# Learning dynamically inspired bases for Koopman and transfer operator approximation

arXiv: https://arxiv.org/abs/2505.05085 · HTML: https://mlsanity.com/paper/2505.05085/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 36 references that are in the map.
- First posted on arXiv 8 May 2025.

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

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (2017)
- [Linearly-Recurrent Autoencoder Networks for Learning Dynamics](https://mlsanity.com/paper/1712.01378/) (2017)
- [Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1708.06850/) (2017)
- [Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems](https://mlsanity.com/paper/2111.14889/) (2021)
- [Deep Koopman Operator with Control for Nonlinear Systems](https://mlsanity.com/paper/2202.08004/) (2022)
- [Deep Learning Models for Global Coordinate Transformations that Linearize PDEs](https://mlsanity.com/paper/1911.02710/) (2019)
- [Carleman Linearization of Nonlinear Systems and Its Finite-Section Approximations](https://mlsanity.com/paper/2207.07755/) (2022)
- [Heterogeneous mixtures of dictionary functions to approximate subspace invariance in Koopman operators](https://mlsanity.com/paper/2206.13585/) (2022)

## Abstract (arXiv)

Transfer and Koopman operator methods offer a framework for representing complex, nonlinear dynamical systems via linear transformations, enabling a deeper understanding of the underlying dynamics. The spectra of these operators provide important insights into system predictability and emergent behaviour, although efficiently estimating them from data can be challenging. We approach this issue through the lens of general operator and representational learning, in which we approximate these linear operators using efficient finite-dimensional representations. Specifically, we machine-learn orthonormal basis functions that are dynamically tailored to the system. This learned basis provides a particularly accurate approximation of the operator's action and enables efficient recovery of eigenfunctions and invariant measures. We illustrate our approach with examples that showcase the retrieval of spectral properties from the estimated operator, and emphasise the dynamically adaptive quality of the machine-learned basis.