# Data-driven system identification using quadratic embeddings of nonlinear dynamics

arXiv: https://arxiv.org/abs/2501.08202 · HTML: https://mlsanity.com/paper/2501.08202/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 48 references that are in the map.
- First posted on arXiv 14 Jan 2025.

## Builds on (most cited first, 16 of 16)

- [Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control](https://mlsanity.com/paper/1611.03537/) (2016)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [Sparse learning of stochastic dynamic equations](https://mlsanity.com/paper/1712.02432/) (2017)
- [Koopman Operator Dynamical Models: Learning, Analysis and Control](https://mlsanity.com/paper/2102.02522/) (2021)
- [Deep Neural Networks for Nonlinear Model Order Reduction of Unsteady Flows](https://mlsanity.com/paper/2007.00936/) (2020)
- [Koopman operator-based model reduction for switched-system control of PDEs](https://mlsanity.com/paper/1710.06759/) (2017)
- [Linear identification of nonlinear systems: A lifting technique based on the Koopman operator](https://mlsanity.com/paper/1605.04457/) (2016)
- [Physics-informed regularization and structure preservation for learning stable reduced models from data with operator inference](https://mlsanity.com/paper/2107.02597/) (2021)
- [Nonlinear Model Order Reduction via Lifting Transformations and Proper Orthogonal Decomposition](https://mlsanity.com/paper/1808.02086/) (2018)
- [Guaranteed Stable Quadratic Models and their applications in SINDy and Operator Inference](https://mlsanity.com/paper/2308.13819/) (2023)
- [Generalized Quadratic Embeddings for Nonlinear Dynamics using Deep Learning](https://mlsanity.com/paper/2211.00357/) (2022)
- [Dynamical systems and complex networks: A Koopman operator perspective](https://mlsanity.com/paper/2405.08940/) (2024)
- [Stability Domains for Quadratic-Bilinear Reduced-Order Models](https://mlsanity.com/paper/2009.02769/) (2020)
- [Abstraction of Elementary Hybrid Systems by Variable Transformation](https://mlsanity.com/paper/1403.7022/) (2014)
- [Koopman Spectral Linearization vs. Carleman Linearization: A Computational Comparison Study](https://mlsanity.com/paper/2310.19078/) (2023)
- [Dissipative quadratizations of polynomial ODE systems](https://mlsanity.com/paper/2311.02508/) (2023)

## Built on by (most cited first, 1 of 1)

- [Integral Formulation of QENDy for Robust Nonlinear System Identification](https://mlsanity.com/paper/2606.11629/) (2026)

## Abstract (arXiv)

We propose a novel data-driven method called QENDy (Quadratic Embedding of Nonlinear Dynamics) that not only allows us to learn quadratic representations of highly nonlinear dynamical systems, but also to identify the governing equations. The approach is based on an embedding of the system into a higher-dimensional feature space in which the dynamics become quadratic. Just like SINDy (Sparse Identification of Nonlinear Dynamics), our method requires trajectory data, time derivatives for the training data points, which can also be estimated using finite difference approximations, and a set of preselected basis functions, called dictionary. We illustrate the efficacy and accuracy of QENDy with the aid of various benchmark problems and compare its performance with SINDy and a deep learning method for identifying quadratic embeddings. Furthermore, we analyze the convergence of QENDy and SINDy in the infinite data limit, highlight their similarities and main differences, and compare the quadratic embedding with linearization techniques based on the Koopman operator.