Consistency requirement of data-driven subgrid-scale modeling in large-eddy simulation

Data-driven subgrid-scale (SGS) modeling in the large-eddy simulations (LES) suffers from the inconsistency between the \textit{a priori} tests and the a posteriori tests, which make training accurate SGS models a difficult task. We study the difference in filtered high-fidelity data and LES to identify the numerical deviation between the two cases, which is a combined impact of commutation error, numerical errors, and error coupling. The impact of the numerical deviation is examined through two SGS model formulations: the eddy-viscosity and the complex nonlinear models. By incorporating numerical deviations into model training, we enhance consistency, stabilize simulations, and improve predictions of energy spectra in a posteriori tests. Our findings highlight that data-driven methods introduce significant nonlinearity and equation coupling, exacerbating inconsistencies compared to non-data-driven approaches. Finally, while the impact of the numerical deviation can be generalized, achieving accurate model predictions necessitates a physically grounded model form and an optimal filter width.