Subgrid Stress Modelling with Multi-dimensional State Space Sequence Models

TL;DR

This work addresses subgrid stress modelling in LES by introducing a grid-spacing aware neural network built on state-space sequence models (S4/S4ND) combined with a Tensor Basis Neural Network (TBNN). The model learns a continuous, grid-aware representation of the subgrid stress tensor, enabling extrapolation to coarser grid spacings without retraining and robust performance under extreme Reynolds numbers, as demonstrated in forced HIT and channel flows. A priori analyses show strong correlation with DNS-derived stresses and stable extrapolation across grid scales, while a posteriori LES tests reveal improved spectra, correlations, and Reynolds-stress predictions compared to traditional SGS models and other neural nets, including at unseen grid widths up to $64\Delta_{DNS}$. The approach remains stable even when Reynolds numbers are magnified by factors up to $5\times 10^5$ from the training set, highlighting practical significance for industrial-scale LES where grid resolution and Reynolds numbers vary widely.

Abstract

Large Eddy Simulations (LES) are becoming increasingly viable due to the growth in computational power the last few decades, and subgrid stress modelling plays a large role in the accuracy of LES. A new class of neural network models, S4 and S4ND models, allow for learning a continuous representation of the discrete dataset, which facilitates a principled approach to incorporating grid dependence in neural network subgrid stress modelling. A S4ND Unet neural network architecture is proposed and trained on both forced Homogeneous Isotropic Turbulence (HIT) and channel flow, where a priori, it is shown to generalize to grid spacings that are coarser than the training set grid spacings, while simpler artificial neural network (ANN) models fail. A posteriori tests on both forced HIT and channel flow indicate that the S4ND model is more accurate than traditional models and ANN-based models on grid sizes that are in the training set. The S4ND model is also able to generalize to grid sizes that are coarser than the training set a posteriori, and is more accurate than many traditional and neural network subgrid stress models. Finally, the proposed model is also evaluated on flows at increasing Reynolds numbers, where it is seen that the proposed neural network remains stable even at a Reynolds number that is 500,000 times that seen in the training set.

Figures (24)

• Figure 1: S4ND extension of the S4 model. A S4 model is instantiated in each direction, and then an outer product is taken to produce a continuous convolution kernel. When training and during inference, the continuous convolution kernel is discretized.
• Figure 2: The overall neural network architecture. The whole 3D domain is inputted into the neural network, and the output is the corresponding subgrid stress at each point. For each S4ND layer, a state space dimension of 8 was chosen to keep the trainable parameters comparable to existing models in literature. The average pooling operation will downsample all spatial dimensions by a factor of 2. As seen, after the final S4ND layer, the neural network architecture splits into two "prediction heads" to predict the "magnitude" and "structure" of the subgrid stress tensor, and both prediction heads use the tensor basis (each prediction head predicts its own set of coefficients to multiply the tensor basis components). See Wu2025_Git for downloadable version of the model and an example input/output.
• Figure 3: Spatial distribution of $\tau_{11}$ for a 2D slice of the 3D domain for a filter width of 16 $\Delta_{DNS}$. All numerical values are ensemble averaged over 10 ensembles, while the spatial fields are chosen randomly for visual reference. The letter $r$ denotes the correlation coefficient. The correlation coefficients are highest for S4ND, while the other, less complex neural networks have lower correlation coefficients.
• Figure 4: Spatial distribution of $\tau_{ij}\Bar{S}_{ij}$ for a 2D slice of the 3D domain for a filter width of 16 $\Delta_{DNS}$. All numerical values are ensemble averaged over 10 ensembles, while the spatial fields are chosen randomly for visual reference.
• Figure 5: Probability Density Functions for 16 $\Delta_{DNS}$ forced HIT. As seen, the S4ND model is more able to accurately capture the distribution of $\tau_{11}$ as compared to other models. The S4ND model and NN model (convolution-based) both capture more backscatter than traditional models, although it is still less than FDNS.