# Learning nonlinear dynamical systems from a single trajectory

arXiv: https://arxiv.org/abs/2004.14681 · HTML: https://mlsanity.com/paper/2004.14681/

## Key facts

- Cited by 46 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Apr 2020.

## Built on by (most cited first, 30 of 46)

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- [Near-optimal Offline and Streaming Algorithms for Learning Non-Linear Dynamical Systems](https://mlsanity.com/paper/2105.11558/) (2021)
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- [Safe Adaptive Learning-based Control for Constrained Linear Quadratic Regulators with Regret Guarantees](https://mlsanity.com/paper/2111.00411/) (2021)
- [Smoothed Online Learning for Prediction in Piecewise Affine Systems](https://mlsanity.com/paper/2301.11187/) (2023)
- [Learning the Uncertainty Sets for Control Dynamics via Set Membership: A Non-Asymptotic Analysis](https://mlsanity.com/paper/2309.14648/) (2023)
- [Online Nonstochastic Control with Adversarial and Static Constraints](https://mlsanity.com/paper/2302.02426/) (2023)
- [Oracle-Efficient Smoothed Online Learning for Piecewise Continuous Decision Making](https://mlsanity.com/paper/2302.05430/) (2023)
- [Leveraging Neural Koopman Operators to Learn Continuous Representations of Dynamical Systems from Scarce Data](https://mlsanity.com/paper/2303.06972/) (2023)
- [Non-asymptotic System Identification for Linear Systems with Nonlinear Policies](https://mlsanity.com/paper/2306.10369/) (2023)
- [PAC-Bayes Generalisation Bounds for Dynamical Systems Including Stable RNNs](https://mlsanity.com/paper/2312.09793/) (2023)
- [NeoRL: Efficient Exploration for Nonepisodic RL](https://mlsanity.com/paper/2406.01175/) (2024)
- [Modeling Latent Non-Linear Dynamical System over Time Series](https://mlsanity.com/paper/2412.08114/) (2024)
- [Outlier-Robust Linear System Identification Under Heavy-tailed Noise](https://mlsanity.com/paper/2501.00421/) (2024)
- [In-Context Learning of Linear Dynamical Systems with Transformers: Approximation Bounds and Depth-Separation](https://mlsanity.com/paper/2502.08136/) (2025)
- [Distributed Online System Identification for LTI Systems Using Reverse Experience Replay](https://mlsanity.com/paper/2207.01062/) (2022)
- [Learning Linearized Models from Nonlinear Systems with Finite Data](https://mlsanity.com/paper/2309.08805/) (2023)
- [Rate-Optimal Non-Asymptotics for the Quadratic Prediction Error Method](https://mlsanity.com/paper/2404.07937/) (2024)
- [Long-Context Linear System Identification](https://mlsanity.com/paper/2410.05690/) (2024)
- [Adaptive Symmetry Discovery for Dynamical System Identification](https://mlsanity.com/paper/2608.08091/) (2026)

## Abstract (arXiv)

We introduce algorithms for learning nonlinear dynamical systems of the form xt+1=σ(Thetastarxt)+varepsilont, where Thetastar is a weight matrix, σ is a nonlinear link function, and varepsilont is a mean-zero noise process. We give an algorithm that recovers the weight matrix Thetastar from a single trajectory with optimal sample complexity and linear running time. The algorithm succeeds under weaker statistical assumptions than in previous work, and in particular i) does not require a bound on the spectral norm of the weight matrix Thetastar (rather, it depends on a generalization of the spectral radius) and ii) enjoys guarantees for non-strictly-increasing link functions such as the ReLU. Our analysis has two key components: i) we give a general recipe whereby global stability for nonlinear dynamical systems can be used to certify that the state-vector covariance is well-conditioned, and ii) using these tools, we extend well-known algorithms for efficiently learning generalized linear models to the dependent setting.