  Chaotic systems, such as turbulent flows, are ubiquitous in science and
engineering. However, their study remains a challenge due to the large range
scales, and the strong interaction with other, often not fully understood,
physics. As a consequence, the spatiotemporal resolution required for accurate
simulation of these systems is typically computationally infeasible,
particularly for applications of long-term risk assessment, such as the
quantification of extreme weather risk due to climate change. While data-driven
modeling offers some promise of alleviating these obstacles, the scarcity of
high-quality simulations results in limited available data to train such
models, which is often compounded by the lack of stability for long-horizon
simulations. As such, the computational, algorithmic, and data restrictions
generally imply that the probability of rare extreme events is not accurately
captured. In this work we present a general strategy for training neural
network models to non-intrusively correct under-resolved long-time simulations
of chaotic systems. The approach is based on training a post-processing
correction operator on under-resolved simulations nudged towards a
high-fidelity reference. This enables us to learn the dynamics of the
underlying system directly, which allows us to use very little training data,
even when the statistics thereof are far from converged. Additionally, through
the use of probabilistic network architectures we are able to leverage the
uncertainty due to the limited training data to further improve extrapolation
capabilities. We apply our framework to severely under-resolved simulations of
quasi-geostrophic flow and demonstrate its ability to accurately predict the
anisotropic statistics over time horizons more than 30 times longer than the
data seen in training.
