# Two methods to approximate the Koopman operator with a reservoir computer

arXiv: https://arxiv.org/abs/2008.10263 · HTML: https://mlsanity.com/paper/2008.10263/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Aug 2020.

## Built on by (most cited first, 7 of 7)

- [Koopman-informed recurrent neural networks](https://mlsanity.com/paper/2410.23467/) (2024)
- [Learning dynamical systems from data: Gradient-based dictionary optimization](https://mlsanity.com/paper/2411.04775/) (2024)
- [How deep is your network? Deep vs. shallow learning of transfer operators](https://mlsanity.com/paper/2509.19930/) (2025)
- [Restrictions on Physical Stochastic Reservoir Computers](https://mlsanity.com/paper/2307.14474/) (2023)
- [Taming Epilepsy: Mean Field Control of Whole-Brain Dynamics](https://mlsanity.com/paper/2603.18035/) (2026)
- [Dictionary learning for Kernel EDMD](https://mlsanity.com/paper/2604.25572/) (2026)
- [Koopman Identification of Nonlinear Systems via Reservoir Liftings](https://mlsanity.com/paper/2605.04917/) (2026)

## Abstract (arXiv)

The Koopman operator provides a powerful framework for data-driven analysis of dynamical systems. In the last few years, a wealth of numerical methods providing finite-dimensional approximations of the operator have been proposed (e.g. extended dynamic mode decomposition (EDMD) and its variants). While convergence results for EDMD require an infinite number of dictionary elements, recent studies have shown that only few dictionary elements can yield an efficient approximation of the Koopman operator, provided that they are well-chosen through a proper training process. However, this training process typically relies on nonlinear optimization techniques. In this paper, we propose two novel methods based on a reservoir computer to train the dictionary. These methods rely solely on linear convex optimization. We illustrate the efficiency of the method with several numerical examples in the context of data reconstruction, prediction, and computation of the Koopman operator spectrum. These results pave the way to the use of the reservoir computer in the Koopman operator framework.