What lies beyond the cut

How much turbulence can a coarse grid know about?

Approaching the optimal closure: equivariance, inductive bias,
and Reynolds-number generalization in data-driven LES

Syver Døving Agdestein · Benjamin Sanderse
ECCOMAS 2026 · MS184A Theory-guided Design of Deep Learning-based Surrogates
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filtered z-velocity slice
what an LES grid sees of a turbulent flow
Agdestein & Sanderse

The same flow, on two grids

DNS velocity, z component
DNS · velocity u
810³ — resolves everything
filtered velocity, z component
LES · filtered velocity ū
128³ — 250× fewer points, misses the small scales

the missing scales still act on the resolved ones — that action is the closure's job

Ground · Two grids

Where the energy lives — and where the grid stops

wavenumber k (log) energy E(k) (log) cut k = π/Δ data — DNS spectra · ν = 2.5·10⁻⁴ / 1.0·10⁻⁴

the grid resolves everything left of the cut;
the shaded energy is invisible

coarsen the filter (Δ ↑) — more energy hidden

raise the Reynolds number (ν ↓) — the spectrum grows a longer tail

one resolved-scale number for
how much lies beyond the cut

Ground · Sub-filter energy

Why this number? Scale the equations and see what survives

velocity slice, original scale
a turbulent solution
velocity slice, rescaled
…rescaled — still a solution

Navier–Stokes maps to Navier–Stokes — a two-parameter symmetry.
The fixed LES grid keeps only the a² = b family.

along that family, exactly one dimensionless group built from ū is invariant:
ReΔ — the symmetry analysis hands the closure its Reynolds input

Ground · Scaling symmetry

Energy crosses the cut in both directions

resolved scales sub-filter scales the cut E(k), k < π/Δ E(k), k > π/Δ forward transfer — dissipation backscatter — 22%
a closure must get this net flux right — at every Reynolds number
Ground · Dissipation & backscatter

Classical closures pick one side of this trade

pointwise SFS dissipation −τ:S̄ density (log) backscatter data — KDE at held-out (ν, Δ) reference (DNS) dyn. Smagorinsky Clark
functional · Smagorinsky

dissipation ≈ right · structure wrong
backscatter 0.0004 — can't flow backward

structural · Clark

structure ≈ right · dissipation 0.6× too weak
backscatter 0.25 — close to the true 0.22

and notice: ν appears in neither formula no ν

Ground · Functional vs structural

The learned closures are just as blind

no ν

the normalization that grants scale-invariance also erases the Reynolds number —
the dissipation calibration freezes at the training regime

fix: train across viscosities, and give every model ReΔ as one extra input

with ν
Main arc · The blind spot

Tested away from training: who keeps their calibration?

training window true dissipation ReΔ at the test point (log) → data — median SFS dissipation / DNS reference · held-out ν and Δ · 5 seeds Reynolds-blind MLP dynamic Smagorinsky — also blind + ReΔ — holds at held-out ν and Δ
Main arc · Calibration

A flow it has never seen: Taylor–Green becoming turbulent

TGV vorticity, turbulent state
vorticity · t = 0
decaying, transitional — out of distribution twice over

laminar → transition → decay
press → to watch

all closures stay stable — but over-dissipate

ReΔ at peak dissipation (log) → 1.0 data — SFS dissipation / reference, 3 filter widths blind + ReΔ

ReΔ corrects the Reynolds-driven share of the excess,
and honestly leaves the share owed to different flow physics

Main arc · Out of distribution

Couldn't a better architecture have fixed this? We checked.

strain invariant · tr S̄² / |Ā|² SFS dissipation −τ:S̄ (normalized) backscatter ↓ the conditional mean — the optimal one-point closure
real data — one slice · one input value ↦ a cloud of true stresses
parameters (log) error data — a-priori error, 5 seeds its error = the floor MLP G-CNN · TBNN
every architecture saturates to the same one-point optimum

so the Reynolds fix could never come from the architecture — it had to come from the input, available to all of them

Secondary arc · The floor
Takeaway

A closure that knows what it's missing works where it has never been.

Agdestein & Sanderse · CWI Amsterdam · “Approaching the optimal closure” · preprint & code available

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