BAMCAFE: A Bayesian Machine Learning Advanced Forecast Ensemble Method for Complex Nonlinear Turbulent Systems with Partial Observations

Ensemble forecast based on physics-informed models is one of the most widely used forecast algorithms for complex nonlinear 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 overcome 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 accurate quantification of 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.