# How deep is your network? Deep vs. shallow learning of transfer operators

arXiv: https://arxiv.org/abs/2509.19930 · HTML: https://mlsanity.com/paper/2509.19930/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 57 references that are in the map.
- First posted on arXiv 24 Sep 2025.

## Builds on (most cited first, 13 of 13)

- [VAMPnets: Deep learning of molecular kinetics](https://mlsanity.com/paper/1710.06012/) (2017)
- [Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1708.06850/) (2017)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [Variational approach for learning Markov processes from time series data](https://mlsanity.com/paper/1707.04659/) (2017)
- [Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/1712.01572/) (2017)
- [Deeptime: a Python library for machine learning dynamical models from time series data](https://mlsanity.com/paper/2110.15013/) (2021)
- [Escaping Saddle Points with Adaptive Gradient Methods](https://mlsanity.com/paper/1901.09149/) (2019)
- [Random vector functional link network: recent developments, applications, and future directions](https://mlsanity.com/paper/2203.11316/) (2022)
- [Singular Value Decomposition of Operators on Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/1807.09331/) (2018)
- [Kernel methods for detecting coherent structures in dynamical data](https://mlsanity.com/paper/1904.07752/) (2019)
- [Dynamical systems and complex networks: A Koopman operator perspective](https://mlsanity.com/paper/2405.08940/) (2024)
- [Two methods to approximate the Koopman operator with a reservoir computer](https://mlsanity.com/paper/2008.10263/) (2020)
- [Learning dynamical systems from data: Gradient-based dictionary optimization](https://mlsanity.com/paper/2411.04775/) (2024)

## Built on by (most cited first, 2 of 2)

- [Optimization of randomized neural networks for transfer operator approximation](https://mlsanity.com/paper/2605.23689/) (2026)
- [Bilevel optimization for data-driven learning of Koopman embeddings using kernel-based autoencoders](https://mlsanity.com/paper/2610.12370/) (2026)

## Abstract (arXiv)

We propose a randomized neural network approach called RaNNDy for learning transfer operators and their spectral decompositions from data. The weights of the hidden layers of the neural network are randomly selected and only the output layer is trained. The main advantage is that without a noticeable reduction in accuracy, this approach significantly reduces the training time and resources while avoiding common problems associated with deep learning such as sensitivity to hyperparameters and slow convergence. Additionally, the proposed framework allows us to compute a closed-form solution for the output layer which directly represents the eigenfunctions of the operator. Moreover, it is possible to estimate uncertainties associated with the computed spectral properties via ensemble learning. We present results for different dynamical operators, including Koopman and Perron-Frobenius operators, which have important applications in analyzing the behavior of complex dynamical systems, and the Schr"odinger operator. The numerical examples, which highlight the strengths but also weaknesses of the proposed framework, include several stochastic dynamical systems, protein folding processes, and the quantum harmonic oscillator.