  Ensemble forecast based on physics-informed models is one of the most widely
used forecast algorithms for complex turbulent systems. A major difficulty in
such a method is the model error that is ubiquitous in practice. Data-driven
machine learning (ML) forecasts can reduce the model error but they often
suffer from the partial and noisy observations. In this paper, a simple but
effective Bayesian machine learning advanced forecast ensemble (BAMCAFE) method
is developed, which combines an available imperfect physics-informed model with
data assimilation (DA) to facilitate the ML ensemble forecast. In the BAMCAFE
framework, a Bayesian ensemble DA is applied to create the training data of the
ML model, which reduces the intrinsic error in the imperfect physics-informed
model simulations and provides the training data of the unobserved variables.
Then a generalized DA is employed for the initialization of the ML ensemble
forecast. In addition to forecasting the optimal point-wise value, the BAMCAFE
also provides an effective approach of quantifying the forecast uncertainty
utilizing a non-Gaussian probability density function that characterizes the
intermittency and extreme events. It is shown using a two-layer Lorenz 96 model
that the BAMCAFE method can significant improve the forecasting skill compared
to the typical reduced-order imperfect models with bare truncation or
stochastic parameterization for both the observed and unobserved large-scale
variables. It is also shown via a nonlinear conceptual model that the BAMCAFE
leads to a comparable non-Gaussian forecast uncertainty as the perfect model
while the associated imperfect physics-informed model suffers from large
forecast biases.
