从单条轨迹学习分数阶动力学

中文摘要 AI 辅助

许多现实世界的过程表现出长期依赖性，其中当前状态依赖于过去状态的缓慢衰减轨迹，而非仅依赖于最近的状态。本文研究了从长度为$t$的单条观测轨迹中对离散时间分数阶线性时不变系统进行系统辨识，这一设定通过Grünwald--Letnikov差分算子捕捉此类非马尔可夫动力学。与马尔可夫系统不同，分数阶系统将估计耦合在整个历史中，这使得统计分析和实际辨识都更具挑战性。我们提出了分数阶普通最小二乘网格搜索（FO-GS），一种简单的两阶段估计器，利用分数阶差分算子的对角结构将辨识问题按行解耦。在稳定性假设下，我们建立了在异质设定中估计分数阶和系统矩阵的高概率非渐近误差界，两种估计误差均按$\mathcal{O}(t^{-1/2})$缩放。通过实验，我们展示了FO-GS在恢复分数阶和底层系统动力学方面优于现有基线方法。

英文摘要

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 Grünwald--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 \emph{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 \(\mathcal{O}(t^{-1/2})\). Through experiments, we show that \emph{FO-GS} outperforms existing baselines in recovering both the fractional order and the underlying system dynamics.

发表机构

• University of Southern California（南加州大学）
• Carnegie Mellon University（卡内基梅隆大学）

机构由 AI 辅助整理，请以论文原文为准。