# Data-driven Koopman mode approximation: A neural power iteration algorithm

arXiv: https://arxiv.org/abs/2608.26943 · HTML: https://mlsanity.com/paper/2608.26943/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 30 references that are in the map.
- First posted on arXiv 27 Aug 2026.

## Builds on (most cited first, 17 of 17)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [World Models](https://mlsanity.com/paper/1803.10122/) (2018)
- [Self-Supervised Learning from Images with a Joint-Embedding Predictive Architecture](https://mlsanity.com/paper/2301.08243/) (2023)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (2017)
- [Data-driven discovery of Koopman eigenfunctions for control](https://mlsanity.com/paper/1707.01146/) (2017)
- [Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems](https://mlsanity.com/paper/2111.14889/) (2021)
- [Koopman Operator Dynamical Models: Learning, Analysis and Control](https://mlsanity.com/paper/2102.02522/) (2021)
- [The Multiverse of Dynamic Mode Decomposition Algorithms](https://mlsanity.com/paper/2312.00137/) (2023)
- [Neural Tangent Kernel: A Survey](https://mlsanity.com/paper/2208.13614/) (2022)
- [Deep Learning Enhanced Dynamic Mode Decomposition](https://mlsanity.com/paper/2108.04433/) (2021)
- [A Lightweight Library for Energy-Based Joint-Embedding Predictive Architectures](https://mlsanity.com/paper/2602.03604/) (2026)
- [Efficient Dynamics Modeling in Interactive Environments with Koopman Theory](https://mlsanity.com/paper/2306.11941/) (2023)
- [An Introductory Guide to Koopman Learning](https://mlsanity.com/paper/2510.22002/) (2025)
- [Learning State-Space Models of Dynamic Systems from Arbitrary Data using Joint Embedding Predictive Architectures](https://mlsanity.com/paper/2508.10489/) (2025)

## Abstract (arXiv)

This paper proposes a novel data-driven algorithm to approximate the dominant eigenfunctions (aka.~modes) of the Koopman operator of nonlinear dynamical systems using neural networks. The relevance of learning the dominant Koopman modes is to approximate nonlinear dynamics by linear ones in a lifted space, thereby enabling simplified control and analysis. To fight the curse of dimensionality arising from using expressive templates (here neural networks) for the mode approximation, the proposed method leverages a power-iteration scheme that directly learns the dominant Koopman modes without explicitly constructing the projection of the Koopman operator on the template of functions. Our approach connects to other approaches in the literature that avoid the curse of dimensionality by learning small dictionaries of functions, but differs from them in that we do not require ``anti-collapse mechanisms'' to ensure that the learned dictionary is expressive enough to approximate the Koopman operator since our power-iteration scheme is designed to converge toward the dominant modes of the projected Koopman operator. The approach is fully data-driven, requiring only sampled state transitions. Theoretical guarantees are provided, showing convergence under increasing sample size and network width (in connection with the neural tangent kernel theorem). Numerical experiments demonstrate that the method achieves accurate and smooth approximations of dominant modes while avoiding the limitations of traditional techniques such as extended dynamic mode decomposition.