# Koopman-based lifting techniques for nonlinear systems identification

arXiv: https://arxiv.org/abs/1709.02003 · HTML: https://mlsanity.com/paper/1709.02003/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Sep 2017.

## Built on by (most cited first, 18 of 18)

- [Learning Stable Koopman Embeddings](https://mlsanity.com/paper/2110.06509/) (2021)
- [Deep Identification of Nonlinear Systems in Koopman Form](https://mlsanity.com/paper/2110.02583/) (2021)
- [Koopman operators with intrinsic observables in rigged reproducing kernel Hilbert spaces](https://mlsanity.com/paper/2403.02524/) (2024)
- [Modeling Unknown Stochastic Dynamical System Subject to External Excitation](https://mlsanity.com/paper/2406.15747/) (2024)
- [On the relationship between Koopman operator approximations and neural ordinary differential equations for data-driven time-evolution predictions](https://mlsanity.com/paper/2411.12940/) (2024)
- [A purely data-driven framework for prediction, optimization, and control of networked processes: application to networked SIS epidemic model](https://mlsanity.com/paper/2108.02005/) (2021)
- [Learning Koopman-based Stability Certificates for Unknown Nonlinear Systems](https://mlsanity.com/paper/2412.02807/) (2024)
- [Koopman operator theory: fundamentals, control, and applications](https://mlsanity.com/paper/2607.01819/) (2026)
- [Estimating Koopman operators for nonlinear dynamical systems: a nonparametric approach](https://mlsanity.com/paper/2103.13752/) (2021)
- [Learned Lifted Linearization Applied to Unstable Dynamic Systems Enabled by Koopman Direct Encoding](https://mlsanity.com/paper/2210.13602/) (2022)
- [Learning Invariant Subspaces of Koopman Operators--Part 1: A Methodology for Demonstrating a Dictionary's Approximate Subspace Invariance](https://mlsanity.com/paper/2212.07358/) (2022)
- [Modeling Nonlinear Dynamics in Continuous Time with Inductive Biases on Decay Rates and/or Frequencies](https://mlsanity.com/paper/2212.13033/) (2022)
- [SINDy vs Hard Nonlinearities and Hidden Dynamics: a Benchmarking Study](https://mlsanity.com/paper/2403.00578/) (2024)
- [Finite-dimensional approximations of push-forwards on locally analytic functionals](https://mlsanity.com/paper/2404.10769/) (2024)
- [Analysis of Truncated Singular Value Decomposition for Koopman Operator-Based Lane Change Model](https://mlsanity.com/paper/2409.18586/) (2024)
- [Tensor-based computation of the Koopman generator via operator logarithm](https://mlsanity.com/paper/2604.07685/) (2026)
- [Flow map learning in nonlinear vector autoregressive models: influence of the feature-library structure on the training error](https://mlsanity.com/paper/2605.31438/) (2026)
- [Data-Driven Learning of Unknown Nonlinear Differential Equations Using Functional Analysis](https://mlsanity.com/paper/2609.04329/) (2026)

## Abstract (arXiv)

We develop a novel lifting technique for nonlinear system identification based on the framework of the Koopman operator. The key idea is to identify the linear (infinitedimensional) Koopman operator in the lifted space of observables, instead of identifying the nonlinear system in the state space, a process which results in a linear method for nonlinear systems identification. The proposed lifting technique is an indirect method that does not require to compute time derivatives and is therefore well-suited to low-sampling rate datasets. Considering different finite-dimensional subspaces to approximate and identify the Koopman operator, we propose two numerical schemes: the main method and the dual method. The main method is a parametric identification technique that can accurately reconstruct the vector field of a broad class of systems (including unstable, chaotic, and system with inputs). The dual method provides estimates of the vector field at the data points and is well-suited to identify high-dimensional systems with small datasets. The present paper describes the two methods, provide theoretical convergence results, and illustrate the lifting techniques with several examples.