Uncertainty Quantification of Autoencoder-based Koopman Operator

TL;DR

This paper introduces a method to quantify uncertainty in autoencoder-based Koopman operators by automatically learning basis functions and analyzing robustness to approximation errors, validated on the Van der Pol model.

Contribution

It presents a novel approach combining autoencoders and robustness certification to quantify uncertainty in Koopman operator approximations.

Findings

Trajectory remains within the uncertainty set in simulations

Autoencoder effectively learns optimal basis functions

Robustness analysis improves confidence in predictions

Abstract

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.