Data-driven Reconstruction of Nonlinear Dynamics from Sparse Observation
This addresses the challenge of modeling complex systems with limited observations, which is incremental as it builds on existing ESN methods.
The authors tackled the problem of reconstructing nonlinear dynamics from sparse time series data by proposing a fixed-point echo state network (ESN) model, achieving reconstruction from only 5-10% of data for chaotic systems and 1-2% for simpler systems.
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.