  We present a comprehensive inter-comparison of linear regression (LR),
stochastic, and deep-learning approaches for reduced-order statistical
emulation of ocean circulation. The reference dataset is provided by an
idealized, eddy-resolving, double-gyre ocean circulation model. Our goal is to
conduct a systematic and comprehensive assessment and comparison of skill,
cost, and complexity of statistical models from the three methodological
classes.
  The model based on LR is considered as a baseline. Additionally, we
investigate its additive white noise augmentation and a multi-level stochastic
approach, deep-learning methods, hybrid frameworks (LR plus deep-learning), and
simple stochastic extensions of deep-learning and hybrid methods. The
assessment metrics considered are: root mean squared error, anomaly
cross-correlation, climatology, variance, frequency map, forecast horizon, and
computational cost.
  We found that the multi-level linear stochastic approach performs the best
for both short- and long-timescale forecasts. The deep-learning hybrid models
augmented by additive state-dependent white noise came second, while their
deterministic counterparts failed to reproduce the characteristic frequencies
in climate-range forecasts. Pure deep learning implementations performed worse
than LR and its noise augmentations. Skills of LR and its white noise extension
were similar on short timescales, but the latter performed better on long
timescales, while LR-only outputs decay to zero for long simulations.
  Overall, our analysis promotes multi-level LR stochastic models with memory
effects, and hybrid models with linear dynamical core augmented by additive
stochastic terms learned via deep learning, as a more practical, accurate, and
cost-effective option for ocean emulation than pure deep-learning solutions.
