A posteriori closure of turbulence models: Are symmetries preserved?

Turbulence modeling remains a longstanding challenge in fluid dynamics. Recent advances in data-driven methods have led to a surge of novel approaches aimed at addressing this problem. This work builds upon our recent work [Phys. Rev. Fluids 10, 044602 (2025)], where we introduced a new closure for a shell model of turbulence using an a posteriori (or solver-in-the-loop) approach. Unlike most deep learning-based models, our method explicitly incorporates physical equations into the neural network framework, ensuring that the closure remains constrained by the underlying physics benefiting from enhanced stability and generalizability. In this paper, we further analyze the learned closure, probing its capabilities and limitations. In particular, we look at joint probability density functions between resolved and unresolved variables, as well as the scale invariance of multipliers (ratios between adjacent shells) within the inertial range. Although our model excels in reproducing high-order statistical moments, it breaks this known symmetry near the cutoff, indicating a fundamental limitation. We discuss the implications of these findings for subgrid-scale modeling in 3D turbulence and outline directions for future research.