  The present paper treats the identification of nonlinear dynamical systems
using Koopman-based deep state-space encoders. Through this method, the usual
drawback of needing to choose a dictionary of lifting functions a priori is
circumvented. The encoder represents the lifting function to the space where
the dynamics are linearly propagated using the Koopman operator. An
input-affine formulation is considered for the lifted model structure and we
address both full and partial state availability. The approach is implemented
using the the deepSI toolbox in Python. To lower the computational need of the
simulation error-based training, the data is split into subsections where
multi-step prediction errors are calculated independently. This formulation
allows for efficient batch optimization of the network parameters and, at the
same time, excellent long term prediction capabilities of the obtained models.
The performance of the approach is illustrated by nonlinear benchmark examples.
