# Data-driven Reconstruction of Nonlinear Dynamics from Sparse Observation

## TL;DR

This paper introduces a fixed-point echo state network method to reconstruct complex nonlinear dynamical systems from extremely sparse time series data, demonstrating high accuracy even with minimal observations.

## Contribution

It proposes a novel fixed-point ESN approach for reconstructing nonlinear dynamics from sparse data, leveraging the universal approximation capability of ESNs.

## Key findings

- Reconstructs chaotic systems from up to 95% missing data
- Successfully reconstructs simple systems from only 1-2% data
- Outperforms traditional methods in data-scarce scenarios

## Abstract

We present a data-driven model to reconstruct nonlinear dynamics from a very sparse times series data, which relies on the strength of the echo state network (ESN) in learning nonlinear representation of data. With an assumption of the universal function approximation capability of ESN, it is shown that the reconstruction problem can be formulated as a fixed-point problem, in which the trajectory of the dynamical system is a fixed point of the ESN. An under-relaxed fixed-point iteration is proposed to reconstruct the nonlinear dynamics from a sparse observation. The proposed fixed-point ESN is tested against both univariate and multivariate chaotic dynamical systems by randomly removing up to 95% of the data. It is shown that the fixed-point ESN is able to reconstruct the complex dynamics from only 5 ~ 10% of the data. For a relatively simple non-chaotic dynamical system, the numerical experiments on a forced van der Pol oscillator show that it is possible to reconstruct the nonlinear dynamics from only 1~2% of the data.