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The Koopman operator framework provides a perspective that non-lineardynamics can be described through the lens of linear operators acting onfunction spaces. As the framework naturally yields linear embedding models,there have been extensive efforts to utilize it for control, where linearcontroller designs in the embedded space can be applied to control possiblynonlinear dynamics. However, it is challenging to successfully deploy thismodeling procedure in a wide range of applications. In this work, some of thefundamental limitations of linear embedding models are addressed. We show anecessary condition for a linear embedding model to achieve zero modelingerror, highlighting a trade-off relation between the model expressivity and arestriction on the model structure to allow the use of linear systems theoriesfor nonlinear dynamics. To address these limitations, neural network-basedmodeling is proposed based on linear embedding with oblique projection, whichis derived from a weak formulation of projection-based linear operatorlearning. We train the proposed model using a two-stage learning procedure,wherein the features and operators are initialized with orthogonal projection,followed by the main training process in which test functions characterizingthe oblique projection are learned from data. The first stage achieves anoptimality ensured by the orthogonal projection and the second stage improvesthe generalizability to various tasks by optimizing the model with the obliqueprojection. We demonstrate the effectiveness of the proposed method over otherdata-driven modeling methods by providing comprehensive numerical evaluationswhere four tasks are considered targeting three different systems.

Extracting Koopman Operators for Prediction and Control of Non-linear Dynamics Using Two-stage Learning and Oblique Projections