Koopman

For a general nonlinear system with state x and flow F_t, the idea is to identify a suitable set of nonlinear functions, called observables, p(x). The time evolution of these observables is then described by the Koopman operator K_t. It can be shown that the Koopman operator is a linear infinite-dimensional operator. This approach leads to an equivalent system representation consisting of a nonlinear operator (the vector of observables) and a linear dynamical system (the generator of the Koopman operator). Both parts can be learned using neural networks. In particular, Deep Operator Networks (DeepONet) \cite{Lu21} show promising potential for future research, as they can represent the underlying infinite-dimensional function space of the observables. From a control engineering perspective, the main advantage lies in the linearity of the system dynamics, which allows for straightforward linear controller design.