# Deep Learning Enhanced Dynamic Mode Decomposition

arXiv: https://arxiv.org/abs/2108.04433 · HTML: https://mlsanity.com/paper/2108.04433/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 39 references that are in the map.
- First posted on arXiv 10 Aug 2021.

## Builds on (most cited first, 9 of 9)

- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (2017)
- [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)
- [Forecasting Sequential Data using Consistent Koopman Autoencoders](https://mlsanity.com/paper/2003.02236/) (2020)
- [Physics-informed Autoencoders for Lyapunov-stable Fluid Flow Prediction](https://mlsanity.com/paper/1905.10866/) (2019)
- [Learning Compositional Koopman Operators for Model-Based Control](https://mlsanity.com/paper/1910.08264/) (2019)
- [Deep Learning of Conjugate Mappings](https://mlsanity.com/paper/2104.01874/) (2021)

## Built on by (most cited first, 19 of 19)

- [Koopman neural operator as a mesh-free solver of non-linear partial differential equations](https://mlsanity.com/paper/2301.10022/) (2023)
- [Koopman operator learning using invertible neural networks](https://mlsanity.com/paper/2306.17396/) (2023)
- [KoopmanLab: machine learning for solving complex physics equations](https://mlsanity.com/paper/2301.01104/) (2023)
- [LETS Forecast: Learning Embedology for Time Series Forecasting](https://mlsanity.com/paper/2506.06454/) (2025)
- [Enhancing Predictive Capabilities in Data-Driven Dynamical Modeling with Automatic Differentiation: Koopman and Neural ODE Approaches](https://mlsanity.com/paper/2310.06790/) (2023)
- [DLKoopman: A deep learning software package for Koopman theory](https://mlsanity.com/paper/2211.08992/) (2022)
- [Quaternion-based dynamic mode decomposition for background modeling in color videos](https://mlsanity.com/paper/2112.13982/) (2021)
- [Why should autoencoders work?](https://mlsanity.com/paper/2310.02250/) (2023)
- [Invertible Koopman neural operator for data-driven modeling of partial differential equations](https://mlsanity.com/paper/2503.19717/) (2025)
- [KIPPO: Koopman-Inspired Proximal Policy Optimization](https://mlsanity.com/paper/2505.14566/) (2025)
- [Accelerating Eigenvalue Dataset Generation via Chebyshev Subspace Filter](https://mlsanity.com/paper/2510.23215/) (2025)
- [RRAEDy: Adaptive Latent Linearization of Nonlinear Dynamical Systems](https://mlsanity.com/paper/2512.07542/) (2025)
- [KoopGen: Koopman Generator Networks for Representing and Predicting Dynamical Systems with Continuous Spectra](https://mlsanity.com/paper/2602.14011/) (2026)
- [Machine Learning Enhanced Hankel Dynamic-Mode Decomposition](https://mlsanity.com/paper/2303.06289/) (2023)
- [Forecasting Local Ionospheric Parameters Using Transformers](https://mlsanity.com/paper/2502.15093/) (2025)
- [Avoiding spectral pollution for transfer operators using residuals](https://mlsanity.com/paper/2507.16915/) (2025)
- [Finding Koopman Invariant Subspaces via Personalized PageRank](https://mlsanity.com/paper/2605.24666/) (2026)
- [Learning the Koopman Operator using Attention Free Transformers](https://mlsanity.com/paper/2606.23957/) (2026)
- [Data-driven Koopman mode approximation: A neural power iteration algorithm](https://mlsanity.com/paper/2608.26943/) (2026)

## Abstract (arXiv)

Koopman operator theory shows how nonlinear dynamical systems can be represented as an infinite-dimensional, linear operator acting on a Hilbert space of observables of the system. However, determining the relevant modes and eigenvalues of this infinite-dimensional operator can be difficult. The extended dynamic mode decomposition (EDMD) is one such method for generating approximations to Koopman spectra and modes, but the EDMD method faces its own set of challenges due to the need of user defined observables. To address this issue, we explore the use of autoencoder networks to simultaneously find optimal families of observables which also generate both accurate embeddings of the flow into a space of observables and submersions of the observables back into flow coordinates. This network results in a global transformation of the flow and affords future state prediction via the EDMD and the decoder network. We call this method the deep learning dynamic mode decomposition (DLDMD). The method is tested on canonical nonlinear data sets and is shown to produce results that outperform a standard DMD approach and enable data-driven prediction where the standard DMD fails.