# Learning Linearized Models from Nonlinear Systems with Finite Data

arXiv: https://arxiv.org/abs/2309.08805 · HTML: https://mlsanity.com/paper/2309.08805/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 34 references that are in the map.
- First posted on arXiv 15 Sep 2023.

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

- [On the Sample Complexity of the Linear Quadratic Regulator](https://mlsanity.com/paper/1710.01688/) (2017)
- [Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification](https://mlsanity.com/paper/1802.08334/) (2018)
- [Near optimal finite time identification of arbitrary linear dynamical systems](https://mlsanity.com/paper/1812.01251/) (2018)
- [Active Learning for Nonlinear System Identification with Guarantees](https://mlsanity.com/paper/2006.10277/) (2020)
- [Non-asymptotic Identification of LTI Systems from a Single Trajectory](https://mlsanity.com/paper/1806.05722/) (2018)
- [Finite Time Identification in Unstable Linear Systems](https://mlsanity.com/paper/1710.01852/) (2017)
- [Learning Linear Dynamical Systems with Semi-Parametric Least Squares](https://mlsanity.com/paper/1902.00768/) (2019)
- [Learning nonlinear dynamical systems from a single trajectory](https://mlsanity.com/paper/2004.14681/) (2020)
- [Non-asymptotic and Accurate Learning of Nonlinear Dynamical Systems](https://mlsanity.com/paper/2002.08538/) (2020)
- [Non-asymptotic Identification of Linear Dynamical Systems Using Multiple Trajectories](https://mlsanity.com/paper/2009.00739/) (2020)
- [Linear identification of nonlinear systems: A lifting technique based on the Koopman operator](https://mlsanity.com/paper/1605.04457/) (2016)
- [Nonparametric Finite Time LTI System Identification](https://mlsanity.com/paper/1902.01848/) (2019)
- [Learning the Dynamics of Autonomous Linear Systems From Multiple Trajectories](https://mlsanity.com/paper/2203.12794/) (2022)
- [Learning Dynamical Systems by Leveraging Data from Similar Systems](https://mlsanity.com/paper/2302.04344/) (2023)
- [On the Sample Complexity of Decentralized Linear Quadratic Regulator with Partially Nested Information Structure](https://mlsanity.com/paper/2110.07112/) (2021)
- [Safely Learning Dynamical Systems from Short Trajectories](https://mlsanity.com/paper/2011.12257/) (2020)

## Built on by (most cited first, 3 of 3)

- [Combining Federated Learning and Control: A Survey](https://mlsanity.com/paper/2407.11069/) (2024)
- [Online Change Points Detection for Linear Dynamical Systems with Finite Sample Guarantees](https://mlsanity.com/paper/2311.18769/) (2023)
- [Learning Linearized Models from Nonlinear Systems under Initialization Constraints with Finite Data](https://mlsanity.com/paper/2505.04954/) (2025)

## Abstract (arXiv)

Identifying a linear system model from data has wide applications in control theory. The existing work on finite sample analysis for linear system identification typically uses data from a single system trajectory under i.i.d random inputs, and assumes that the underlying dynamics is truly linear. In contrast, we consider the problem of identifying a linearized model when the true underlying dynamics is nonlinear. We provide a multiple trajectories-based deterministic data acquisition algorithm followed by a regularized least squares algorithm, and provide a finite sample error bound on the learned linearized dynamics. Our error bound demonstrates a trade-off between the error due to nonlinearity and the error due to noise, and shows that one can learn the linearized dynamics with arbitrarily small error given sufficiently many samples. We validate our results through experiments, where we also show the potential insufficiency of linear system identification using a single trajectory with i.i.d random inputs, when nonlinearity does exist.