arxiv: 2604.23874 · v2 · submitted 2026-04-26 · ⚛️ physics.flu-dyn · cs.LG· math.DS· physics.comp-ph· physics.geo-ph

Recognition: unknown

Deep Learning of Solver-Aware Turbulence Closures from Nudged LES Dynamics

Ashwin Suriyanarayanan , Dibyajyoti Chakraborty , Romit Maulik

Authors on Pith no claims yet

Pith reviewed 2026-05-08 05:14 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn cs.LGmath.DSphysics.comp-phphysics.geo-ph

keywords turbulence modelinglarge eddy simulationdeep learningnudgingcontinuous data assimilationclosure modelsfluid dynamicsnumerical stability

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The pith

Nudging sparse DNS observations into LES lets neural networks learn stable turbulence closures without backpropagating through the solver.

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

The paper introduces a training method for deep learning turbulence closure models in large eddy simulations that draws from continuous data assimilation. By nudging the coarse LES simulation towards sparse direct numerical simulation data, the neural network learns the forcing needed to reproduce ground-truth statistics. This a-priori approach avoids the high cost and solver modifications of embedding the network inside the LES solver for gradient backpropagation. The resulting models maintain long-term stability and adapt to different numerical schemes and discretizations, unlike previous a-posteriori methods that suffer from instabilities due to filter and discretization mismatches.

Core claim

The authors show that a neural network closure can be trained a-priori by using nudging to incorporate sparse DNS data as observations in the LES equations, allowing the model to learn the required subgrid forcing that captures correct statistics and ensures stability over extended simulations, while generalizing across various numerical and temporal schemes without the need for adjoints or solver differentiation.

What carries the argument

The nudging-based forcing term derived from continuous data assimilation, which the deep learning model learns to provide as the turbulence closure for coarse-grid LES.

If this is right

The trained closures can be deployed in existing LES solvers without any modifications or differentiability requirements.
Models trained this way remain stable and accurate for long integration times across different numerical discretizations.
Generalization to unseen schemes becomes possible, reducing the need for scheme-specific retraining.
Comparisons show improved handling of numerical errors compared to traditional algebraic closure models.

Where Pith is reading between the lines

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

This method could extend to training closures for flows with complex boundaries where full DNS is prohibitive.
Integrating such learned closures with operational data assimilation systems might improve forecast accuracy in atmospheric modeling.
The lack of backpropagation might enable faster iteration in exploring different network architectures for closure modeling.

Load-bearing premise

Nudging with sparse DNS observations trains closures that correct for both filtering and numerical discretization errors without causing new instabilities or requiring per-scheme adjustments.

What would settle it

Running long-term LES with the learned closure on a new numerical scheme and observing whether the solution remains bounded and matches DNS statistics or diverges.

Figures

Figures reproduced from arXiv: 2604.23874 by Ashwin Suriyanarayanan, Dibyajyoti Chakraborty, Romit Maulik.

**Figure 1.** Figure 1: A schematic of the proposed data-driven closure modeling using nudging view at source ↗

**Figure 2.** Figure 2: Single-scheme results for the two-dimensional case. (a) Turbulent kinetic energy evolution for DNS, SMAG, D-SMAG, view at source ↗

**Figure 3.** Figure 3: Multi-scheme results for the two-dimensional case using FiLM-based numerical-scheme conditioning. (a) Turbulent kinetic view at source ↗

**Figure 4.** Figure 4: Representative coarse-grid x-component velocity fields at different times. SMAG and D-SMAG are visibly more dissipative than the NN rollouts. The predicted closure fields for the two schemes are also different, reflecting the different levels of numerical dissipation associated with the underlying discretizations. (a) (b) (c) view at source ↗

**Figure 5.** Figure 5: Comparison of the predicted forcing associated with the VL and UPWIND schemes for the same synchronized coarse-grid view at source ↗

**Figure 6.** Figure 6: Three-dimensional results for the model trained on the C2, C4 and C6 schemes. (a) Turbulent kinetic energy evolution. view at source ↗

**Figure 7.** Figure 7: Representative coarse-grid x-component velocity fields at different times. 10 view at source ↗

**Figure 8.** Figure 8: Comparison of the predicted correction during three-dimensional rollout. (a) Representative momentum-forcing fields view at source ↗

**Figure 9.** Figure 9: Comparison of convective-term and model-prediction differences for the two-dimensional case. (a) Convective-flux fields view at source ↗

**Figure 10.** Figure 10: Comparison of convective-term and predicted forcing differences for the three-dimensional case. (a) Visual comparison of view at source ↗

**Figure 11.** Figure 11: Comparison of the forcing-term spectra: (a) two-dimensional case; (b) three-dimensional case. The SMAG and D-SMAG view at source ↗

**Figure 12.** Figure 12: Visualization of the component-wise forcing for SMAG, D-SMAG, NN and NUDGE. NN and NUDGE provide additive view at source ↗

**Figure 13.** Figure 13: Generalization across time-stepping schemes in the two- and three-dimensional cases. (a,c) Turbulent kinetic energy view at source ↗

**Figure 14.** Figure 14: Close-up comparison of temporal-integrator effects on turbulent kinetic energy: (a) two-dimensional case; (b) three view at source ↗

**Figure 15.** Figure 15: Error comparison of partial nudging for two-dimensional setup. (a,b) view at source ↗

**Figure 16.** Figure 16: Error comparison of partial nudging for three-dimensional setup. (a,b,c) Error compared with view at source ↗

**Figure 17.** Figure 17: Turbulent kinetic energy and time-averaged energy spectrum for partial nudging: (a,b) two-dimensional case; (c,d) view at source ↗

**Figure 18.** Figure 18: Turbulent kinetic energy and RMS error for discrete nudging: (a,b) two-dimensional case; (c,d) three-dimensional case. view at source ↗

**Figure 19.** Figure 19: Dependence of the three-dimensional discrete-nudging error on the nudging coefficient view at source ↗

read the original abstract

Deep learning approaches have shown remarkable promise in turbulence closure modeling for large eddy simulations (LES). The differentiable physics paradigm uses the so-called a-posteriori approach for learning by embedding a neural network closure directly inside the solver and optimizing its learnable parameters against ground truth time-series data which may be observed sparsely. This addresses a key limitation of a-priori learning where direct numerical simulation (DNS) data is used to approximate the subgrid stress with the assumption of a filter. However, closures that are trained in this manner frequently lead to unstable deployments due to the mismatch between the assumed filter and the effect of numerical discretizations. However, a-posteriori learning incurs high computational costs due to the need to backpropagate gradients through an LES solver. Furthermore, a-posteriori methods are challenging to apply broadly since they require significant modification of existing solvers. Finally, these approaches have also been observed to be limited when generalization is desired across different numerical schemes. In this work, we discuss a novel approach for the deep learning of turbulence closure models motivated by the continuous data assimilation (CDA) approach (also known as nudging). Our approach enables a-priori training of closures for coarse-grid LES, treating DNS data as sparse observations. This approach enables the deep learning model to successfully learn the required forcing to capture the ground-truth statistics while maintaining long term stability without needing adjoints or backpropagation through the solver. We train and evaluate the model's ability to adapt to different numerical and temporal schemes. Additionally, we analyse the model behavior with varying numerical discretization errors and compare its predictions to traditional closure models.

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.

Desk Editor's Note private letter to a colleague

Nudging from DNS lets them train NN closures a-priori with claimed stability and scheme adaptability, but the results need to show the closure works once nudging is turned off at inference.

read the letter

The core idea here is using continuous data assimilation, or nudging, to pull coarse-grid LES states toward sparse DNS observations during training. This lets the neural net learn a closure that matches ground-truth statistics while staying stable, all without backpropagating through the solver or modifying existing codes. They also test how well the model adapts when the numerical scheme or time step changes and look at sensitivity to discretization errors. That setup directly targets the filter-mismatch instability that kills many pure a-priori models and the computational cost that limits a-posteriori ones. The approach is new enough in this specific combination to be worth attention for anyone trying to make data-driven closures practical in production CFD codes. They give credit to the CDA literature and standard nudging concepts, which keeps the framing honest. What the paper does well is lay out a training loop that avoids adjoints and still claims long-term stability plus cross-scheme generalization. The evaluation plan—different numerical and temporal discretizations plus comparison to traditional models—addresses real deployment questions. The soft spots are in the evidence and the central assumption. The abstract states success on stability and statistics but supplies no numbers, error bars, or direct comparisons, so the magnitude of the improvement is unclear. The bigger issue is whether those stable statistics survive after the nudging term is removed at test time. If the nudging is continuously correcting drift that the learned closure would otherwise produce, then the method has not yet shown a standalone solver-aware model. The nudging coefficient is listed as a free parameter, which could hide case-by-case tuning. This work is aimed at CFD researchers who already run LES and want to plug in learned closures without rewriting solvers. A reader focused on hybrid physics-ML turbulence modeling will find the method description and the scheme-adaptation tests useful even if the quantitative claims need tightening. It deserves a serious referee because the idea engages a genuine practical gap and the authors have thought through the a-priori versus a-posteriori trade-offs. I would send it out for review with the expectation that the stability-without-nudging test and some concrete metrics will be required in revision.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a nudging-based a-priori training method for neural network turbulence closures in LES. DNS data are treated as sparse observations to learn a forcing term that reproduces ground-truth statistics and maintains long-term stability, without requiring adjoints or backpropagation through the solver. The approach is evaluated for its ability to adapt across numerical and temporal schemes and is compared against traditional closures.

Significance. If the results hold, the method offers a computationally attractive route to stable, solver-aware closures that avoid the cost and implementation burden of differentiable-physics training. The explicit focus on generalization across discretizations and the avoidance of solver backpropagation are practical strengths that could broaden the applicability of data-driven LES modeling.

major comments (2)

[Method] The training procedure (described in the method section) incorporates an explicit nudging term that continuously corrects the coarse-grid state toward DNS observations. The manuscript does not state whether this term remains active when long-term stability and statistical fidelity are evaluated at inference. If stability is demonstrated only while nudging is present, the central claim of a deployable, solver-aware closure is not yet supported.
[Results] Results on adaptation to different numerical and temporal schemes report qualitative agreement but lack quantitative stability metrics (e.g., energy spectra drift rates or time-to-blowup) for the learned closure when the nudging term is removed and the discretization is altered. These metrics are needed to substantiate the generalization claim.

minor comments (2)

The abstract uses both 'analyse' and 'analyze'; standardize spelling throughout.
[Figures] Figure captions should explicitly distinguish the NN-predicted closure from the nudging forcing term to avoid reader confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment below and have revised the paper to provide the requested clarifications and quantitative metrics.

read point-by-point responses

Referee: [Method] The training procedure (described in the method section) incorporates an explicit nudging term that continuously corrects the coarse-grid state toward DNS observations. The manuscript does not state whether this term remains active when long-term stability and statistical fidelity are evaluated at inference. If stability is demonstrated only while nudging is present, the central claim of a deployable, solver-aware closure is not yet supported.

Authors: We thank the referee for this important point. The nudging term is used exclusively during the a-priori training phase to enable the neural network to learn a forcing that reproduces DNS statistics from sparse observations. At inference, when the learned closure is deployed in the LES solver for long-term stability and statistical evaluations, the nudging term is fully disabled. This is the basis for our claim of a deployable, solver-aware closure that does not require solver modifications or adjoints. We have added explicit statements in the revised Methods and Results sections to clarify the distinction between training and inference phases. revision: yes
Referee: [Results] Results on adaptation to different numerical and temporal schemes report qualitative agreement but lack quantitative stability metrics (e.g., energy spectra drift rates or time-to-blowup) for the learned closure when the nudging term is removed and the discretization is altered. These metrics are needed to substantiate the generalization claim.

Authors: We agree that quantitative stability metrics strengthen the generalization claims. The original manuscript emphasized qualitative agreement and visual inspection of time series to demonstrate stability without nudging. In the revision, we have added quantitative metrics including kinetic energy spectra drift rates (computed over 1000 time units) and time-to-blowup (or confirmation of no blow-up) for the learned closure across altered numerical and temporal schemes. These results are now reported in the revised Results section and supplementary figures, showing that the closure maintains stability and outperforms traditional models in several cases. revision: yes

Circularity Check

0 steps flagged

No significant circularity; training uses external DNS observations via standard nudging

full rationale

The paper's central method trains an NN closure a-priori by treating DNS data as sparse observations inside a nudging (CDA) framework. This is an application of an established external technique to a new context (solver-aware closures), not a reduction of any claimed prediction or uniqueness result to the paper's own fitted values or self-citations. No equations are shown that define the target statistics in terms of the learned forcing by construction, and no self-citation chain is invoked to justify the core premise. The reported stability and cross-scheme adaptation are presented as empirical outcomes evaluated after training, keeping the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard fluid dynamics assumptions plus the novel application of nudging; one free parameter for nudging strength.

free parameters (1)

Nudging coefficient
Parameter controlling the strength of data assimilation, likely tuned during training to balance correction and model learning.

axioms (1)

domain assumption DNS data serves as sparse observations for nudging in LES
Core to the CDA motivation described in the abstract.

pith-pipeline@v0.9.0 · 5619 in / 1224 out tokens · 66134 ms · 2026-05-08T05:14:08.649297+00:00 · methodology

discussion (0)

Reference graph

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