# Kernel Dynamic Mode Decomposition For Sparse Reconstruction of Closable Koopman Operators

arXiv: https://arxiv.org/abs/2505.06806 · HTML: https://mlsanity.com/paper/2505.06806/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 67 references that are in the map.
- First posted on arXiv 11 May 2025.

## Builds on (most cited first, 15 of 15)

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- [UniFormer: Unified Transformer for Efficient Spatiotemporal Representation Learning](https://mlsanity.com/paper/2201.04676/) (2022)
- [Traffic Graph Convolutional Recurrent Neural Network: A Deep Learning Framework for Network-Scale Traffic Learning and Forecasting](https://mlsanity.com/paper/1802.07007/) (2018)
- [On the Similarity between the Laplace and Neural Tangent Kernels](https://mlsanity.com/paper/2007.01580/) (2020)
- [Deep Bidirectional and Unidirectional LSTM Recurrent Neural Network for Network-wide Traffic Speed Prediction](https://mlsanity.com/paper/1801.02143/) (2018)
- [Long-Range Transformers for Dynamic Spatiotemporal Forecasting](https://mlsanity.com/paper/2109.12218/) (2021)
- [Low-Rank Autoregressive Tensor Completion for Spatiotemporal Traffic Data Imputation](https://mlsanity.com/paper/2104.14936/) (2021)
- [Kernel-based approximation of the Koopman generator and Schr"odinger operator](https://mlsanity.com/paper/2005.13231/) (2020)
- [The Multiverse of Dynamic Mode Decomposition Algorithms](https://mlsanity.com/paper/2312.00137/) (2023)
- [Kernel Learning for Robust Dynamic Mode Decomposition: Linear and Nonlinear Disambiguation Optimization (LANDO)](https://mlsanity.com/paper/2106.01510/) (2021)
- [Koopman Kernel Regression](https://mlsanity.com/paper/2305.16215/) (2023)
- [A kernel-based approach to molecular conformation analysis](https://mlsanity.com/paper/1809.11092/) (2018)
- [Composition operators on reproducing kernel Hilbert spaces with analytic positive definite functions](https://mlsanity.com/paper/1911.11992/) (2019)
- [Kernel Machines Beat Deep Neural Networks on Mask-based Single-channel Speech Enhancement](https://mlsanity.com/paper/1811.02095/) (2018)

## Abstract (arXiv)

Spatial temporal reconstruction of dynamical system is indeed a crucial problem with diverse applications ranging from climate modeling to numerous chaotic and physical processes. These reconstructions are based on the harmonious relationship between the Koopman operators and the choice of dictionary, determined implicitly by a kernel function. This leads to the approximation of the Koopman operators in a reproducing kernel Hilbert space (RKHS) associated with that kernel function. Data-driven analysis of Koopman operators demands that Koopman operators be closable over the underlying RKHS, which still remains an unsettled, unexplored, and critical operator-theoretic challenge. We aim to address this challenge by investigating the embedding of the Laplacian kernel in the measure-theoretic sense, giving rise to a rich enough RKHS to settle the closability of the Koopman operators. We leverage Kernel Extended Dynamic Mode Decomposition with the Laplacian kernel to reconstruct the dominant spatial temporal modes of various diverse dynamical systems. After empirical demonstration, we concrete such results by providing the theoretical justification leveraging the closability of the Koopman operators on the RKHS generated by the Laplacian kernel on the avenues of Koopman mode decomposition and the Koopman spectral measure. Such results were explored from both grounds of operator theory and data-driven science, thus making the Laplacian kernel a robust choice for spatial-temporal reconstruction.