Universal Learning of Nonlinear Dynamics
This work addresses a fundamental challenge in control theory and machine learning for modeling complex systems, though it appears incremental as it generalizes existing spectral filtering methods.
The paper tackles the problem of learning unknown nonlinear dynamical systems with marginal stability, presenting an algorithm based on spectral filtering that achieves vanishing prediction error for systems with finitely many such modes, with rates defined by a new learnability measure.
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.