Autoencoding Reduced Order Models for Control through the Lens of Dynamic Mode Decomposition
Abstract: Modeling and control of high-dimensional dynamical systems often involve some dimensionality reduction techniques to construct a lower-order model that makes the associated task computationally feasible or less demanding. In recent years, two techniques have become
widely popular for analysis and reduced order modeling of high-dimensional dynamical systems: (1) dynamic mode decomposition and (2) deep autoencoding learning. This paper establishes a connection between dynamic mode decomposition and autoencoding learning
for controlled dynamical systems. Specifically, we first show that an optimization objective for learning a linear autoencoding reduced order model can be formulated such that its solution closely resembles the solution obtained by the dynamic mode decomposition with control algorithm. The linear autoencoding architecture is then extended to a deep autoencoding architecture to learn a nonlinear reduced order model. Finally, the learned reduced order model is used to design a controller utilizing stability-constrained deep neural networks. The studied framework does not require knowledge of the governing equations of the underlying system and learns the model and controller solely from time series data of observations and actuations. We provide empirical analysis on modeling and control of spatiotemporal high-dimensional systems, including fluid flow control.
Submission Length: Long submission (more than 12 pages of main content)
Changes Since Last Submission: Our revision includes the following changes in response to the reviewers' feedback. The changes in the revised version are highlighted in blue.
1. We have revised the Related Work section to highlight the difference between our work and prior works.
2. We have modified Figure 1 and its caption for better readability and visualization.
3. We have revised Section 3.2 to enhance the organization of the necessary preliminary concepts and clarify the assumptions.
4. We have subdivided Section 4.1 into smaller subsections to improve the readability of the content.
5. We have clarified the motivation behind using the loss $L_{recon}$ in Section 4.1.
6. In Section 4.1., we have provided a clearer explanation for the choice of continuous-time formulation for the DeepROM and the use of a numerical integrator.
7. We have moved the discussion on the semi-orthogonality constraint from the experiment section to the LAROM method section (4.1.2.).
8. We have clarified the purpose of Theorem 4.2.1
9. We have added a discussion on relating the modes of LAROM to DMDc in section 5.2.1.
10. We have added the explanations for choosing the specific dataset/experiments.
11. We have included a new baseline for comparison (section 5.1) and updated the Figures. 5, 6, 7, 9, 10, 11 by adding the new baseline.
12. We have added details on the design choices of the networks in Appendix B.2 and C.2, and Figures 12 and 13.
13. We have added a new Appendix D to discuss the details of the new baseline.
14. We have removed the claim of gradient descent-based modal decomposition using LAROM from the conclusion.
15. We have replaced the term "model order reduction" with "reduced order modeling".
16. We have substituted acronyms with their corresponding full phrases wherever it was deemed appropriate.
Assigned Action Editor: ~Jeffrey_Pennington1
Submission Number: 1019
Loading