  Simulator imperfection, often known as model error, is ubiquitous in
practical data assimilation problems. Despite the enormous efforts dedicated to
addressing this problem, properly handling simulator imperfection in data
assimilation remains to be a challenging task. In this work, we propose an
approach to dealing with simulator imperfection from a point of view of
functional approximation that can be implemented through a certain machine
learning method, such as kernel-based learning adopted in the current work. To
this end, we start from considering a class of supervised learning problems,
and then identify similarities between supervised learning and variational data
assimilation. These similarities found the basis for us to develop an
ensemble-based learning framework to tackle supervised learning problems, while
achieving various advantages of ensemble-based methods over the variational
ones. After establishing the ensemble-based learning framework, we proceed to
investigate the integration of ensemble-based learning into an ensemble-based
data assimilation framework to handle simulator imperfection. In the course of
our investigations, we also develop a strategy to tackle the issue of
multi-modality in supervised-learning problems, and transfer this strategy to
data assimilation problems to help improve assimilation performance. For
demonstration, we apply the ensemble-based learning framework and the
integrated, ensemble-based data assimilation framework to a supervised learning
problem and a data assimilation problem with an imperfect forward simulator,
respectively. The experiment results indicate that both frameworks achieve good
performance in relevant case studies, and that functional approximation through
machine learning may serve as a viable way to account for simulator
imperfection in data assimilation problems.
