Dynamic models with $p$ parameters are identified by $2p+1$ random features

2026-07-17

AI Summary

This work addresses the challenge of modeling dynamical systems subject to state-dependent, non-i.i.d., and non-Gaussian noise by proposing a general identification framework. By integrating dynamical system embedding theory with random feature mappings, the method extends classical noise-free system identification approaches to complex stochastic environments. It establishes that only \(2p+1\) random features are sufficient to uniquely identify continuous or discrete-time dynamical models containing \(p\) parameters. Theoretical analysis provides identifiability guarantees for a broad class of stochastic dynamical systems, while numerical experiments on the Lorenz-63 system and Hénon map demonstrate the method’s efficacy in accurately recovering underlying system structures from observations corrupted by strongly correlated, non-Gaussian noise.

Abstract

A foundational principle in nonlinear dynamics is that the structure of a dynamical system can be recovered from a small number of generic measurements or coordinates. We develop an analogous principle for the identification of dynamic models for time series {\em with noise}, which builds on previous identification results for noiseless dynamical systems. The noise is allowed to be non-iid, non-Gaussian, and dependent on the state. Our results cover noisily observed differential equations and discrete-time dynamical systems, as well as stochastic models with process noise. We illustrate the utility of this identification principle using a Lorenz-63 model and a Hénon map model, both with observational noise.

Problem

• dynamic models
• system identification
• observational noise
• nonlinear dynamics
• time series

Innovation

• dynamic model identification
• random features
• noisy time series
• non-Gaussian noise
• state-dependent noise