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arxiv: 2604.18536 · v1 · submitted 2026-04-20 · 🧮 math.NA · cs.NA· physics.flu-dyn

A differentiable software suite for accelerated simulation of turbulent flows

Syver D{\o}ving Agdestein , Benjamin Sanderse This is my paper

Pith reviewed 2026-05-10 03:25 UTC · model grok-4.3

classification 🧮 math.NA cs.NAphysics.flu-dyn
keywords incompressible Navier-StokesJulia packageautomatic differentiationlarge-eddy simulationturbulent channel flowstaggered gridGPU accelerationneural network closure
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The pith

A Julia package for incompressible Navier-Stokes simulations supplies hand-written adjoint kernels that make the full solver differentiable for neural network training.

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

The paper introduces IncompressibleNavierStokes.jl, an open-source Julia package that solves the incompressible Navier-Stokes equations on staggered Cartesian grids. It supplies matrix-free kernels written once and compiled for either multi-threaded CPUs or GPUs, plus custom adjoint kernels for every discrete operator. These adjoints support efficient reverse-mode automatic differentiation across the entire time-stepping procedure. As a result, neural network models that close the turbulence equations can be trained while they remain embedded inside a large-eddy simulation. Memory optimizations also let the code run double-precision direct numerical simulations at resolutions up to 840 cubed on a single GPU, with results validated on turbulent channel flow.

Core claim

The package achieves efficient reverse-mode automatic differentiation through the entire solver by using hand-written adjoint kernels for all discrete operators, combined with matrix-free, hardware-agnostic implementations compiled for multi-threaded CPU or GPU execution, allowing neural network closure models to be trained a-posteriori while embedded in large-eddy simulation.

What carries the argument

Hand-written adjoint kernels for the discrete staggered-grid operators, which supply the exact discrete derivatives required for reverse-mode automatic differentiation of the full solver.

If this is right

  • Neural network closure models can be trained end-to-end inside large-eddy simulations without needing separate forward and adjoint solvers.
  • Direct numerical simulations of incompressible turbulence reach 840 cubed resolution in double precision on a single GPU.
  • Sensitivity analysis and optimization of flow quantities become practical because the entire discrete operator is differentiable.
  • The single-source kernel design reduces the effort required to maintain or extend the code across CPU and GPU platforms.

Where Pith is reading between the lines

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

  • The same pattern of hand-written adjoints could be added to other existing CFD codes to make them compatible with embedded machine-learning models.
  • The differentiability opens the door to solving inverse problems that infer boundary data or model parameters from measured flow statistics.
  • Extending the memory optimizations to distributed multi-GPU runs would allow even larger grids while preserving the single-source code structure.

Load-bearing premise

The hand-written adjoint kernels compute derivatives that match the discrete operators closely enough that they do not introduce significant errors into the simulation or into the training of any embedded neural network.

What would settle it

Compute the gradient of a simple output functional with respect to an input parameter using the adjoint kernels, then recompute the same gradient with finite differences on an identical small grid; a mismatch larger than discretization error would falsify the claim that the adjoints are accurate.

Figures

Figures reproduced from arXiv: 2604.18536 by Benjamin Sanderse, Syver D{\o}ving Agdestein.

Figure 1
Figure 1. Figure 1: Different horizontal grid spacing profiles. From left to right: uni [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: fig. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: Overview of the software project components and their interactions. The source code (center) is imported by the test suite and example scripts, [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Rendered documentation from a docstring. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Generated documentation website [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Different example scripts included in the documentation with [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Snapshot of the turbulent channel flow during the transition [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 11
Figure 11. Figure 11: Third order cross-moment of the fluctuations [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Second order cross-moment of the fluctuations [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗
Figure 10
Figure 10. Figure 10: Fourth moment of 𝑥-velocity fluctuations ⟨ 𝑢′𝑢′𝑢′𝑢′ ⟩ for DNS of turbulent channel flow with 512 × 1024 × 256 grid points on a uniform grid, compared with reference data from Vreman and Kuerten [33]. y + 100 101 102 ⟨u'u'v'⟩ 0.0 0.5 1.0 Vreman and Kuerten (2014) IncompressibleNavierStokes.jl [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 13
Figure 13. Figure 13: shows the mean eddy-viscosity ⟨𝜈̄ ∆⟩∕𝜈 normal￾ized by the kinematic viscosity 𝜈. The DNS and no-model simulations do not produce any eddy-viscosity (we set 𝜈 ∆ = 0). The Smagorinsky viscosity does not decay to zero at the wall, which is a well-known issue with that model. All the other models decay to zero at the wall, with the WALE model show￾ing the fastest decay. In the linear layer, Smagorinsky has th… view at source ↗
Figure 14
Figure 14. Figure 14: Mean streamwise velocity profile ⟨𝑢̄⟩ for LES of turbulent channel flow with 128 × 64 × 64 grid points on a non-uniform grid, compared with reference data from Vreman and Kuerten [33]. emphasize that all the eddy-viscosities are scaled by the re￾spective model constants. For different choices of constants, the magnitude of the eddy-viscosities can change, but not the shape of the profiles. Figures 14 to 1… view at source ↗
Figure 15
Figure 15. Figure 15: Average squared velocity fluctuations in the [PITH_FULL_IMAGE:figures/full_fig_p013_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Third order cross-moment of the fluctuations [PITH_FULL_IMAGE:figures/full_fig_p013_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: fig. 17 [PITH_FULL_IMAGE:figures/full_fig_p014_17.png] view at source ↗
Figure 17
Figure 17. Figure 17: Finite difference errors for the derivative of [PITH_FULL_IMAGE:figures/full_fig_p015_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Convergence of the numerical solution to the analytical solution for the velocity field of the 2D Taylor-Green vortex at [PITH_FULL_IMAGE:figures/full_fig_p015_18.png] view at source ↗
read the original abstract

We present IncompressibleNavierStokes.jl, an open-source Julia package for solving the incompressible Navier--Stokes equations on staggered Cartesian grids. The package features matrix-free, hardware-agnostic kernels that are compiled from a single source for multi-threaded CPU or GPU execution, and hand-written adjoint kernels for all discrete operators, enabling efficient reverse-mode automatic differentiation through the entire solver. This differentiability allows neural network closure models to be trained a-posteriori while embedded in a large-eddy simulation. Memory optimizations permit double-precision direct numerical simulations at resolutions up to $840^3$ on a single GPU. The software design, numerical methods, hardware performance, and integration of neural network closure models are described, and results for turbulent channel flow are validated against reference data.

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 introduces IncompressibleNavierStokes.jl, an open-source Julia package for solving the incompressible Navier-Stokes equations on staggered Cartesian grids. It features matrix-free, hardware-agnostic kernels compiled from a single source for multi-threaded CPU or GPU execution, hand-written adjoint kernels for all discrete operators to enable reverse-mode automatic differentiation through the full solver, memory optimizations supporting double-precision DNS up to 840^3 on a single GPU, and integration of neural network closure models for a-posteriori training in large-eddy simulations, with validation results for turbulent channel flow against reference data.

Significance. If the differentiability and performance claims hold, the work provides a practical open-source platform for accelerated turbulent flow simulations that embeds neural network models directly in the solver loop. The single-source kernel design and high-resolution single-GPU capability are concrete strengths for reproducible CFD research.

major comments (2)
  1. Abstract: The central claim that hand-written adjoint kernels for all discrete operators enable reliable reverse-mode AD through the solver is load-bearing for the a-posteriori NN training application, yet no verification is reported (e.g., the adjoint consistency test v·(J u) = (Jᵀ v)·u within machine epsilon for random vectors u,v on each operator, or a finite-difference gradient check on the full solver). Without such checks, implementation errors in the adjoints could silently corrupt gradients used for NN closure training.
  2. Validation section (implied by abstract): The turbulent channel flow results are stated to be validated against reference data, but the manuscript provides no quantitative error metrics, grid resolutions used in the comparison, or assessment of how memory optimizations affect accuracy at high Reynolds numbers; this information is required to substantiate the accuracy claims that underpin the package's utility.
minor comments (1)
  1. The abstract and methods description would benefit from explicit cross-references to the specific discrete operators (e.g., divergence, gradient, convection) for which adjoints were implemented.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and positive assessment of the work's significance. We address each major comment below, indicating the revisions that will be incorporated to strengthen the manuscript.

read point-by-point responses

  1. Referee: Abstract: The central claim that hand-written adjoint kernels for all discrete operators enable reliable reverse-mode AD through the solver is load-bearing for the a-posteriori NN training application, yet no verification is reported (e.g., the adjoint consistency test v·(J u) = (Jᵀ v)·u within machine epsilon for random vectors u,v on each operator, or a finite-difference gradient check on the full solver). Without such checks, implementation errors in the adjoints could silently corrupt gradients used for NN closure training.

    Authors: We agree that explicit verification of the adjoint kernels is essential to substantiate the reliability of reverse-mode AD for a-posteriori NN training. The manuscript describes the hand-written adjoints but does not report numerical checks. We will add a dedicated subsection (likely in the Numerical Methods or Software Design section) presenting adjoint consistency tests of the form v·(J u) = (Jᵀ v)·u within machine epsilon for random vectors on each operator, along with a finite-difference gradient check on the full solver for a representative test case. These additions will directly address the concern regarding potential silent errors in the gradients. revision: yes

  2. Referee: Validation section (implied by abstract): The turbulent channel flow results are stated to be validated against reference data, but the manuscript provides no quantitative error metrics, grid resolutions used in the comparison, or assessment of how memory optimizations affect accuracy at high Reynolds numbers; this information is required to substantiate the accuracy claims that underpin the package's utility.

    Authors: We acknowledge that the current manuscript states validation against reference data without providing the requested quantitative details. We will revise the Validation section to include the specific grid resolutions employed in the comparisons, quantitative error metrics (such as L2 norms of mean velocity profiles and skin-friction coefficients relative to reference DNS), and a short assessment of the impact of the memory optimizations on accuracy, particularly at the higher Reynolds numbers considered. These changes will better support the accuracy claims. revision: yes

Circularity Check

0 steps flagged

Software implementation paper with no mathematical derivation chain

full rationale

The manuscript is a description of the IncompressibleNavierStokes.jl package, its matrix-free kernels, hand-written adjoints, memory optimizations, and integration with neural-network closures for a-posteriori training. No first-principles derivation, parameter fitting, or predictive claim is presented that could reduce to its own inputs by construction. Validation is performed against external reference data for turbulent channel flow. No self-citation load-bearing steps, ansatz smuggling, or renaming of known results appear in the provided text. The hand-written adjoint kernels are an implementation choice whose correctness is an engineering verification issue rather than a circularity issue in any derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work builds on standard numerical methods for CFD and AD techniques without introducing new physical entities or fitted parameters beyond implementation choices.

axioms (2)
  • domain assumption Incompressible Navier-Stokes equations accurately model the fluid flows of interest.
    This is the foundational physical model assumed for the simulations.
  • domain assumption The discrete operators on staggered grids can be differentiated via hand-written adjoints.
    Assumed for the AD functionality.

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