high order to accurately approximate fundamentally nonlinear behaviors. This issue poses an inherent risk of overfitting to training data, thereby limiting predictive capabilities. By contrast, this work explores strategies for nonlinear data -driven system identification using strategies inspired by Koopman analysis. General strategies that yield nonlinear models are presented for systems both with and without control. Subsequent pro-jection of the resulting nonlinear equations onto a low-rank basis yields a low-order representation for the underlying dynamical system. In both computational and experimental examples considered in this work, linear estimators of the Koopman operator are generally only able to provide short-term predictions for the observable dynamics, while comparable nonlinear estimators for the system dynamics provide accurate predictions on substantially longer timescales and replicate infinite-time behaviors that linear predictors cannot.