# Non-asymptotic and Accurate Learning of Nonlinear Dynamical Systems

arXiv: https://arxiv.org/abs/2002.08538 · HTML: https://mlsanity.com/paper/2002.08538/

## Key facts

- Cited by 39 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Feb 2020.

## Built on by (most cited first, 30 of 39)

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- [A Tutorial on the Non-Asymptotic Theory of System Identification](https://mlsanity.com/paper/2309.03873/) (2023)
- [Optimal Exploration for Model-Based RL in Nonlinear Systems](https://mlsanity.com/paper/2306.09210/) (2023)
- [Streaming Linear System Identification with Reverse Experience Replay](https://mlsanity.com/paper/2103.05896/) (2021)
- [Near-optimal Offline and Streaming Algorithms for Learning Non-Linear Dynamical Systems](https://mlsanity.com/paper/2105.11558/) (2021)
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- [Finite Sample Identification of Bilinear Dynamical Systems](https://mlsanity.com/paper/2208.13915/) (2022)
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- [Statistical Safety and Robustness Guarantees for Feedback Motion Planning of Unknown Underactuated Stochastic Systems](https://mlsanity.com/paper/2212.06874/) (2022)
- [Smoothed Online Learning for Prediction in Piecewise Affine Systems](https://mlsanity.com/paper/2301.11187/) (2023)
- [Active Learning for Control-Oriented Identification of Nonlinear Systems](https://mlsanity.com/paper/2404.09030/) (2024)
- [Strong Consistency and Rate of Convergence of Switched Least Squares System Identification for Autonomous Markov Jump Linear Systems](https://mlsanity.com/paper/2112.10753/) (2021)
- [Continuized Acceleration for Quasar Convex Functions in Non-Convex Optimization](https://mlsanity.com/paper/2302.07851/) (2023)
- [Learning Linear Dynamics from Bilinear Observations](https://mlsanity.com/paper/2409.16499/) (2024)
- [Oracle-Efficient Smoothed Online Learning for Piecewise Continuous Decision Making](https://mlsanity.com/paper/2302.05430/) (2023)
- [Non-asymptotic System Identification for Linear Systems with Nonlinear Policies](https://mlsanity.com/paper/2306.10369/) (2023)
- [Finite Sample Identification of Partially Observed Bilinear Dynamical Systems](https://mlsanity.com/paper/2501.07652/) (2025)
- [NeoRL: Efficient Exploration for Nonepisodic RL](https://mlsanity.com/paper/2406.01175/) (2024)
- [Online Stochastic Gradient Descent Learns Linear Dynamical Systems from A Single Trajectory](https://mlsanity.com/paper/2102.11822/) (2021)
- [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)
- [Concentration Phenomenon for Random Dynamical Systems: An Operator Theoretic Approach](https://mlsanity.com/paper/2212.03670/) (2022)
- [The Complexity of Sequential Prediction in Dynamical Systems](https://mlsanity.com/paper/2402.06614/) (2024)
- [Exact Recovery Guarantees for Parameterized Nonlinear System Identification Problem under Sparse Disturbances or Semi-Oblivious Attacks](https://mlsanity.com/paper/2409.00276/) (2024)

## Abstract (arXiv)

We consider the problem of learning stabilizable systems governed by nonlinear state equation ht+1=φ(ht,ut;θ)+wt. Here θ is the unknown system dynamics, ht  is the state, ut is the input and wt is the additive noise vector. We study gradient based algorithms to learn the system dynamics θ from samples obtained from a single finite trajectory. If the system is run by a stabilizing input policy, we show that temporally-dependent samples can be approximated by i.i.d. samples via a truncation argument by using mixing-time arguments. We then develop new guarantees for the uniform convergence of the gradients of empirical loss. Unlike existing work, our bounds are noise sensitive which allows for learning ground-truth dynamics with high accuracy and small sample complexity. Together, our results facilitate efficient learning of the general nonlinear system under stabilizing policy. We specialize our guarantees to entry-wise nonlinear activations and verify our theory in various numerical experiments