Koopman-Based Learning of Infinitesimal Generators without Operator Logarithm

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.