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Simulator imperfection, often known as model error, is ubiquitous inpractical data assimilation problems. Despite the enormous efforts dedicated toaddressing this problem, properly handling simulator imperfection in dataassimilation remains to be a challenging task. In this work, we propose anapproach to dealing with simulator imperfection from a point of view offunctional approximation that can be implemented through a certain machinelearning method, such as kernel-based learning adopted in the current work. Tothis end, we start from considering a class of supervised learning problems,and then identify similarities between supervised learning and variational dataassimilation. These similarities found the basis for us to develop anensemble-based learning framework to tackle supervised learning problems, whileachieving various advantages of ensemble-based methods over the variationalones. After establishing the ensemble-based learning framework, we proceed toinvestigate the integration of ensemble-based learning into an ensemble-baseddata assimilation framework to handle simulator imperfection. In the course ofour investigations, we also develop a strategy to tackle the issue ofmulti-modality in supervised-learning problems, and transfer this strategy todata assimilation problems to help improve assimilation performance. Fordemonstration, we apply the ensemble-based learning framework and theintegrated, ensemble-based data assimilation framework to a supervised learningproblem and a data assimilation problem with an imperfect forward simulator,respectively. The experiment results indicate that both frameworks achieve goodperformance in relevant case studies, and that functional approximation throughmachine learning may serve as a viable way to account for simulatorimperfection in data assimilation problems.

Ensemble-based kernel learning for a class of data assimilation problems with imperfect forward simulators