Attractor learning in synchronized chaotic systems in the presence of unresolved scales

Recently, supermodels consisting of an ensemble of interacting models, synchronizing on a common solution, have been proposed as an alternative to the common non-interactive multi-model ensembles in order to improve climate predictions. The connection terms in the interacting ensemble are to be optimized based on the data. The supermodel approach has been successfully demonstrated in a number of simulation experiments with an assumed ground truth and a set of good, but imperfect models. The supermodels were optimized with respect to their short-term prediction error. Nevertheless, they produced long-term climatological behavior that was close to the long-term behavior of the assumed ground truth, even in cases where the long-term behavior of the imperfect models was very different. In these supermodel experiments, however, a perfect model class scenario was assumed, in which the ground truth and imperfect models belong to the same model class and only differ in parameter setting. In this paper, we consider the imperfect model class scenario, in which the ground truth model class is more complex than the model class of imperfect models due to unresolved scales. We perform two supermodel experiments in two toy problems. The first one consists of a chaotically driven Lorenz 63 oscillator ground truth and two Lorenz 63 oscillators with constant forcings as imperfect models. The second one is more realistic and consists of a global atmosphere model as ground truth and imperfect models that have perturbed parameters and reduced spatial resolution. In both problems, we find that supermodel optimization with respect to short-term prediction error can lead to a long-term climatological behavior that is worse than that of the imperfect models. However, we also show that attractor learning can remedy this problem, leading to supermodels with long-term behavior superior to the imperfect models.