Title: A posteriori closure of turbulence models: are symmetries preserved ? Show Chinese title

Title: 湍流模型的后验闭合：对称性是否被保留？

Abstract: 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 previous study (arXiv:2411.13194), 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 to assess whether cross-correlations are well preserved or if just the mean behavior is captured. Additionally, we investigate the scale invariance of multipliers - ratios between adjacent shells - within the inertial range. Although our model excels in reproducing high-order observables such as flatness, 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. Show Chinese abstract

Abstract: 湍流建模一直是流体力学中的一个长期挑战。 近年来，数据驱动方法的最新进展催生了许多旨在解决这一问题的新方法。 这项工作建立在我们之前的研究基础上（arXiv:2411.13194），在其中我们利用后验（或求解器循环）方法为湍流壳模型引入了一种新的闭合形式。 与大多数基于深度学习的模型不同，我们的方法明确地将物理方程纳入神经网络框架，确保闭合形式受到底层物理的约束，从而提高稳定性并增强泛化能力。 在本文中，我们进一步分析了学到的闭合形式，探讨其能力和局限性。 特别是，我们研究联合概率密度函数以评估是否很好地保留了交叉相关性，或者只是捕获了平均行为。 此外，我们还研究了惯性范围内相邻壳层之间倍增器的比例的尺度不变性。 尽管我们的模型在再现高阶观测值（如平坦度）方面表现出色，但在截止点附近破坏了已知的对称性，这表明存在基本限制。 我们讨论了这些发现对三维湍流亚网格尺度建模的影响，并概述了未来研究的方向。