# Generalizing Dynamic Mode Decomposition: Balancing Accuracy and Expressiveness in Koopman Approximations

arXiv: https://arxiv.org/abs/2108.03712 · HTML: https://mlsanity.com/paper/2108.03712/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 51 references that are in the map.
- First posted on arXiv 8 Aug 2021.

## Builds on (most cited first, 18 of 18)

- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (2017)
- [Linearly-Recurrent Autoencoder Networks for Learning Dynamics](https://mlsanity.com/paper/1712.01378/) (2017)
- [Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control](https://mlsanity.com/paper/1611.03537/) (2016)
- [Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1708.06850/) (2017)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator](https://mlsanity.com/paper/1703.04680/) (2017)
- [Data-driven discovery of Koopman eigenfunctions for control](https://mlsanity.com/paper/1707.01146/) (2017)
- [Optimal construction of Koopman eigenfunctions for prediction and control](https://mlsanity.com/paper/1810.08733/) (2018)
- [Koopman operator-based model reduction for switched-system control of PDEs](https://mlsanity.com/paper/1710.06759/) (2017)
- [Derivative-Based Koopman Operators for Real-Time Control of Robotic Systems](https://mlsanity.com/paper/2010.05778/) (2020)
- [Learning Stable Models for Prediction and Control](https://mlsanity.com/paper/2005.04291/) (2020)
- [Sparsity-promoting algorithms for the discovery of informative Koopman invariant subspaces](https://mlsanity.com/paper/2002.10637/) (2020)
- [A Class of Logistic Functions for Approximating State-Inclusive Koopman Operators](https://mlsanity.com/paper/1712.03132/) (2017)
- [Parallel Learning of Koopman Eigenfunctions and Invariant Subspaces For Accurate Long-Term Prediction](https://mlsanity.com/paper/2005.06138/) (2020)
- [A convex data-driven approach for nonlinear control synthesis](https://mlsanity.com/paper/2006.15477/) (2020)
- [Koopman Operator Based Modeling for Quadrotor Control on SE(3)](https://mlsanity.com/paper/2103.03363/) (2021)
- [Handling plant-model mismatch in Koopman Lyapunov-based model predictive control via offset-free control framework](https://mlsanity.com/paper/2010.07239/) (2020)

## Built on by (most cited first, 5 of 5)

- [SafEDMD: A Koopman-based data-driven controller design framework for nonlinear dynamical systems](https://mlsanity.com/paper/2402.03145/) (2024)
- [Temporal Forward-Backward Consistency, Not Residual Error, Measures the Prediction Accuracy of Extended Dynamic Mode Decomposition](https://mlsanity.com/paper/2207.07719/) (2022)
- [Koopman operator theory: fundamentals, control, and applications](https://mlsanity.com/paper/2607.01819/) (2026)
- [Finding Koopman Invariant Subspaces via Personalized PageRank](https://mlsanity.com/paper/2605.24666/) (2026)
- [Flow map learning in nonlinear vector autoregressive models: influence of the feature-library structure on the training error](https://mlsanity.com/paper/2605.31438/) (2026)

## Abstract (arXiv)

This paper tackles the data-driven approximation of unknown dynamical systems using Koopman-operator methods. Given a dictionary of functions, these methods approximate the projection of the action of the operator on the finite-dimensional subspace spanned by the dictionary. We propose the Tunable Symmetric Subspace Decomposition algorithm to refine the dictionary, balancing its expressiveness and accuracy. Expressiveness corresponds to the ability of the dictionary to describe the evolution of as many observables as possible and accuracy corresponds to the ability to correctly predict their evolution. Based on the observation that Koopman-invariant subspaces give rise to exact predictions, we reason that prediction accuracy is a function of the degree of invariance of the subspace generated by the dictionary and provide a data-driven measure to measure invariance proximity. The proposed algorithm iteratively prunes the initial functional space to identify a refined dictionary of functions that satisfies the desired level of accuracy while retaining as much of the original expressiveness as possible. We provide a full characterization of the algorithm properties and show that it generalizes both Extended Dynamic Mode Decomposition and Symmetric Subspace Decomposition. Simulations on planar systems show the effectiveness of the proposed methods in producing Koopman approximations of tunable accuracy that capture relevant information about the dynamical system.