Today's leading projections of climate change predicate on Atmospheric General Circulation Models (GCMs). Since the atmosphere consists of a staggering range of scales that impact global trends, but computational constraints prevent many of these scales from being directly represented in numerical simulations, GCMs require "parameterisations'' - models for the influence of unresolved processes on the resolved scales. State-of-the-art parameterisations are commonly based on combinations of phenomenological arguments and physics, and are of considerably lower fidelity than the resolved simulation. In particular, the parameterisation of low-altitude stratocumulus clouds that result from small-scale processes in sub-tropical marine boundary layers is widely considered the largest source of uncertainty that remains in contemporary GCMs' prediction of the temperature response to a global increase in CO2. Improvements in the capacity of machine learning algorithms and the increasing availability of high-fidelity datasets from global satellite data and local Large Eddy Simulations (LES) have identified data-driven parameterisations as a high-potential option to break the deadlock. However, early contributions in this field still rely on inconsistent multiscale model formulations and are plagued by instability. To sketch a clearer picture on the sources of the accuracy and instability of data-driven parameterisations, this work proposes a framework in which no assumptions on the model form are made, building on recent work at the TU Delft. It uses Artificial Neural Networks (ANNs) to infer exact projections of the unresolved scales processes on the resolved degrees of freedom. These ``interaction terms'' naturally arise from Variational Multiscale (VMS) model formulations. The resulting VMM-ANN framework limits error to the data-driven interaction term approximations, offering explicit insight into their functioning.
The model is assessed in the context of a statistically stationary convective boundary layer turbulence problem, which is further reduced to a one-dimensional, forced inviscid Burgers' equation. Simple, feedforward ANNs with relatively local input stencils that are trained on error-free data a priori to inserting them in forward simulations (offline) are capable of predicting the interaction terms of this problem much better than traditional, algebraic VMS closures in offline settings at various levels of discretisation; they also generalise well to uncorrelated instances of the turbulence. However, this performance does not translate to simulations of forward problems. It is shown that the model suffers from instability due to i) ill-posed nonlinear solution procedures and ii) self-inflicted error accumulation. These correspond to two dimensions of forward simulations that are not accounted for by offline training on error-free data. The first instability mode can be dealt with by reformulating the VMM-ANN model architecture; the second is conjectured to demand training strategies that account for the self-induced errors. Finally, despite scaling well, the framework is still found to be comparatively computationally expensive. In all, appreciable challenges therefore remain in order to capitalise on the promise offered by ANNs to improve the parameterisation of clouds in GCMs in particular and turbulence in general.