Modeling Partially Observed Nonlinear Dynamical Systems and Efficient Data Assimilation via Conditional Gaussian Koopman Network

A conditional Gaussian Koopman network (CGKN) is developed in this work to learn surrogate models that can perform efficient state forecast and data assimilation (DA) for high-dimensional complex dynamical systems, e.g., systems governed by nonlinear partial differential equations (PDEs). Focusing on nonlinear partially observed systems that are common in many engineering and earth science applications, this work exploits Koopman embedding to discover a proper latent representation of the unobserved system states, such that the dynamics of the latent states are conditional linear, i.e., linear with the given observed system states, with which the analytical formulae of DA facilitate the incorporation of DA performance into the learning process of the modeled system, leading to a framework that unifies scientific machine learning (SciML) and data assimilation. The performance of CGKN is demonstrated on several canonical problems governed by nonlinear PDEs with intermittency and turbulent features. The CGKN framework also serves as an example to illustrate unifying the development of SciML models and their other outer-loop applications of digital twins, such as design optimization, inverse problems, and optimal control.