Learning switched non-linear dynamical systems from a single trajectory

2026-07-26

AI Summary

This work addresses the lack of non-asymptotic theoretical guarantees for learning time-varying switched nonlinear dynamical systems from a single trajectory. Under assumptions of system stability and i.i.d. switching, the authors employ empirical risk minimization and derive a non-asymptotic upper bound on the prediction risk by leveraging the metric entropy of the underlying function class. This analysis provides the first explicit non-asymptotic theoretical guarantee for this class of problems, quantitatively linking convergence rates to the effective sample size. Explicit convergence rates are established for both Hölder continuous and linear function classes, and numerical experiments corroborate the theoretical findings.

Abstract

We study empirical risk minimization for learning non-linear dynamical systems whose transition dynamics may switch over time. Under stability assumptions, and i.i.d switching over a set of $K$ modes, we derive non-asymptotic bounds on the prediction risk expressed in terms of the metric entropy of the underlying function class. We instantiate our general result for Hölder and linear function classes, obtaining explicit convergence rates that depend on the effective sample size $Tp_i$, where $T$ is the trajectory length and $p_i$ is the probability of observing mode $i$. Numerical simulations support our theoretical findings. To the best of our knowledge, these results are the first non-asymptotic guarantees for learning switched nonlinear dynamical systems from a single trajectory.

Problem

• switched dynamical systems
• non-linear dynamics
• single trajectory learning
• mode switching
• prediction risk

Innovation

• switched dynamical systems
• non-asymptotic guarantees
• metric entropy
• single trajectory learning
• nonlinear dynamics