Interpreting neural network models of residual scalar flux

We show that, in addition to providing effective and competitive closures, when analysed in terms of the dynamics and physically relevant diagnostics, artificial neural networks (ANNs) can be both interpretable and provide useful insights into 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 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. A priori analysis of the subfilter scalar variance transfer rate demonstrates that learnt ANN models outperform optimised turbulent Prandtl number closures and Clark-type gradient models. Next, a posteriori solutions are obtained with each model over several integral time scales. 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 a 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.