  The Koopman operator framework provides a perspective that non-linear
dynamics can be described through the lens of linear operators acting on
function spaces. As the framework naturally yields linear embedding models,
there have been extensive efforts to utilize it for control, where linear
controller designs in the embedded space can be applied to control possibly
nonlinear dynamics. However, it is challenging to successfully deploy this
modeling procedure in a wide range of applications. In this work, some of the
fundamental limitations of linear embedding models are addressed. We show a
necessary condition for a linear embedding model to achieve zero modeling
error, highlighting a trade-off relation between the model expressivity and a
restriction on the model structure to allow the use of linear systems theories
for nonlinear dynamics. To address these limitations, neural network-based
modeling is proposed based on linear embedding with oblique projection, which
is derived from a weak formulation of projection-based linear operator
learning. 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 characterizing
the oblique projection are learned from data. The first stage achieves an
optimality ensured by the orthogonal projection and the second stage improves
the generalizability to various tasks by optimizing the model with the oblique
projection. We demonstrate the effectiveness of the proposed method over other
data-driven modeling methods by providing comprehensive numerical evaluations
where four tasks are considered targeting three different systems.
