Deep Koopman Learning using Noisy Data
Abstract: This paper proposes a data-driven framework to learn a finite-dimensional approximation
of a Koopman operator for approximating the state evolution of a dynamical system under
noisy observations. To this end, our proposed solution has two main advantages. First, the
proposed method only requires the measurement noise to be bounded. Second, the proposed
method modifies the existing deep Koopman operator formulations by characterizing the
effect of the measurement noise on the Koopman operator learning and then mitigating it
by updating the tunable parameter of the observable functions of the Koopman operator,
making it easy to implement. The performance of the proposed method is demonstrated on
several standard benchmarks. We then compare the presented method with similar methods
proposed in the latest literature on Koopman learning.
Submission Length: Regular submission (no more than 12 pages of main content)
Changes Since Last Submission: We sincerely thank all reviewers for their time and for providing valuable comments and suggestions. We greatly appreciate your thorough review and have carefully addressed your concerns. Below, we summarize the main revisions made to our manuscript, which can be found in the revised draft via the latest uploaded draft. All major changes are highlighted in blue for clarity:
1. We have added the recommended references in the latest draft.
2. We have included relevant remarks in the manuscript to clarify potential points of confusion.
3. We have refined the Challenges and Key Ideas section to focus solely on the key ideas of our work. Additionally, we have introduced Remark 1 to further clarify the fundamental mechanism of our approach. These changes are clearly marked in the draft.
4. We have improved the algorithm, re-executed the simulations, and reorganized the plots for better clarity and coherence.
5. The estimation errors across all gradient-based methods remain marginal over the training data, as they are terminated based on the same terminal accuracy criterion. We have explicitly highlighted this point in the revised draft, with the changes marked in blue.
6. To streamline notation, we have removed unnecessary symbols. For example, we have moved the Lipschitz constant from the main text to the appendix, where it is explicitly defined when necessary. Similar refinements have been applied to other notations.
7. We have addressed minor comments and revised the manuscript to align with the standards suggested by the reviewers.
8. We have revised the proof section, particularly the transition between Equations (33–35). The updated proof is now more straightforward, as we focus on reformulating the optimization problem by solely identifying the upper bound and eliminating constant terms.
9. We have updated the surface vehicle example to incorporate data from real surface vehicle robots. The details of this revision can be found in the Simulation Section of the revised manuscript, marked in blue.
We appreciate the reviewers' insightful feedback, which has helped us improve the clarity, rigor, and overall quality of our work.
Assigned Action Editor: ~William_T_Redman1
Submission Number: 4069
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