Conditional Gaussian Koopman Operator for Modeling Complex Systems and Data Assimilation

Koopman theory has been actively explored in the context of data-driven modeling during the past few years. The key advantage is the promise of linear dynamics in the Koopman embedding space, which facilitates the learning and control of a complex dynamical system. However, for many real applications of complex dynamical systems, e.g., autonomous systems and the surrounding environment, linear dynamics may demand a very high-dimensional Koopman embedding space and could not make full use of some existing knowledge of nonlinear physics. In this work, we propose a conditional Gaussian Koopman operator (CG-Koopman) framework that leverages both known nonlinear physics and the Koopman embedding of unknown physics for modeling a complex dynamical system. The framework also facilitates an analytical formula for data assimilation (DA) of unknown physics in both the Koopman embedding space and the original physical space. We demonstrate that the analytical formula of DA can be used in data-driven modeling as an additional computationally affordable loss. This DA loss function promotes the proposed framework to capture the interactions between state variables and thus advances its modeling skills. Numerical experiments based on chaotic systems with intermittency and strong non-Gaussian features indicate that the proposed framework outperforms the standard Koopman operator models that strictly aim at a linear model in the embedding space, and the DA loss of the unknown physics further enhances the modeling skills of the CG-Koopman framework.