  The Koopman operator has gained significant attention in recent years for its
ability to verify evolutionary properties of continuous-time nonlinear systems
by lifting state variables into an infinite-dimensional linear vector space.
The challenge remains in providing estimations for transitional properties
pertaining to the system's vector fields based on discrete-time observations.
To retrieve such infinitesimal system transition information, leveraging the
structure of Koopman operator learning, current literature focuses on
developing techniques free of time derivatives through the use of the Koopman
operator logarithm. However, the soundness of these methods has so far been
demonstrated only for maintaining effectiveness within a restrictive function
space, together with knowledge of the operator spectrum properties. To better
adapt to the practical applications in learning and control of unknown systems,
we propose a logarithm-free technique for learning the infinitesimal generator
without disrupting the Koopman operator learning framework. This approach
claims compatibility with other system verification tools using the same set of
training data. We provide numerical examples to demonstrate its effectiveness
in applications of system identification and stability prediction.
