  This work presents a systematic framework for improving the predictions of
statistical quantities for turbulent systems, with a focus on correcting
climate simulations obtained by coarse-scale models. While high resolution
simulations or reanalysis data are available, they cannot be directly used as
training datasets to machine learn a correction for the coarse-scale climate
model outputs, since chaotic divergence, inherent in the climate dynamics,
makes datasets from different resolutions incompatible. To overcome this
fundamental limitation we employ coarse-resolution model simulations nudged
towards high quality climate realizations, here in the form of ERA5 reanalysis
data. The nudging term is sufficiently small to not pollute the coarse-scale
dynamics over short time scales, but also sufficiently large to keep the
coarse-scale simulations close to the ERA5 trajectory over larger time scales.
The result is a compatible pair of the ERA5 trajectory and the weakly nudged
coarse-resolution E3SM output that is used as input training data to machine
learn a correction operator. Once training is complete, we perform free-running
coarse-scale E3SM simulations without nudging and use those as input to the
machine-learned correction operator to obtain high-quality (corrected) outputs.
The model is applied to atmospheric climate data with the purpose of predicting
global and local statistics of various quantities of a time-period of a decade.
Using datasets that are not employed for training, we demonstrate that the
produced datasets from the ML-corrected coarse E3SM model have statistical
properties that closely resemble the observations. Furthermore, the corrected
coarse-scale E3SM output for the frequency of occurrence of extreme events,
such as tropical cyclones and atmospheric rivers are presented. We present
thorough comparisons and discuss limitations of the approach.
