  This paper proposes a method for uncertainty quantification of an
autoencoder-based Koopman operator. The main challenge of using the Koopman
operator is to design the basis functions for lifting the state. To this end,
this paper builds an autoencoder to automatically search the optimal lifting
basis functions with a given loss function. We approximate the Koopman operator
in a finite-dimensional space with the autoencoder, while the approximated
Koopman has an approximation uncertainty. To resolve the problem, we compute a
robust positively invariant set for the approximated Koopman operator to
consider the approximation error. Then, the decoder of the autoencoder is
analyzed by robustness certification against approximation error using the
Lipschitz constant in the reconstruction phase. The forced Van der Pol model is
used to show the validity of the proposed method. From the numerical simulation
results, we confirmed that the trajectory of the true state stays in the
uncertainty set centered by the reconstructed state.
