  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.
