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arxiv: 2606.01150 · v1 · pith:KIJQDDJSnew · submitted 2026-05-31 · ⚛️ physics.flu-dyn

End-to-end optimization of subgrid scale models for discontinuous spectral element schemes based on the discrete adjoint method

Nicola Clinco , Niccol\`o Tonicello , Paola Cinnella , Gianluigi Rozza This is my paper

Pith reviewed 2026-06-28 16:36 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords large eddy simulationsubgrid scale modelingdiscrete adjoint methodspectral difference schemeparameter optimizationhomogeneous isotropic turbulenceTaylor-Green vortex
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The pith

A discrete adjoint framework optimizes subgrid-scale model parameters inside a spectral difference LES solver.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops an end-to-end optimization approach for subgrid-scale models in discontinuous spectral element methods for large-eddy simulation. It employs the discrete adjoint method within a Spectral Difference solver to tune model parameters using filtered DNS data from homogeneous isotropic turbulence. The objective function is the averaged decay rate of Legendre modal coefficients, chosen for stability in chaotic flows. Optimized versions of the Smagorinsky model and nonlinear tensor models are tested on multiple configurations and show better performance than standard closures on both training and out-of-sample cases such as decaying turbulence and Taylor-Green vortex.

Core claim

The central claim is that optimizing a limited set of parameters in classical and nonlinear SGS models through a discrete adjoint framework inside the SD discretization produces models that achieve significant improvements over baseline closures in forced and decaying homogeneous isotropic turbulence as well as the Taylor-Green vortex, with robustness across resolutions, polynomial orders, and Reynolds numbers.

What carries the argument

The discrete adjoint method applied to the Spectral Difference scheme, with the objective function defined as the spatio-temporally averaged decay of Legendre modal coefficients.

If this is right

  • The optimized SGS models generalize to out-of-sample flow configurations including decaying homogeneous isotropic turbulence and the Taylor-Green vortex.
  • Performance gains hold across variations in grid resolution, polynomial order, and Reynolds number.
  • Both the Smagorinsky model and non-linear tensor-basis formulations benefit from the parameter optimization.
  • The methodology applies to both one-dimensional Burgers turbulence and three-dimensional cases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The framework could extend to other high-order discontinuous methods for turbulence modeling.
  • Different objective functions might further improve optimization stability in other chaotic flow regimes.
  • Such tuned models could reduce the need for manual calibration when applying LES to engineering flows.

Load-bearing premise

The spatio-temporally averaged decay of the Legendre modal coefficients provides a suitable and stable objective function for the optimization in chaotic LES systems.

What would settle it

An independent test case in which the optimized SGS model produces larger errors than the baseline model in key turbulence statistics would falsify the improvement claim.

Figures

Figures reproduced from arXiv: 2606.01150 by Gianluigi Rozza, Niccol\`o Tonicello, Nicola Clinco, Paola Cinnella.

Figure 1
Figure 1. Figure 1: Dispersion curves for Spectral Difference (left) and Discontinuous Galerkin (right) methods for order from p = 3 to p = 7 (color gradient from ocher to black). Dashed line indicates the analytical solution. In this section we introduced the SD scheme for a general one-dimensional conservation law including both first- and second-order terms. We also briefly mentioned the more simplified setting of linear a… view at source ↗
Figure 2
Figure 2. Figure 2: Disssipation curves for Spectral Difference (left) and Discontinuous Galerkin (right) methods for order from p = 3 to p = 7 (color gradient from ocher to black). Dashed line indicates the analytical solution. designed to act as an implicit SGS model, providing an appropriate level of numerical dissipation at small scales. Another part of the turbulence modeling community instead relies on numerical schemes… view at source ↗
Figure 3
Figure 3. Figure 3: A schematic diagram of the optimization algorithm. of introducing spurious effects due to the chaotic nature of the system increases. As a result, a tradeoff between these limit cases needs to be considered. At the same time, constructing a loss function based only on quantities sampled at the final time is not effective because of the chaotic nature of the solution. Identifying the mechanisms underlying a… view at source ↗
Figure 4
Figure 4. Figure 4: A schematic drawing of the computational stencil for the spectral difference scheme. Stencil for the inviscid (left) and viscous (right) operators. The wider stencil arises from the coupling introduced by the gradient computation. In fact, as ex￾plained in Section 3, the computation of second-order terms requires an additional interpolation step and an extra correction at the interfaces between elements wi… view at source ↗
Figure 5
Figure 5. Figure 5: Assembly of the right hand side operator and reverse-mode mode compu￾tation of the vector-jacobian product. For the computation of the right hand side, following the green arrows: the state vector is interpolated to flux points, where invis￾cid and viscous fluxes are evaluated and combined to form the residual. Each of these steps is schematically shown on the right side of the scheme. Following the blue a… view at source ↗
Figure 6
Figure 6. Figure 6: Kinetic energy spectra for various resolutions and for different polynomial orders considered in the study. Symbols indicate the reference [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: shows the modal energy (36) for different polynomial orders and different resolutions in the Burgers case. In this framework, Legendre modes can be interpreted similarly to Fourier modes: low-order modes are associated with large-scale structures, whereas higher-order modes represent progressively finer scales. Note that lower resolutions present a less pronounced modal decay with respect to higher resolut… view at source ↗
Figure 8
Figure 8. Figure 8: Comparison between solutions at t = 400s for p = 7 for the coarsest (Ndof = 1024) and the finest resolution (Ndof = 4096). The solution with Ndof = 1024 contains many wiggles as a consequence of under-resolution. In the previous paragraphs, we discussed the role played by numerics in this particular setting. As representative parameter associated to the discretization, we considered the polynomial order, n… view at source ↗
Figure 9
Figure 9. Figure 9 [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of the kinetic energy spectrum between ILESs and optimized Smagorinsky for different polynomial orders for Ndof = 2048. Solid lines, optimized Smagorinsky; dotted lines, ILES. Symbols indicate the reference. 6.1. Optimization of the Smagorinsky constant. Like in the forced-Burgers test case, we start by optimizing the Smagorinsky constant within a statistically steady state setting (i.e., FHIT)… view at source ↗
Figure 11
Figure 11. Figure 11: Comparison of the kinetic energy spectrum between ILESs and the opti￾mized Smagorinsky for different polynomial orders for the coarsest (Ndof = 1024) and the finest resolution (Ndof = 4096). Solid lines, optimized Smagorinsky; dotted lines, ILES. Symbols indicate the reference. the methodology is shown in figure 12. After projection, the LES solution goes through a short transient before recovering the ex… view at source ↗
Figure 12
Figure 12. Figure 12: Schematic of the 3D FHIT methodology: a forward run over ∆T1 is first performed, after which sensitivities are computed using the adjoint loop over the interval ∆T. 100 101 102 k 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 E ( k ) DNS fDNS k−5/3 [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Example of instantaneous vorticity field (left) and spatio-temporally aver￾aged kinetic energy spectrum (right) for the FHIT case. The filtered DNS, shown in the right figure in red, refers to the cutoff ∆LES = 4∆DNS which corresponds to a total of number of degrees of freedom of Ndof = 643 . is small enough to make the backward loop for the adjoint stable. As a consequence, the time window used for the o… view at source ↗
Figure 14
Figure 14. Figure 14: Normalized loss functions for different polynomial orders for Ndof = 64 (left) and for Ndof = 80 (right). about 30 iterations for both resolutions. The loss function decreases smoothly for all polynomial orders indicating that the overall procedure is stable and, with limited number of iterations, gives significant improvements in terms of objective function minimization. A slightly different trend is obs… view at source ↗
Figure 15
Figure 15. Figure 15: Comparison of the modal decay obtained at the beginning of the optimiza￾tion and the optimized one for the various polynomial orders. Top row, Ndof = 64; Bottom row, Ndof = 80. Solid lines, optimized Smagorinsky model; dotted lines, base￾line model (Cs = 0.187). Lines with symbols indicate the filtered DNS data. Ndof p = 6 p = 5 p = 4 64 0.1032 0.104 0.0995 80 0.0915 0.0845 0.0805 [PITH_FULL_IMAGE:figure… view at source ↗
Figure 16
Figure 16. Figure 16: Time and space-averaged modal energy for different polynomial orders. Solid lines, optimized Smagorinsky; dotted lines ILES. Line with symbols indicate the filtered DNS. 1 2 3 4 5 N 10−3 10−2 10−1 100 bEi 1 2 3 4 5 6 N 1 2 3 4 5 6 7 N p = 4 p = 5 p = 6 [PITH_FULL_IMAGE:figures/full_fig_p022_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Time and space-averaged modal energy for different polynomial orders. Solid lines, ∆T = 1; dash-dotted lines ∆T = 0.02. Line with symbols indicate the filtered DNS. from the projection of the filtered DNS onto the LES grid. Consequently, all the analyses on the modal energy should be considered valid for all the modes except the last one, which is the one most likely affected by the projection error. 6.1.… view at source ↗
Figure 18
Figure 18. Figure 18: Time evolution of the kinetic energy (left) and viscous dissipation (right) for NDoF = 64. Solid lines, optimized Smagorinsky model; dotted lines, ILES. Symbols indicate the filtered DNS data at the given resolution. In contrast, the optimized model shows good agreement with the reference data for both quantities of interest. Small differences can still be identified, particularly in the viscous dissipati… view at source ↗
Figure 19
Figure 19. Figure 19: Kinetic energy (left) and viscous dissipation (right) spectra for Ndof = 64 computed at t = 1. Solid lines, optimized Smagorinsky model; dotted lines, ILES. Symbols indicate the filtered DNS data at the given resolution. 0 5 10 15 20 t 0.000 0.001 0.002 0.003 0.004 0.005  v p = 4 p = 5 p = 6 0 5 10 15 20 t 0 5 10 15 20 t [PITH_FULL_IMAGE:figures/full_fig_p024_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Averaged viscous dissipation for the TGV case at Re = 5000 using the optimized Smagorinsky model (solid lines) and baseline Smagorinsky (dashed lines) for different resolutions and different polynomial orders. From left to right: 643 , 803 and 963 . Symbols indicate the filtered DNS data at the given resolution. 3 compensate for the different numerical characteristics of the specific polynomial order. Thi… view at source ↗
Figure 21
Figure 21. Figure 21: Averaged viscous dissipation for the TGV case at Re = 1600 for differ￾ent resolutions. From left to right: 643 , 803 and 963 . Purple solid lines, optimized Smagorinsky model; purple dashed lines, baseline Smagorinsky model. Symbols indi￾cate the filtered DNS data at the given resolution. 0 5 10 15 20 t 0.02 0.04 0.06 0.08 0.10 0.12 E k 0 5 10 15 20 t 0 5 10 15 20 t [PITH_FULL_IMAGE:figures/full_fig_p025… view at source ↗
Figure 22
Figure 22. Figure 22: Averaged kinetic energy for the TGV case at Re = 1600 for different reso￾lutions. From left to right: 643 , 803 and 963 . Purple solid lines, optimized Smagorinsky model; purple dashed lines, baseline Smagorinsky model. Symbols indicate the filtered DNS data at the given resolution. on the corresponding filtered DNS data (first column), while the Smagorinsky model optimized on the 803 forced HIT configura… view at source ↗
Figure 23
Figure 23. Figure 23: Averaged viscous dissipation for the TGV case at Re = 5000 for different resolutions and polynomial orders. From left to right: 643 , 803 and 963 . From top to bottom: p = 4, p = 5 and p = 6. Solid lines, optimized Smagorinsky model at resolution 643 ; dash-dotted lines, optimized Smagorinsky model at resolution 803 . Symbols indicate the filtered DNS data at the given resolution. with Seij = 1 2 Å ∂uei ∂… view at source ↗
Figure 24
Figure 24. Figure 24: TB3 model. Normalized-loss functions for different polynomial orders for Ndof = 64 (left) and for Ndof = 80 (right). This observation establishes a link between the proposed tensor-basis model and the Clark model, showing that the latter can be interpreted as a specific instance of the expansion with fixed coefficients. In this context, it is interesting to note that the second coefficient is greater, in … view at source ↗
Figure 25
Figure 25. Figure 25: Averaged viscous dissipation for the TGV case at Re = 5000 for different resolutions and polynomial orders. From left to right: 643 , 803 , 963 . Solid lines, TB3 model at resolution 803 ; dashed lines, TB3 model at resolution 643 ; blue dash-dotted lines, GEP2 from [92]; blue dotted lines, GEP1 from [92]. Symbols indicate the filtered DNS data at the given resolution. 0 5 10 15 20 t 0.00 0.02 0.04 0.06 0… view at source ↗
Figure 26
Figure 26. Figure 26: Averaged kinetic energy for the TGV case at Re = 5000 for different resolutions and polynomial orders. From left to right: 643 , 803 , 963 . Solid lines, TB3 model at resolution 803 ; dashed lines, TB3 model at resolution 643 ; blue dash-dotted lines, GEP2 from [92]; blue dotted lines, GEP1 from [92]. Symbols indicate the filtered DNS data at the given resolution. 7. Conclusions In this work, we have pres… view at source ↗
read the original abstract

In computational fluid dynamics, Large Eddy Simulation (LES) offers a compelling balance between accuracy and computational cost by resolving large-scale flow structures while modeling unresolved subgrid scales. However, its predictive capacity is critically dependent on the choice and calibration of subgrid-scale (SGS) models, which often involve problem-dependent parameters and exhibit intricate interactions with the numerical discretization. In this work, we propose a discrete-adjoint framework to optimize SGS model parameters in the loop, leveraging automatic differentiation within a high-order Spectral Difference (SD) solver. Coarse-grained simulations of Forced Homogeneous Isotropic Turbulence (FHIT), together with filtered Direct Numerical Simulation (DNS) data, are used to optimize a limited set of parameters for classical SGS models, including the Smagorinsky model and non-linear tensor-basis formulations. For chaotic systems such as LES, the choice of objective function plays a crucial role in the stability and accuracy of the optimization. Here, we consider the spatio-temporally averaged decay of the Legendre modal coefficients as the quantity of interest for the SD scheme. The optimization is performed across different grid resolutions and polynomial orders, highlighting the impact of numerical discretization on model performance. The methodology is applied to both one-dimensional Burgers turbulence and fully three-dimensional turbulence. The trained models are subsequently assessed on out-of-sample configurations, including Decaying Homogeneous Isotropic Turbulence (DHIT) and the Taylor-Green vortex. Variations in polynomial order, grid resolution, and Reynolds number are considered to evaluate robustness and generalization. In all test cases, the optimized models demonstrate significant improvements over baseline SGS closures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript develops a discrete-adjoint optimization framework, leveraging automatic differentiation in a high-order Spectral Difference solver, to calibrate parameters of classical SGS models (Smagorinsky and nonlinear tensor-basis forms) against filtered DNS data. The objective is the spatio-temporally averaged decay rate of Legendre modal coefficients; optimization is performed on forced HIT at varying resolutions and polynomial orders, with out-of-sample assessment on decaying HIT and Taylor-Green vortex. The central claim is that the resulting models yield significant improvements over baseline closures in all tested configurations.

Significance. If the central claim holds, the work offers a systematic route to SGS-model calibration that accounts for the interaction with a specific high-order discretization, which is a recognized challenge in LES. The explicit use of the discrete adjoint for end-to-end optimization in chaotic turbulent flows is a methodological strength that could be adopted more broadly if the chosen objective is shown to produce physically consistent models.

major comments (2)
  1. [objective-function paragraph] The paragraph describing the objective function (abstract and methods): the manuscript asserts that the spatio-temporally averaged decay of Legendre modal coefficients is a suitable and stable objective for chaotic LES optimization, yet provides no demonstration that minimization of this scalar produces correct inter-scale energy transfer, kinetic-energy spectra, or dissipation rates on the out-of-sample DHIT and Taylor-Green cases. If the functional primarily damps high-mode energy without enforcing spectral shape, the reported improvements may be limited to the training objective and not generalize to physically relevant diagnostics.
  2. [out-of-sample assessment] Results section on out-of-sample tests: the claim of 'significant improvements over baseline SGS closures' in all test cases is load-bearing for the central contribution, but the manuscript supplies no quantitative metrics (e.g., relative error reduction in spectra or structure functions, convergence histories of the adjoint optimization, or error bars across realizations). Without these data it is impossible to judge whether the improvements are robust or merely marginal.
minor comments (1)
  1. [methods] Notation for the modal-coefficient decay rate should be defined explicitly with an equation number rather than described only in prose.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [objective-function paragraph] The paragraph describing the objective function (abstract and methods): the manuscript asserts that the spatio-temporally averaged decay of Legendre modal coefficients is a suitable and stable objective for chaotic LES optimization, yet provides no demonstration that minimization of this scalar produces correct inter-scale energy transfer, kinetic-energy spectra, or dissipation rates on the out-of-sample DHIT and Taylor-Green cases. If the functional primarily damps high-mode energy without enforcing spectral shape, the reported improvements may be limited to the training objective and not generalize to physically relevant diagnostics.

    Authors: The objective was selected because matching modal decay rates from filtered DNS directly encodes the correct inter-scale transfer for the SD discretization. We agree additional evidence is needed and will add kinetic-energy spectra, dissipation-rate comparisons, and inter-scale transfer diagnostics for the out-of-sample cases in the revision. revision: yes

  2. Referee: [out-of-sample assessment] Results section on out-of-sample tests: the claim of 'significant improvements over baseline SGS closures' in all test cases is load-bearing for the central contribution, but the manuscript supplies no quantitative metrics (e.g., relative error reduction in spectra or structure functions, convergence histories of the adjoint optimization, or error bars across realizations). Without these data it is impossible to judge whether the improvements are robust or merely marginal.

    Authors: We agree that quantitative support is required. The revised manuscript will report relative error reductions on spectra and structure functions, adjoint optimization convergence histories, and error bars from multiple realizations. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper optimizes a small set of SGS parameters by minimizing a modal-decay objective against filtered DNS data within a discrete-adjoint loop, then evaluates the resulting models on explicitly out-of-sample configurations (DHIT, Taylor-Green vortex) at varied resolutions and Reynolds numbers. No derivation step equates a claimed prediction to its own fitted inputs by construction, nor does any load-bearing premise rest on a self-citation chain; the reported improvements are therefore externally falsifiable and independent of the training data used for optimization.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Ledger constructed from abstract only; full paper would allow more precise identification of fitted quantities and background assumptions.

free parameters (1)
  • SGS model parameters (Smagorinsky constant and nonlinear coefficients)
    A limited set of parameters for classical SGS models are optimized against filtered DNS data.
axioms (1)
  • domain assumption The spatio-temporally averaged decay of Legendre modal coefficients is an appropriate objective for chaotic turbulent flows.
    Abstract explicitly states that the choice of objective function plays a crucial role in stability and accuracy for LES.

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