Approaching the optimal closure: equivariance, inductive bias, and Reynolds-number generalization in data-driven LES
Data-driven closures for large-eddy simulation (LES) are commonly built to respect the symmetries of the Navier--Stokes equations, on the premise that this improves accuracy and generalization. We test this premise in a controlled comparison of three data-driven LES closures that share a pointwise, Galilean-invariant velocity-gradient construction but span non-equivariant, octahedral-equivariant, and tensor-basis designs: an unconstrained multi-layer perceptron (MLP), a group-convolutional network whose exactly equivariant weights we synthesize in closed form, and a tensor-basis neural network (TBNN). The designs follow from an analysis of which symmetries survive discretization on a uniform grid, where the continuous orthogonal group reduces to the 48-element octahedral group. Across a range of network sizes the three closures saturate to the same a priori and a posteriori accuracy, and a direct conditional-mean estimate identifies the a priori floor as the one-point optimal closure of Langford and Moser. The equivariant and tensor-basis models reach this floor with $25$ times fewer parameters than the MLP: the inductive bias buys parameter efficiency rather than a lower error floor. Finally, we train the closures across several viscosities and supply the global filter-scale Reynolds number $\operatorname{Re}_Δ= Δ^2 \| \nabla \bar{u} \| / ν$ as an input, a scaling-invariant feature dictated by the same symmetry analysis. The closures then generalize across Reynolds number: they hold their dissipation calibration at held-out viscosities and filter ratios where Reynolds-blind closures mis-dissipate, and partially correct it on an out-of-distribution Taylor--Green flow. Reynolds-number generalization is thus largely a calibration that the right input feature supplies.