# Dynamic mode decomposition in vector-valued reproducing kernel Hilbert spaces for extracting dynamical structure among observables

arXiv: https://arxiv.org/abs/1808.10551 · HTML: https://mlsanity.com/paper/1808.10551/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Aug 2018.

## Built on by (most cited first, 12 of 12)

- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Reproducing kernel Hilbert C*-module and kernel mean embeddings](https://mlsanity.com/paper/2101.11410/) (2021)
- [Koopman operators with intrinsic observables in rigged reproducing kernel Hilbert spaces](https://mlsanity.com/paper/2403.02524/) (2024)
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- [A case study of spatiotemporal forecasting techniques for weather forecasting](https://mlsanity.com/paper/2209.14782/) (2022)
- [A purely data-driven framework for prediction, optimization, and control of networked processes: application to networked SIS epidemic model](https://mlsanity.com/paper/2108.02005/) (2021)
- [An Interpretable Approach to Load Profile Forecasting in Power Grids using Galerkin-Approximated Koopman Pseudospectra](https://mlsanity.com/paper/2304.07832/) (2023)
- [Learning Linear Embeddings for Non-Linear Network Dynamics with Koopman Message Passing](https://mlsanity.com/paper/2305.09060/) (2023)
- [Operator-Based Detecting, Learning, and Stabilizing Unstable Periodic Orbits of Chaotic Attractors](https://mlsanity.com/paper/2310.12156/) (2023)
- [Kernel Operator-Theoretic Bayesian Filter for Nonlinear Dynamical Systems](https://mlsanity.com/paper/2411.00198/) (2024)
- [Bayesian Transfer Operators in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2509.22482/) (2025)

## Abstract (arXiv)

Understanding nonlinear dynamical systems (NLDSs) is challenging in a variety of engineering and scientific fields. Dynamic mode decomposition (DMD), which is a numerical algorithm for the spectral analysis of Koopman operators, has been attracting attention as a way of obtaining global modal descriptions of NLDSs without requiring explicit prior knowledge. However, since existing DMD algorithms are in principle formulated based on the concatenation of scalar observables, it is not directly applicable to data with dependent structures among observables, which take, for example, the form of a sequence of graphs. In this paper, we formulate Koopman spectral analysis for NLDSs with structures among observables and propose an estimation algorithm for this problem. This method can extract and visualize the underlying low-dimensional global dynamics of NLDSs with structures among observables from data, which can be useful in understanding the underlying dynamics of such NLDSs. To this end, we first formulate the problem of estimating spectra of the Koopman operator defined in vector-valued reproducing kernel Hilbert spaces, and then develop an estimation procedure for this problem by reformulating tensor-based DMD. As a special case of our method, we propose the method named as Graph DMD, which is a numerical algorithm for Koopman spectral analysis of graph dynamical systems, using a sequence of adjacency matrices. We investigate the empirical performance of our method by using synthetic and real-world data.