Universal Learning of Nonlinear Dynamics

TL;DR

This paper introduces a spectral filtering algorithm for learning marginally stable nonlinear dynamical systems, providing provable guarantees and extending previous methods to noisy and asymmetric systems.

Contribution

It presents a novel spectral filtering approach with theoretical guarantees for learning a broad class of nonlinear dynamical systems, including noisy and marginally stable cases.

Findings

Proves vanishing prediction error for systems with finitely many marginally stable modes

Develops a spectral filtering algorithm for linear systems that handles noise and asymmetry

Generalizes previous spectral filtering techniques to more complex dynamical systems

Abstract

We study the fundamental problem of learning a marginally stable unknown nonlinear dynamical system. We describe an algorithm for this problem, based on the technique of spectral filtering, which learns a mapping from past observations to the next based on a spectral representation of the system. Using techniques from online convex optimization, we prove vanishing prediction error for any nonlinear dynamical system that has finitely many marginally stable modes, with rates governed by a novel quantitative control-theoretic notion of learnability. The main technical component of our method is a new spectral filtering algorithm for linear dynamical systems, which incorporates past observations and applies to general noisy and marginally stable systems. This significantly generalizes the original spectral filtering algorithm to both asymmetric dynamics as well as incorporating noise correction, and is of independent interest.