Learning Fractional-Order Dynamics from a Single Trajectory

arXiv:2609.18127v1 Announce Type: new Abstract: 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 Grunwald--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.

Source: arXiv cs.LG | 2026-09-17