# Safely Learning Dynamical Systems from Short Trajectories

arXiv: https://arxiv.org/abs/2011.12257 · HTML: https://mlsanity.com/paper/2011.12257/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Nov 2020.

## Built on by (most cited first, 5 of 5)

- [Learning Linearized Models from Nonlinear Systems with Finite Data](https://mlsanity.com/paper/2309.08805/) (2023)
- [Learning Linearized Models from Nonlinear Systems under Initialization Constraints with Finite Data](https://mlsanity.com/paper/2505.04954/) (2025)
- [Safely Learning Dynamical Systems](https://mlsanity.com/paper/2305.12284/) (2023)
- [Stability properties of gradient flow dynamics for the symmetric low-rank matrix factorization problem](https://mlsanity.com/paper/2411.15972/) (2024)
- [Safely Learning Controlled Stochastic Dynamics](https://mlsanity.com/paper/2506.02754/) (2025)

## Abstract (arXiv)

A fundamental challenge in learning to control an unknown dynamical system is to reduce model uncertainty by making measurements while maintaining safety. In this work, we formulate a mathematical definition of what it means to safely learn a dynamical system by sequentially deciding where to initialize the next trajectory. In our framework, the state of the system is required to stay within a given safety region under the (possibly repeated) action of all dynamical systems that are consistent with the information gathered so far. For our first two results, we consider the setting of safely learning linear dynamics. We present a linear programming-based algorithm that either safely recovers the true dynamics from trajectories of length one, or certifies that safe learning is impossible. We also give an efficient semidefinite representation of the set of initial conditions whose resulting trajectories of length two are guaranteed to stay in the safety region. For our final result, we study the problem of safely learning a nonlinear dynamical system. We give a second-order cone programming based representation of the set of initial conditions that are guaranteed to remain in the safety region after one application of the system dynamics.