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Universal Learning of Nonlinear Dynamics
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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.
Forward citations
Cited by 2 Pith papers
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Spectral Distillation: From Nonlinear Dynamics to Linear State-Space Models
A convex spectral learner followed by spectral-to-LDS distillation extracts compact linear state-space predictors from nonlinear dynamics with a provable one-step error decomposition.
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SFO: Learning PDE Operators via Spectral Filtering
A neural operator that expands PDE kernels in fixed Hilbert-matrix eigenmodes achieves state-of-the-art benchmark accuracy with substantially fewer parameters.
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