Physics-Constrained Neural Closure for Lattice Boltzmann Large-Eddy Simulation

Abstract

We present a physics-constrained, data-driven subgrid-scale (SGS) stress closure for large-eddy simulation (LES) in the lattice Boltzmann method (LBM). Trained on filtered-downsampled (FD) data from LBM direct numerical simulation (DNS) of forced homogeneous isotropic turbulence (FHIT) spanning multiple filter widths, a compact neural network maps nine macroscopic derivative inputs - six strain-rate and three vorticity components - to the six independent components of the SGS stress tensor; a deviatoric projection is applied post-inference to obtain the traceless stress used in the solver. Training combines a stress data loss with physics terms for SGS energy-transfer (Pi) matching, rotational equivariance under cube rotations, and compatibility of the implied SGS forcing with the divergence-based coupling. The predicted stress is coupled to the solver through a split strategy: a dissipative, strain-aligned contribution is represented through an effective-viscosity projection, while the remaining anisotropic residual is applied through a forcing term. This construction is intended to retain both backscatter (via the effective viscosity) and non-dissipative anisotropic effects (via the residual forcing), while remaining compatible with LBM deployment. In the cases considered here, a priori results show good agreement with FD references across stress components and SGS-transfer statistics, and a posteriori rollouts improve several energetic and statistical measures relative to static and dynamic Smagorinsky baselines. A preliminary transfer test in turbulent channel flow is also reported without retraining. Finally, we demonstrate production deployment via ONNX Runtime, with throughput comparable to a dynamic Smagorinsky baseline in the tested configuration.

Figures (11)

• Figure 1: Neural-network architecture for SGS stress prediction (9$\rightarrow$64$\rightarrow$64$\rightarrow$6). A deviatoric projection is applied post-inference.
• Figure 2: A priori validation scatter plots for SGS stress components. Each panel compares the predicted SGS stress component $\tau_{ij}^{\mathrm{pred}}$ against the FD reference $\tau_{ij}^{\mathrm{true}}$ over a validation dataset: FD grid $128^3$ ($256^3$ DNS, $\Delta=2$), $\nu=0.001$. The dashed line indicates $y=x$. Reported values are the coefficient of determination $R^2$ and Pearson correlation $\rho$ computed per component.
• Figure 3: A priori p.d.f. of the SGS energy transfer $\Pi$ comparing FD reference, ML prediction, and static Smagorinsky baseline (a priori). The ML model closely matches FD across the peak and tails, including backscatter (negative $\Pi$) and intermittent forward transfer (positive $\Pi$). For Smagorinsky, the non-negative eddy viscosity yields zero p.d.f. for $\Pi<0$ and underpredicts the positive-$\Pi$ tail.
• Figure 4: Characterization and verification of the hybrid SGS stress-splitting closure versus normalized time $t/t_0$. (a) Residual fraction $\langle\|\boldsymbol{\tau}^{\mathrm{res}}\|\rangle / \langle\|\boldsymbol{\tau}^{\mathrm{ML}}\|\rangle$, quantifying the portion of the learned SGS stress not representable by an eddy-viscosity-aligned contribution. (b) Global stress-alignment proxy between the learned stress and its dissipative component, computed from domain-averaged inner products and magnitudes. (c) Energy consistency: residual dissipation $\Pi^{\mathrm{res}} = -\tau^{\mathrm{res}}_{ij} S_{ij}$ and resolved-scale work of the residual forcing $\langle \mathbf{u}\cdot\mathbf{F}^{\mathrm{res}}\rangle$, with $\mathbf{F}^{\mathrm{res}}=-\nabla\cdot\boldsymbol{\tau}^{\mathrm{res}}$. (d) Backscatter fraction, defined as the fraction of grid cells where $\Pi=-\tau^{\mathrm{ML}}_{ij}S_{ij}<0$. (e) Mean magnitude of the residual forcing $\langle|\mathbf{F}^{\mathrm{res}}|\rangle$, indicating the activity level of the explicit anisotropic SGS contribution. (f) Discrete orthogonality shown on a logarithmic scale; both $|\langle\Pi^{\mathrm{res}}\rangle|$ and $|\langle\mathbf{u}\cdot\mathbf{F}^{\mathrm{res}}\rangle|$ remain at numerical noise levels. Panels (c) and (f) demonstrate that the residual stress is non-dissipative and that its explicit injection via forcing is energy-consistent at the discrete level. Data from a representative FHIT rollout ($128^3$, $\nu=0.001$).
• Figure 5: A posteriori p.d.f. of $\Pi$ comparing FD reference with ML inference deployed in LBM with MRT and BGK collision operators. The BGK configuration remains statistically consistent with FD, indicating transferability beyond the collision model used to generate the training data.