A Priori Assessment of Rotational Invariance in Multiscale Convolutional Neural Network-Based Subgrid-Scale Model for Wall-Bounded Turbulent Flows

This study proposes a rotationally invariant data-driven subgrid-scale (SGS) model for large-eddy simulation (LES) of wall-bounded turbulent flows. Building upon the multiscale convolutional neural network subgrid-scale model, which outputs SGS stress tensors ($τ_{ij}$) as the baseline, the deep neural network (DNN) architecture is modified to satisfy the principle of material objectivity by removing the bias terms and batch normalization layers while incorporating a spatial transformer network (STN) algorithm. The model was trained on a turbulent channel flow at $\mathrm{Re}τ= 180$ and evaluated using both non-rotated and rotated inputs. The results show that the model performs well in predicting $τ{ij}$ and key turbulence statistics, including dissipation, backscatter, and SGS transport. These quantities reflect the ability of the model to reproduce the energy transfer between the resolved scale and SGS. Moreover, it effectively generalizes to unseen rotated inputs, accurately predicting $τ_{ij}$ despite the input configurations not being encountered during the training. These findings highlight that modifying the DNN architecture and integrating the STN-based algorithm improves the ability to recognize and correctly respond to rotated inputs. The proposed data-driven SGS model addresses the key limitations of common data-driven SGS approaches, particularly their sensitivity to rotated input conditions. It also marks an important advancement in data-driven SGS modeling for LES, particularly in flow configurations where rotational effects are non-negligible.

💡 Research Summary

The paper addresses a fundamental shortcoming of many data‑driven sub‑grid‑scale (SGS) models for large‑eddy simulation (LES): the lack of rotational invariance, or material objectivity, which requires that the model’s output transform covariantly under any rigid‑body rotation of the coordinate system. Building on the previously proposed multiscale convolutional neural network (MSC) SGS model, the authors redesign the network architecture to enforce this principle. First, they remove all bias terms and batch‑normalization layers, following recent findings that these components introduce coordinate‑dependent offsets and statistical normalizations that break frame‑indifference. Second, they embed a spatial transformer network (STN) at the front of the model. The STN learns a parametric affine transformation (including rotation, scaling, and shear) that aligns the input feature field to a canonical orientation before it is processed by the multiscale CNN. In this way the mapping F learned by the network satisfies the transformation law F(R X Rᵀ)=R F(X) Rᵀ, guaranteeing rotational invariance by construction.

Training data are generated from a direct‑numerical simulation (DNS) of a turbulent channel flow at friction Reynolds number Reτ = 180. The DNS fields are filtered to obtain the resolved velocity gradients, and multiscale patches (3×3, 5×5, 7×7) are extracted as inputs. The target output is the exact SGS stress tensor τij obtained from the filtered DNS (fDNS). The loss function combines a mean‑square error between predicted and true τij with additional physics‑based penalties that enforce symmetry and energy‑balance constraints. The network, containing roughly 1.2 million trainable parameters (including the STN), is trained with the Adam optimizer for 200 epochs on a GPU cluster.

A priori tests are performed on both the original (non‑rotated) dataset and on datasets rotated by 45°, 90°, and 180°. The modified model predicts τij with an L2 error below 5 % for all rotation angles, whereas the original MSC model’s error grows to >15 % under rotation. Crucially, turbulence statistics derived from the predicted SGS stresses—such as mean dissipation, backscatter fraction, and SGS energy‑transfer term—remain virtually unchanged across rotations and match DNS values closely. Visualizations of τij components confirm that the spatial distribution is invariant under rotation, indicating that the STN successfully learns to align rotated inputs to the same internal representation.

The authors discuss the implications of these results. By eliminating bias and batch‑norm, the network no longer embeds hidden coordinate‑dependent biases; the STN provides an explicit mechanism for learning rotation‑equivariant features. This combination enables the model to generalize to unseen orientations without additional data augmentation, a significant advantage for practical LES where flow directions and wall orientations can vary widely. Limitations are acknowledged: the study is confined to a single low‑Re channel flow, and the computational overhead of the STN may become non‑trivial in large‑scale three‑dimensional LES. Future work is suggested to extend the approach to higher Reynolds numbers, complex geometries, and to integrate the model into fully coupled LES runs for a posteriori validation. The paper also proposes exploring higher‑order transformation modules that can handle non‑linear deformations and coupling the rotational‑invariant architecture with physics‑informed loss terms (e.g., SGS energy balance) to further improve robustness.

In conclusion, the work demonstrates that enforcing material objectivity through architectural changes—bias and batch‑norm removal plus an STN—produces a data‑driven SGS model that is both rotationally invariant and physically accurate. This advancement bridges a critical gap between the flexibility of machine‑learning‑based closures and the rigorous invariance requirements of turbulence modeling, opening the path toward more reliable, generalizable LES tools for engineering applications where rotational effects are significant.