# Sparsity-promoting algorithms for the discovery of informative Koopman invariant subspaces

arXiv: https://arxiv.org/abs/2002.10637 · HTML: https://mlsanity.com/paper/2002.10637/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Feb 2020.

## Built on by (most cited first, 10 of 10)

- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Neural Implicit Flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal data](https://mlsanity.com/paper/2204.03216/) (2022)
- [Benchmarking sparse system identification with low-dimensional chaos](https://mlsanity.com/paper/2302.10787/) (2023)
- [A toolkit for data-driven discovery of governing equations in high-noise regimes](https://mlsanity.com/paper/2111.04870/) (2021)
- [PyKoopman: A Python Package for Data-Driven Approximation of the Koopman Operator](https://mlsanity.com/paper/2306.12962/) (2023)
- [Generalizing Dynamic Mode Decomposition: Balancing Accuracy and Expressiveness in Koopman Approximations](https://mlsanity.com/paper/2108.03712/) (2021)
- [Randomized Projection Learning Method forDynamic Mode Decomposition](https://mlsanity.com/paper/2110.01718/) (2021)
- [Ensemble forecasts in reproducing kernel Hilbert space family](https://mlsanity.com/paper/2207.14653/) (2022)
- [Modal Analysis of Spatiotemporal Data via Multivariate Gaussian Process Regression](https://mlsanity.com/paper/2403.13118/) (2024)
- [Bayesian Transfer Operators in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2509.22482/) (2025)

## Abstract (arXiv)

Koopman decomposition is a non-linear generalization of eigen-decomposition, and is being increasingly utilized in the analysis of spatio-temporal dynamics. Well-known techniques such as the dynamic mode decomposition (DMD) and its linear variants provide approximations to the Koopman operator, and have been applied extensively in many fluid dynamic problems. Despite being endowed with a richer dictionary of nonlinear observables, nonlinear variants of the DMD, such as extended/kernel dynamic mode decomposition (EDMD/KDMD) are seldom applied to large-scale problems primarily due to the difficulty of discerning the Koopman invariant subspace from thousands of resulting Koopman eigenmodes. To address this issue, we propose a framework based on multi-task feature learning to extract the most informative Koopman invariant subspace by removing redundant and spurious Koopman triplets. In particular, we develop a pruning procedure that penalizes departure from linear evolution. These algorithms can be viewed as sparsity promoting extensions of EDMD/KDMD. Further, we extend KDMD to a continuous-time setting and show a relationship between the present algorithm, sparsity-promoting DMD, and an empirical criterion from the viewpoint of non-convex optimization. The effectiveness of our algorithm is demonstrated on examples ranging from simple dynamical systems to two-dimensional cylinder wake flows at different Reynolds numbers and a three-dimensional turbulent ship air-wake flow. The latter two problems are designed such that very strong nonlinear transients are present, thus requiring an accurate approximation of the Koopman operator. Underlying physical mechanisms are analyzed, with an emphasis on characterizing transient dynamics. The results are compared to existing theoretical expositions and numerical approximations.