# Learning Fractional-Order Dynamics from a Single Trajectory

arXiv: https://arxiv.org/abs/2609.18127 · HTML: https://mlsanity.com/paper/2609.18127/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 37 references that are in the map.
- First posted on arXiv 16 Sep 2026.

## Builds on (most cited first, 12 of 12)

- [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)
- [Non-asymptotic Identification of LTI Systems from a Single Trajectory](https://mlsanity.com/paper/1806.05722/) (2018)
- [Non-asymptotic Identification of Linear Dynamical Systems Using Multiple Trajectories](https://mlsanity.com/paper/2009.00739/) (2020)
- [Finite-time Identification of Stable Linear Systems: Optimality of the Least-Squares Estimator](https://mlsanity.com/paper/2003.07937/) (2020)
- [Single Trajectory Nonparametric Learning of Nonlinear Dynamics](https://mlsanity.com/paper/2202.08311/) (2022)
- [Fractal Patterns May Illuminate the Success of Next-Token Prediction](https://mlsanity.com/paper/2402.01825/) (2024)
- [A Sampling Complexity-aware Framework for Discrete-time Fractional-Order Dynamical System Identification](https://mlsanity.com/paper/2501.17499/) (2025)
- [A Tale of Two Structures: Do LLMs Capture the Fractal Complexity of Language?](https://mlsanity.com/paper/2502.14924/) (2025)
- [On Learning Discrete-Time Fractional-Order Dynamical Systems](https://mlsanity.com/paper/2103.14975/) (2021)
- [Learning to Stabilize Unknown LTI Systems on a Single Trajectory under Stochastic Noise](https://mlsanity.com/paper/2406.00234/) (2024)
- [End-to-End Learning Framework for Solving Non-Markovian Optimal Control](https://mlsanity.com/paper/2502.04649/) (2025)

## Abstract (arXiv)

Many real-world processes exhibit long-range dependence, where the current state depends on a slowly decaying trace of past states rather than on the most recent state alone. This paper studies system identification for discrete-time fractional-order linear time-invariant systems from a single observed trajectory of length t, a setting that captures such non-Markovian dynamics through the Gr"unwald--Letnikov difference operator. Unlike Markovian systems, fractional-order systems couple estimation across the entire history, making both statistical analysis and practical identification more challenging. We propose Fractional-Order Ordinary-Least-Squares Grid-Search (FO-GS), a simple two-stage estimator that exploits the diagonal structure of the fractional-difference operator to decouple the identification problem row-wise. Under the stability assumption, we establish high-probability, non-asymptotic error bounds for estimating both the fractional order and the system matrix in the heterogeneous setting, with both estimation errors scaling as (O(t-1/2)). Through experiments, we show that FO-GS outperforms existing baselines in recovering both the fractional order and the underlying system dynamics.