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The Koopman operator has become an essential tool for data-driven analysis,prediction and control of complex systems, the main reason being the enormouspotential of identifying linear function space representations of nonlineardynamics from measurements. Until now, the situation where for large-scalesystems, we (i) only have access to partial observations (i.e., measurements,as is very common for experimental data) or (ii) deliberately perform coarsegraining (for efficiency reasons) has not been treated to its full extent. Inthis paper, we address the pitfall associated with this situation, that theclassical EDMD algorithm does not automatically provide a Koopman operatorapproximation for the underlying system if we do not carefully select thenumber of observables. Moreover, we show that symmetries in the system dynamicscan be carried over to the Koopman operator, which allows us to massivelyincrease the model efficiency. We also briefly draw a connection to domaindecomposition techniques for partial differential equations and presentnumerical evidence using the Kuramoto--Sivashinsky equation.

Partial observations, coarse graining and equivariance in Koopman operator theory for large-scale dynamical systems