  We show that in addition to providing effective and competitive closures,
when analysed in terms of dynamics and physically-relevant diagnostics,
artificial neural networks (ANNs) can be both interpretable and provide useful
insights in the on-going task of developing and improving turbulence closures.
In the context of large-eddy simulations (LES) of a passive scalar in
homogeneous isotropic turbulence, exact subfilter fluxes obtained by filtering
direct numerical simulations (DNS) are used both to train deep ANN models as a
function of filtered variables, and to optimise the coefficients of a turbulent
Prandtl number LES closure. \textit{A-priori} analysis of the subfilter scalar
variance transfer rate demonstrates that learnt ANN models out-perform
optimised turbulent Prandtl number closures and Clark-type gradient models.
Next, \textit{a-posteriori} solutions are obtained with each model over several
integral timescales. These experiments reveal, with single- and multi-point
diagnostics, that ANN models temporally track exact resolved scalar variance
with greater accuracy compared to other subfilter flux models for a given
filter length scale. Finally, we interpret the artificial neural networks
statistically with differential sensitivity analysis to show that the ANN
models feature dynamics reminiscent of so-called "mixed models", where mixed
models are understood as comprising both a structural and functional component.
Besides enabling enhanced-accuracy LES of passive scalars henceforth, we
anticipate this work to contribute to utilising neural network models as a tool
in interpretability, robustness and model discovery.
