# Reconstructing nonlinear networks subject to fast-varying noises by using linearization with expanded variables

## TL;DR

This paper introduces a linearization method using expanded variables to reconstruct nonlinear networks affected by fast-varying noise, enabling accurate recovery from low-frequency data.

## Contribution

It presents a novel approach that linearizes nonlinear differential equations with expanded variables and uses two-time correlations for noise decomposition, improving network reconstruction.

## Key findings

- Successfully reconstructs nonlinear networks with all nonlinearities and noise characteristics.
- Effective at low sampling frequencies, maintaining reconstruction accuracy.
- Numerical results confirm the theoretical validity of the method.

## Abstract

Reconstructing noise-driven nonlinear networks from time series of output variables is a challenging problem, which turns to be very difficult when nonlinearity of dynamics, strong noise impacts and low measurement frequencies jointly affect. In this Letter, we propose a general method that introduces a number of nonlinear terms of the measurable variables as artificial and new variables, and uses the expanded variables to linearize nonlinear differential equations. Moreover, we use two-time correlations to decompose effects of system dynamics and noise driving. With these transformations, reconstructing nonlinear dynamics of the original system is equivalent to solving linear dynamics of the expanded system at the least squares approximations. We can well reconstruct nonlinear networks, including all dynamic nonlinearities, network links, and noise statistical characteristics, as sampling frequency is rather low. Numerical results fully verify the validity of theoretical derivations.