The Split Closure

Large-eddy simulation needs a closure model — an approximation for the subgrid-scale stress that the coarse grid cannot resolve. In the lattice Boltzmann method, this closure must respect the solver’s specific coupling structure, which differs from conventional Navier-Stokes discretizations.

arXiv:2603.15992 trains a compact neural network to map nine macroscopic derivative inputs — six strain-rate and three vorticity components — to the six independent components of the subgrid-scale stress tensor. The training combines a stress data loss with physics terms enforcing rotational equivariance under cube rotations, energy-transfer matching, and compatibility with the divergence-based coupling.

The critical design choice: the predicted stress is split into two parts before coupling to the solver. A dissipative, strain-aligned contribution enters through an effective-viscosity projection. The remaining anisotropic residual enters through a forcing term. This split is not a convenience — it is structurally necessary.

The effective-viscosity part retains backscatter: the model can reduce the effective viscosity below the molecular value, allowing energy to flow from small scales to large. The forcing part captures non-dissipative anisotropic effects that no eddy-viscosity model can represent. Neither part alone captures the full physics. Together, they decompose the closure into what the solver’s existing machinery can absorb (viscosity) and what requires a new channel (forcing).

A posteriori rollouts improve energy spectra and statistical measures relative to both static and dynamic Smagorinsky baselines. The model transfers to turbulent channel flow without retraining. The split between dissipative and anisotropic is not just a solver accommodation — it is a physically meaningful decomposition that the solver architecture made visible.