# Learned Lifted Linearization Applied to Unstable Dynamic Systems Enabled by Koopman Direct Encoding

arXiv: https://arxiv.org/abs/2210.13602 · HTML: https://mlsanity.com/paper/2210.13602/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 22 references that are in the map.
- First posted on arXiv 24 Oct 2022.

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

- [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)
- [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)
- [Optimal construction of Koopman eigenfunctions for prediction and control](https://mlsanity.com/paper/1810.08733/) (2018)
- [Deep Learning of Koopman Representation for Control](https://mlsanity.com/paper/2010.07546/) (2020)
- [Koopman-based lifting techniques for nonlinear systems identification](https://mlsanity.com/paper/1709.02003/) (2017)
- [Learning Stable Koopman Embeddings](https://mlsanity.com/paper/2110.06509/) (2021)
- [Diffeomorphically Learning Stable Koopman Operators](https://mlsanity.com/paper/2112.04085/) (2021)
- [Learning Stable Models for Prediction and Control](https://mlsanity.com/paper/2005.04291/) (2020)
- [A Class of Logistic Functions for Approximating State-Inclusive Koopman Operators](https://mlsanity.com/paper/1712.03132/) (2017)
- [Global, Unified Representation of Heterogenous Robot Dynamics Using Composition Operators: A Koopman Direct Encoding Method](https://mlsanity.com/paper/2208.00175/) (2022)

## Abstract (arXiv)

This paper presents a Koopman lifting linearization method that is applicable to nonlinear dynamical systems having both stable and unstable regions. It is known that DMD and other standard data-driven methods face a fundamental difficulty in constructing a Koopman model when applied to unstable systems. Here we solve the problem by incorporating knowledge about a nonlinear state equation with a learning method for finding an effective set of observables. In a lifted space, stable and unstable regions are separated into independent subspaces. Based on this property, we propose to find effective observables through neural net training where training data are separated into stable and unstable trajectories. The resultant learned observables are used for constructing a linear state transition matrix using method known as Direct Encoding, which transforms the nonlinear state equation to a state transition matrix through inner product computations with the observables. The proposed method shows a dramatic improvement over existing DMD and data-driven methods.