A posteriori closure of turbulence models: are symmetries preserved?

TL;DR

A new closure for a shell model of turbulence using an a posteriori (or solver-in-the-loop) approach that 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.

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 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.

Figures (8)

• Figure 1: Energy spectrum $|u_n|^2$ as a function of the wavenumber $k_n$. The red-circled markers indicate the reduced system, while black crosses represent the fully resolved system. The inset zooms into the cutoff region, highlighting the nonlinear energy transfer through triadic interactions, with blue and green arrows indicating interactions for the two shells preceding the cutoff. It becomes clear that to close the system, only the two shells after the cutoff are needed to model. The x-axis is labeled in $\log_2 k_n$, along with the Kolmogorov scale $k_\eta$ and maximum resolved wavenumber $k_N$.
• Figure 2: Sketch illustrating the solver-in-the-loop approach. The LES trajectory $\tilde{\bm{u}}$ is evolved over $m$ time steps using the discrete solver $\mathcal{P}$, while the correction operator $\mathcal{C}$, parameterized by a neural network, predicts the unresolved scales. The objective is to iteratively refine $\mathcal{C}$ during training, bringing $\tilde{\bm{u}}$ closer to the reference trajectory $\bm{u}^{\text{ref}}$, which integrates the full system dynamics from the same initial condition. Here, the subscript denotes the time step index of the trajectory, i.e., $\bm{u}_k = \bm{u}(t+k\Delta{t})$, and not the shell index.
• Figure 3: Structure functions $S_n^{(p)}$ as a function of shell index $n$ for different orders $p =1, \ldots,6$. The LES-NN model is compared with the ground truth solution.
• Figure 4: Time evolution of the normalised energy density $\log_2(|u_n|^2 k_n^{2/3})$ for (a) the ground truth and (b) the LES-NN model. (c) Absolute normalised error $\left||u_n|^2 - |\tilde{u}_n|^2\right| k_n^{2/3}$ between the two fields. The x-axis represents the wavenumber index $n$, while the y-axis represents time normalised by the integral time scale $t/\tau_0$. The dashed line represents the cutoff scale.
• Figure 5: Contours of the joint PDF of the summed energy across the input shells for the neural network (x-axis) and the subgrid-scale output shells (y-axis) for both the ground truth and the LES-NN model.