Partial observations, coarse graining and equivariance in Koopman
  operator theory for large-scale dynamical systems
Sebastian Peitz Hans Harder Feliks Nüske Friedrich Philipp Manuel Schaller Karl Worthmann
The Koopman operator has become an essential tool for data-driven analysis,
prediction and control of complex systems, the main reason being the enormous
potential of identifying linear function space representations of nonlinear
dynamics from measurements. Until now, the situation where for large-scale
systems, we (i) only have access to partial observations (i.e., measurements,
as is very common for experimental data) or (ii) deliberately perform coarse
graining (for efficiency reasons) has not been treated to its full extent. In
this paper, we address the pitfall associated with this situation, that the
classical EDMD algorithm does not automatically provide a Koopman operator
approximation for the underlying system if we do not carefully select the
number of observables. Moreover, we show that symmetries in the system dynamics
can be carried over to the Koopman operator, which allows us to massively
increase the model efficiency. We also briefly draw a connection to domain
decomposition techniques for partial differential equations and present
numerical evidence using the Kuramoto--Sivashinsky equation.