# Temporal Forward-Backward Consistency, Not Residual Error, Measures the Prediction Accuracy of Extended Dynamic Mode Decomposition

arXiv: https://arxiv.org/abs/2207.07719 · HTML: https://mlsanity.com/paper/2207.07719/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 33 references that are in the map.
- First posted on arXiv 15 Jul 2022.

## Builds on (most cited first, 14 of 14)

- [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (2017)
- [Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control](https://mlsanity.com/paper/1611.03537/) (2016)
- [On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator](https://mlsanity.com/paper/1703.04680/) (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)
- [Learning Stable Koopman Embeddings](https://mlsanity.com/paper/2110.06509/) (2021)
- [Derivative-Based Koopman Operators for Real-Time Control of Robotic Systems](https://mlsanity.com/paper/2010.05778/) (2020)
- [Diffeomorphically Learning Stable Koopman Operators](https://mlsanity.com/paper/2112.04085/) (2021)
- [Neural Koopman Lyapunov Control](https://mlsanity.com/paper/2201.05098/) (2022)
- [Generalizing Dynamic Mode Decomposition: Balancing Accuracy and Expressiveness in Koopman Approximations](https://mlsanity.com/paper/2108.03712/) (2021)
- [Data-Driven Operator Theoretic Methods for Global Phase Space Learning](https://mlsanity.com/paper/1910.03011/) (2019)
- [Parallel Learning of Koopman Eigenfunctions and Invariant Subspaces For Accurate Long-Term Prediction](https://mlsanity.com/paper/2005.06138/) (2020)
- [Enhancement for Robustness of Koopman Operator-based Data-driven Mobile Robotic Systems](https://mlsanity.com/paper/2103.00812/) (2021)
- [The Challenge of Small Data: Dynamic Mode Decomposition, Redux](https://mlsanity.com/paper/2104.04005/) (2021)

## Built on by (most cited first, 2 of 2)

- [Koopman operator theory: fundamentals, control, and applications](https://mlsanity.com/paper/2607.01819/) (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)

Extended Dynamic Mode Decomposition (EDMD) is a popular data-driven method to approximate the action of the Koopman operator on a linear function space spanned by a dictionary of functions. The accuracy of EDMD model critically depends on the quality of the particular dictionary's span, specifically on how close it is to being invariant under the Koopman operator. Motivated by the observation that the residual error of EDMD, typically used for dictionary learning, does not encode the quality of the function space and is sensitive to the choice of basis, we introduce the novel concept of consistency index. We show that this measure, based on using EDMD forward and backward in time, enjoys a number of desirable qualities that make it suitable for data-driven modeling of dynamical systems: it measures the quality of the function space, it is invariant under the choice of basis, can be computed in closed form from the data, and provides a tight upper-bound for the relative root mean square error of all function predictions on the entire span of the dictionary.