# Deep Identification of Nonlinear Systems in Koopman Form

arXiv: https://arxiv.org/abs/2110.02583 · HTML: https://mlsanity.com/paper/2110.02583/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 21 references that are in the map.
- First posted on arXiv 6 Oct 2021.

## Builds on (most cited first, 10 of 10)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [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)
- [Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control](https://mlsanity.com/paper/1611.03537/) (2016)
- [Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1708.06850/) (2017)
- [Nonlinear state-space identification using deep encoder networks](https://mlsanity.com/paper/2012.07697/) (2020)
- [Koopman-based lifting techniques for nonlinear systems identification](https://mlsanity.com/paper/1709.02003/) (2017)
- [Non-linear State-space Model Identification from Video Data using Deep Encoders](https://mlsanity.com/paper/2012.07721/) (2020)

## Built on by (most cited first, 11 of 11)

- [Deep networks for system identification: a Survey](https://mlsanity.com/paper/2301.12832/) (2023)
- [On the adaptation of recurrent neural networks for system identification](https://mlsanity.com/paper/2201.08660/) (2022)
- [Meta-Learning of Neural State-Space Models Using Data From Similar Systems](https://mlsanity.com/paper/2211.07768/) (2022)
- [Learning neural state-space models: do we need a state estimator?](https://mlsanity.com/paper/2206.12928/) (2022)
- [Meta-Learning for Physically-Constrained Neural System Identification](https://mlsanity.com/paper/2501.06167/) (2025)
- [Koopman Data-Driven Predictive Control with Robust Stability and Recursive Feasibility Guarantees](https://mlsanity.com/paper/2405.01292/) (2024)
- [Initialization Approach for Nonlinear State-Space Identification via the Subspace Encoder Approach](https://mlsanity.com/paper/2304.02119/) (2023)
- [Automated Global Analysis of Experimental Dynamics through Low-Dimensional Linear Embeddings](https://mlsanity.com/paper/2411.00989/) (2024)
- [Koopman Operator Identification of Model Parameter Trajectories for Temporal Domain Generalization (KOMET)](https://mlsanity.com/paper/2603.26923/) (2026)
- [Autoregressive models for biomedical signal processing](https://mlsanity.com/paper/2304.11070/) (2023)
- [Online learning of neural state-space models](https://mlsanity.com/paper/2607.17614/) (2026)

## Abstract (arXiv)

The present paper treats the identification of nonlinear dynamical systems using Koopman-based deep state-space encoders. Through this method, the usual drawback of needing to choose a dictionary of lifting functions a priori is circumvented. The encoder represents the lifting function to the space where the dynamics are linearly propagated using the Koopman operator. An input-affine formulation is considered for the lifted model structure and we address both full and partial state availability. The approach is implemented using the the deepSI toolbox in Python. To lower the computational need of the simulation error-based training, the data is split into subsections where multi-step prediction errors are calculated independently. This formulation allows for efficient batch optimization of the network parameters and, at the same time, excellent long term prediction capabilities of the obtained models. The performance of the approach is illustrated by nonlinear benchmark examples.