# Leveraging Neural Koopman Operators to Learn Continuous Representations of Dynamical Systems from Scarce Data

arXiv: https://arxiv.org/abs/2303.06972 · HTML: https://mlsanity.com/paper/2303.06972/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 26 references that are in the map.
- First posted on arXiv 13 Mar 2023.

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

- [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)
- [Linearly-Recurrent Autoencoder Networks for Learning Dynamics](https://mlsanity.com/paper/1712.01378/) (2017)
- [Forecasting Sequential Data using Consistent Koopman Autoencoders](https://mlsanity.com/paper/2003.02236/) (2020)
- [On orthogonality and learning recurrent networks with long term dependencies](https://mlsanity.com/paper/1702.00071/) (2017)
- [Combining data assimilation and machine learning to emulate a dynamical model from sparse and noisy observations: a case study with the Lorenz 96 model](https://mlsanity.com/paper/2001.01520/) (2020)
- [Deep Dynamical Modeling and Control of Unsteady Fluid Flows](https://mlsanity.com/paper/1805.07472/) (2018)
- [Learning Compositional Koopman Operators for Model-Based Control](https://mlsanity.com/paper/1910.08264/) (2019)
- [Learning nonlinear dynamical systems from a single trajectory](https://mlsanity.com/paper/2004.14681/) (2020)
- [Learning Dynamical Systems from Partial Observations](https://mlsanity.com/paper/1902.11136/) (2019)
- [Learning Stable Models for Prediction and Control](https://mlsanity.com/paper/2005.04291/) (2020)
- [EM-like Learning Chaotic Dynamics from Noisy and Partial Observations](https://mlsanity.com/paper/1903.10335/) (2019)

## Built on by (most cited first, 5 of 5)

- [Neural Koopman prior for data assimilation](https://mlsanity.com/paper/2309.05317/) (2023)
- [Augmented Invertible Koopman Autoencoder for long-term time series forecasting](https://mlsanity.com/paper/2503.12930/) (2025)
- [Learning Sentinel-2 reflectance dynamics for data-driven assimilation and forecasting](https://mlsanity.com/paper/2305.03743/) (2023)
- [Course Correcting Koopman Representations](https://mlsanity.com/paper/2310.15386/) (2023)
- [Koopman Ensembles for Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2403.06757/) (2024)

## Abstract (arXiv)

Over the last few years, several works have proposed deep learning architectures to learn dynamical systems from observation data with no or little knowledge of the underlying physics. A line of work relies on learning representations where the dynamics of the underlying phenomenon can be described by a linear operator, based on the Koopman operator theory. However, despite being able to provide reliable long-term predictions for some dynamical systems in ideal situations, the methods proposed so far have limitations, such as requiring to discretize intrinsically continuous dynamical systems, leading to data loss, especially when handling incomplete or sparsely sampled data. Here, we propose a new deep Koopman framework that represents dynamics in an intrinsically continuous way, leading to better performance on limited training data, as exemplified on several datasets arising from dynamical systems.