# Data-Driven Encoding: A New Numerical Method for Computation of the Koopman Operator

arXiv: https://arxiv.org/abs/2301.06542 · HTML: https://mlsanity.com/paper/2301.06542/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 25 references that are in the map.
- First posted on arXiv 16 Jan 2023.

## Builds on (most cited first, 12 of 12)

- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control](https://mlsanity.com/paper/1611.03537/) (2016)
- [Dynamic mode decomposition with control](https://mlsanity.com/paper/1409.6358/) (2014)
- [Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1708.06850/) (2017)
- [Optimal construction of Koopman eigenfunctions for prediction and control](https://mlsanity.com/paper/1810.08733/) (2018)
- [Learning Stable Koopman Embeddings](https://mlsanity.com/paper/2110.06509/) (2021)
- [Diffeomorphically Learning Stable Koopman Operators](https://mlsanity.com/paper/2112.04085/) (2021)
- [Learning Stable Models for Prediction and Control](https://mlsanity.com/paper/2005.04291/) (2020)
- [Model Predictive Control of a Vehicle using Koopman Operator](https://mlsanity.com/paper/2103.04978/) (2021)
- [Global, Unified Representation of Heterogenous Robot Dynamics Using Composition Operators: A Koopman Direct Encoding Method](https://mlsanity.com/paper/2208.00175/) (2022)
- [Learning of Causal Observable Functions for Koopman-DFL Lifting Linearization of Nonlinear Controlled Systems and Its Application to Excavation Automation](https://mlsanity.com/paper/2104.02004/) (2021)
- [Robust Dynamic Mode Decomposition](https://mlsanity.com/paper/2105.09869/) (2021)

## Built on by (most cited first, 1 of 1)

- [PyKoopman: A Python Package for Data-Driven Approximation of the Koopman Operator](https://mlsanity.com/paper/2306.12962/) (2023)

## Abstract (arXiv)

This paper presents a data-driven method for constructing a Koopman linear model based on the Direct Encoding (DE) formula. The prevailing methods, Dynamic Mode Decomposition (DMD) and its extensions are based on least squares estimates that can be shown to be biased towards data that are densely populated. The DE formula consisting of inner products of a nonlinear state transition function with observable functions does not incur this biased estimation problem and thus serves as a desirable alternative to DMD. However, the original DE formula requires knowledge of the nonlinear state equation, which is not available in many practical applications. In this paper, the DE formula is extended to a data-driven method, Data-Driven Encoding (DDE) of Koopman operator, in which the inner products are calculated from data taken from a nonlinear dynamic system. An effective algorithm is presented for the computation of the inner products, and their convergence to true values is proven. Numerical experiments verify the effectiveness of DDE compared to Extended DMD. The experiments demonstrate robustness to data distribution and the convergent properties of DDE, guaranteeing accuracy improvements with additional sample points. Furthermore, DDE is applied to deep learning of the Koopman operator to further improve prediction accuracy.