# Online Estimation of the Koopman Operator Using Fourier Features

arXiv: https://arxiv.org/abs/2212.01503 · HTML: https://mlsanity.com/paper/2212.01503/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 35 references that are in the map.
- First posted on arXiv 3 Dec 2022.

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

- [Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1708.06850/) (2017)
- [Data-driven discovery of Koopman eigenfunctions for control](https://mlsanity.com/paper/1707.01146/) (2017)
- [Generalizing Koopman Theory to allow for inputs and control](https://mlsanity.com/paper/1602.07647/) (2016)
- [Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/1712.01572/) (2017)
- [Non-Stationary Spectral Kernels](https://mlsanity.com/paper/1705.08736/) (2017)
- [Optimal construction of Koopman eigenfunctions for prediction and control](https://mlsanity.com/paper/1810.08733/) (2018)
- [Active Learning of Dynamics for Data-Driven Control Using Koopman Operators](https://mlsanity.com/paper/1906.05194/) (2019)
- [Modeling and Control of Soft Robots Using the Koopman Operator and Model Predictive Control](https://mlsanity.com/paper/1902.02827/) (2019)
- [Generalized Spectral Kernels](https://mlsanity.com/paper/1506.02236/) (2015)
- [Extended Dynamic Mode Decomposition with Learned Koopman Eigenfunctions for Prediction and Control](https://mlsanity.com/paper/1911.08751/) (2019)
- [Episodic Koopman Learning of Nonlinear Robot Dynamics with Application to Fast Multirotor Landing](https://mlsanity.com/paper/2004.01708/) (2020)
- [A Convex Optimization Approach to Learning Koopman Operators](https://mlsanity.com/paper/2102.03934/) (2021)
- [ACD-EDMD: Analytical Construction for Dictionaries of Lifting Functions in Koopman Operator-based Nonlinear Robotic Systems](https://mlsanity.com/paper/2111.12256/) (2021)
- [Learning and Leveraging Features in Flow-Like Environments to Improve Situational Awareness](https://mlsanity.com/paper/2109.06107/) (2021)

## Built on by (most cited first, 4 of 4)

- [Koopman-informed recurrent neural networks](https://mlsanity.com/paper/2410.23467/) (2024)
- [Spectrally Informed Learning of Fluid Flows](https://mlsanity.com/paper/2408.14407/) (2024)
- [KoopMotion: Learning Almost Divergence Free Koopman Flow Fields for Motion Planning](https://mlsanity.com/paper/2509.09074/) (2025)
- [Dictionary learning for Kernel EDMD](https://mlsanity.com/paper/2604.25572/) (2026)

## Abstract (arXiv)

Transfer operators offer linear representations and global, physically meaningful features of nonlinear dynamical systems. Discovering transfer operators, such as the Koopman operator, require careful crafted dictionaries of observables, acting on states of the dynamical system. This is ad hoc and requires the full dataset for evaluation. In this paper, we offer an optimization scheme to allow joint learning of the observables and Koopman operator with online data. Our results show we are able to reconstruct the evolution and represent the global features of complex dynamical systems.