Deep Koopman-layered Model with Universal Property Based on Toeplitz Matrices
Yuka Hashimoto, Tomoharu Iwata
TL;DR
This paper introduces a deep Koopman-layered model utilizing Toeplitz matrices with universal approximation capabilities, enabling effective analysis of nonautonomous dynamical systems and demonstrating superior eigenvalue estimation performance.
Contribution
The paper presents a novel deep Koopman-layered model with Toeplitz matrices, establishing universality, generalization, and efficient training via Krylov methods, applicable to nonautonomous systems.
Findings
Outperforms existing methods in eigenvalue estimation for Koopman operators.
Demonstrates universality and generalization of the model.
Efficient training through Krylov subspace methods.
Abstract
We propose deep Koopman-layered models with learnable parameters in the form of Toeplitz matrices for analyzing the transition of the dynamics of time-series data. The proposed model has both theoretical solidness and flexibility. By virtue of the universal property of Toeplitz matrices and the reproducing property underlying the model, we show its universality and generalization property. In addition, the flexibility of the proposed model enables the model to fit time-series data coming from nonautonomous dynamical systems. When training the model, we apply Krylov subspace methods for efficient computations, which establish a new connection between Koopman operators and numerical linear algebra. We also empirically demonstrate that the proposed model outperforms existing methods on eigenvalue estimation of multiple Koopman operators for nonautonomous systems.
Peer Reviews
Decision·Submitted to ICLR 2025
1. **Originality** The paper introduces an innovative integration of deep learning techniques with Koopman operator theory. By proposing deep Koopman-layered models that utilize learnable Toeplitz matrices and Fourier functions, the authors offer a fresh perspective for analyzing nonautonomous systems, which are often challenging for traditional methods. The approach of simultaneously estimating multiple Koopman operators is a notable contribution that broadens the model's applicability. 2. **Q
1. To my understanding, the choice of the Fourier basis in the proposed deep Koopman-layered model is motivated by its universality in function representation, desirable theoretical properties for analyzing Koopman operators, flexibility in learning multiple operators simultaneously, and compatibility with efficient Krylov subspace methods for low computational cost. However, it is still unclear: - What specific properties of the Fourier basis make it particularly suitable for capturing the dyn
- Transforming the original space into the space of observables using fixed basis functions (as opposed to learning the space of observables like in Lusch 2017) allows the authors to develop a strong theoretical backing for their approach including showing universality and a generalization bound. I think the theoretical foundation of the work is its strongest feature. - A nice feature of the approach is that one can analyze the eigenvalues of the trained model to interpret the underlying system
- Overall, I found it quite challenging to understand the details of your proposed learning algorithm and I am not confident that I would be able to reproduce your approach using your paper alone (that said, I appreciate the authors providing their code in the supplementary materials). It would have been helpful for me if you had included a section which provides a step-by-step summary or an algorithm block showing how your approach works for a given time-series dataset. - I think the claims you
Constructing models for non-autonomous, non-linear dynamical systems is a very important and very challenging problem. On the n-torus, Fourier series provide a very efficient and useful dictioanry to approximate functions in L2, so the setting the authors chose to present their Koopman approximation is reasonable. The theoretical contributions cover most of the important, first aspects that should be treated when introducing a new model, and seem to be proven appropriately. The numerical example
The authors chose an extremely specific setting (systems on n-tori) and can therefore avoid the most challenging question in Koopman operator approximation - how to choose the proper dictionary for a given problem? The challenging aspect in their setting thus only comes from the fact that the chosen systems are non-autonomous, not that they are non-linear. Unfortunately, the authors do not discuss any existing methods for non-autonomous systems or compare their method in the numerical setting ag
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Taxonomy
TopicsOptical Polarization and Ellipsometry · Neural Networks and Applications