  El Ni\~{n}o-Southern Oscillation (ENSO) is one of the significant climate
phenomena, which appears periodically in the tropic Pacific. The intermediate
coupled ocean-atmosphere Zebiak-Cane (ZC) model is the first and classical one
designed to numerically forecast the ENSO events. Traditionally, the
conditional nonlinear optimal perturbation (CNOP) approach has been used to
capture optimal precursors in practice. In this paper, based on
state-of-the-art statistical machine learning techniques, we investigate the
sampling algorithm proposed in [Shi and Sun, 2023] to obtain optimal precursors
via the CNOP approach in the ZC model. For the ZC model, or more generally, the
numerical models with dimension $\mathrm{O}(10^4-10^5)$, the numerical
performance, regardless of the statically spatial patterns and the dynamical
nonlinear time evolution behaviors as well as the corresponding quantities and
indices, shows the high efficiency of the sampling method by comparison with
the traditional adjoint method. The sampling algorithm does not only reduce the
gradient (first-order information) to the objective function value
(zeroth-order information) but also avoids the use of the adjoint model, which
is hard to develop in the coupled ocean-atmosphere models and the
parameterization models. In addition, based on the key characteristic that the
samples are independently and identically distributed, we can implement the
sampling algorithm by parallel computation to shorten the computation time.
Meanwhile, we also show in the numerical experiments that the important
features of optimal precursors can be still captured even when the number of
samples is reduced sharply.
