Uncertainty Quantification of Autoencoder-based Koopman Operator

TL;DR

This work tackles uncertainty quantification for an autoencoder-based Koopman operator by learning the lifting function with an autoencoder and bounding both finite-dimensional approximation error and decoder reconstruction error. It builds a robust positively invariant set to contain the Koopman residual and uses a Lipschitz-based robustness certificate for the decoder, validated on a forced Van der Pol oscillator where the true state remains within the predicted uncertainty set around the reconstructed state. The method integrates EDMD-style lifting with a learned, overcomplete representation, and provides practical guarantees through Raković-style RPI outer-approximations and SDP-based Lipschitz constant computation. The results demonstrate reliable trajectory tracking within quantified uncertainty bounds, enabling more robust data-driven prediction and control of nonlinear systems.

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.

Figures (3)

• Figure 1: Overall structure of the proposed method
• Figure 2: Multi-step prediction performance with test set. The initial state $\textbf{x}_0$ and the input are available.
• Figure 3: State trajectory for $k \in [0, 7]$ in phase plane: The initial state (i.e., green pentagram) is $\textbf{x}_0^T=[-0.1~-0.5]^T$. Gray area shows set $\mathcal{D}$ at each k=2.1s, 2.9s, 4.7s, and 5.5s.

Theorems & Definitions (2)

• proof
• proof