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arxiv: 2606.25259 · v1 · pith:5NAUGD37new · submitted 2026-06-24 · ⚛️ physics.comp-ph · physics.flu-dyn

A Neural Surrogate Approach for Simulating Natural Convection Problems

Nurshat Menglik , Alex Shao , David Hyde This is my paper

Pith reviewed 2026-06-25 20:38 UTC · model grok-4.3

classification ⚛️ physics.comp-ph physics.flu-dyn
keywords natural convectionBoussinesq approximationcompressible flowFourier neural operatorneural surrogatefinite element methodheat transferfluid dynamics
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The pith

A Fourier neural operator trained on paired Boussinesq and compressible simulations corrects per-channel errors in natural convection predictions with one evaluation per run.

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

The paper trains a neural surrogate on matched pairs of simulations: one using the cheaper Boussinesq model and one using the more expensive compressible model. After training, the surrogate is applied once to any new Boussinesq result to produce fields whose accuracy approaches the compressible reference across velocity, pressure, and temperature. This yields structural similarity indices near 1.0 and mean-squared error reductions of one to three orders of magnitude in both two and three dimensions. The approach therefore supplies higher-fidelity natural-convection data at roughly the cost of the original Boussinesq solve.

Core claim

Using compressible-flow results as high-fidelity references, a single evaluation of a Fourier neural operator trained on matched Boussinesq-compressible pairs raises the per-channel accuracy of Boussinesq predictions to SSIM values close to unity for all flow variables and produces corresponding mean-squared-error reductions of one to nearly three orders of magnitude across test distributions in two and three spatial dimensions.

What carries the argument

Fourier neural operator trained on paired Boussinesq-compressible simulation snapshots, applied once per new Boussinesq run to map the approximate fields onto the higher-fidelity compressible solution.

If this is right

  • Boussinesq-based codes can be post-processed to compressible accuracy without re-deriving or re-solving the full compressible equations.
  • The same paired-training strategy applies to other inexpensive flow approximations whose errors can be learned from a more expensive reference model.
  • Open-source release of the trained operator and the paired data sets allows immediate reuse on new geometries or boundary conditions within the covered regime.
  • Three-dimensional natural-convection problems become practical at the accuracy level previously available only from compressible solvers.

Where Pith is reading between the lines

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

  • The method could be inserted into existing Boussinesq solvers as an optional post-processing step, enabling selective high-fidelity output only where needed.
  • Retraining on new parameter ranges would be required for problems far from the original training distribution, such as very high Rayleigh numbers or complex geometries.
  • Because the surrogate operates on solution snapshots rather than governing equations, it could be combined with other reduced-order modeling techniques for further speed-ups.

Load-bearing premise

The set of paired Boussinesq and compressible simulations used for training covers the parameter space and flow regimes that will appear in future test cases.

What would settle it

A new natural-convection test case whose Rayleigh or Prandtl numbers lie outside the training ranges, run with both models; if the surrogate fails to reduce mean-squared error by at least one order of magnitude relative to the raw Boussinesq result, the generalization claim is false.

Figures

Figures reproduced from arXiv: 2606.25259 by Alex Shao, David Hyde, Nurshat Menglik.

Figure 1
Figure 1. Figure 1: Domain geometry and boundary conditions for differentially heated thermal cavity simulations. A temperature [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Self-convergence with mesh size for Boussinesq and fully compressible formulations. Relative [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Time-step convergence using first-order and second-order implicit time discretization schemes for compressible for [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Temperature profile validation with COMSOL at [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Validation with COMSOL (Boussinesq, 2D). Despite using different discretizations and solvers, the two sets of [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Validation with COMSOL (Compressible, 2D). Despite using different discretizations and solvers, the two sets of [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Validation with COMSOL (Compressible, 3D). Despite using different discretizations and solvers, the two sets of [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Mass conservation diagnostics for the 2D compressible cavity case (Mesh [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Analysis for First & Second Law of Thermodynamics. Top: Energy components and boundary heat fluxes. Middle: [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Strong scaling analysis for the 2D compressible flow simulation ( [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Unsteady MMS duct verification for the fully compressible formulation at [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Mesh convergence for the unsteady MMS duct case (fully compressible formulation). Relative [PITH_FULL_IMAGE:figures/full_fig_p024_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Representative realization of the stochastic Voronoi temperature boundary conditions applied to the 3D thermal [PITH_FULL_IMAGE:figures/full_fig_p027_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The feature extraction pipeline. (Top) Four raw physical fields on [PITH_FULL_IMAGE:figures/full_fig_p028_14.png] view at source ↗
Figure 16
Figure 16. Figure 16: UMAP projection of 500 shape feature vectors [PITH_FULL_IMAGE:figures/full_fig_p029_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: 2D Fourier Neural Operator (FNO2D) architecture. The network maps a Boussinesq prediction to its fully compressible counterpart. Left/Top (Macroscopic Pipeline): The input consists of 6 channels on a 64 × 64 grid (four physical fields ux, uy, T, p and two augmented coordinates x, y). The input is normalized and reflectively padded, then lifted to a latent width of W = 128 via a pointwise 1×1 convolution P… view at source ↗
Figure 18
Figure 18. Figure 18: Training and validation loss (per-channel MSE in normalized [PITH_FULL_IMAGE:figures/full_fig_p032_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: All four flow variables for a single representative in-distribution stochastic sample (the stochastic-trained surrogate [PITH_FULL_IMAGE:figures/full_fig_p033_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Cross-dataset FNO comparison on a single representative sample, shown for [PITH_FULL_IMAGE:figures/full_fig_p035_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Per-channel qualitative comparison of the 3D stochastic-trained FNO surrogate on one in-distribution stochastic [PITH_FULL_IMAGE:figures/full_fig_p036_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Cross test-set qualitative comparison of the 3D stochastic-trained FNO surrogate on the temperature field, for [PITH_FULL_IMAGE:figures/full_fig_p036_22.png] view at source ↗
read the original abstract

This paper presents a neural surrogate approach for improving the accuracy of natural convection problems simulated with a Boussinesq flow model (incompressible flow with heat transfer). Our approach, based on Fourier neural operators, uses training data consisting of matched pairs of simulations run under the computationally cheaper yet less accurate Boussinesq flow model and a more computationally expensive and more accurate compressible flow model. In both cases, we implement our parallelized simulation codes based on an implicit monolithic mixed finite element method (FEM) approach using the open-source FEniCSx framework. Our implementations are validated against a commercial software package, COMSOL, as well as standard test problems from the literature. We include a careful discussion and analysis of data set generation and present learning results in two and three spatial dimensions. Using compressible flow results as high-fidelity reference solutions, our learning approach, with a single model evaluation per simulation, substantially improves the per-channel accuracy of Boussinesq predictions, with structural similarity (SSIM) close to unity across all flow variables and test distributions and corresponding mean-squared error reductions of one to nearly three orders of magnitude. All code and data is released as open-source.

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

1 major / 0 minor

Summary. The paper claims that a Fourier neural operator (FNO) trained on matched pairs of Boussinesq (incompressible) and compressible natural convection simulations, generated via implicit monolithic mixed FEM in FEniCSx, can correct the Boussinesq model to achieve SSIM values near unity and MSE reductions of 1–3 orders of magnitude across flow variables on test distributions, with validation against COMSOL and literature benchmarks; all code and data are released open-source.

Significance. If the reported generalization holds, the approach offers a practical route to high-fidelity natural convection results at the cost of a single low-fidelity simulation plus one FNO evaluation, which would be valuable for parameter studies in computational fluid dynamics. The open-source release of code, data, and the careful validation against independent COMSOL runs and standard test problems are clear strengths that support reproducibility and adoption.

major comments (1)
  1. [Dataset generation and learning results sections] The central generalization claim (SSIM near 1 and 1–3 order MSE reductions on unseen test distributions) is load-bearing and depends on the paired dataset sufficiently covering the relevant parameter space. The manuscript provides no explicit train/test split statistics, parameter histograms (e.g., Rayleigh/Prandtl number ranges, aspect ratios, boundary conditions), or out-of-distribution protocol, leaving open whether the gains reflect interpolation within the convex hull of training data rather than operator learning that corrects the Boussinesq approximation for new regimes.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive comments and for highlighting the strengths of the open-source release and validation. We address the single major comment below.

read point-by-point responses

  1. Referee: [Dataset generation and learning results sections] The central generalization claim (SSIM near 1 and 1–3 order MSE reductions on unseen test distributions) is load-bearing and depends on the paired dataset sufficiently covering the relevant parameter space. The manuscript provides no explicit train/test split statistics, parameter histograms (e.g., Rayleigh/Prandtl number ranges, aspect ratios, boundary conditions), or out-of-distribution protocol, leaving open whether the gains reflect interpolation within the convex hull of training data rather than operator learning that corrects the Boussinesq approximation for new regimes.

    Authors: We agree that explicit documentation of dataset coverage is necessary to support the generalization claims. The manuscript contains a discussion of dataset generation, but does not include the requested train/test split statistics, parameter histograms, or a clear out-of-distribution protocol. In the revised version we will add these elements (histograms of Rayleigh and Prandtl numbers, aspect ratios, boundary conditions, and split details) together with a statement clarifying that the reported test cases are held-out samples drawn from the same parameter ranges used for training. This will allow readers to assess whether the observed improvements constitute interpolation within the training convex hull or operator learning that corrects the Boussinesq model. revision: yes

Circularity Check

0 steps flagged

No circularity: improvements are empirical outputs of training on independent paired simulation data

full rationale

The paper generates training data from independent FEniCSx simulations of Boussinesq and compressible models, trains an FNO on those pairs, and reports empirical SSIM/MSE gains on test cases validated against COMSOL and literature. No equation, ansatz, or self-citation reduces the claimed per-channel accuracy improvement to a quantity defined by the model itself or by prior author work; the derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim depends on the neural operator learning a generalizable correction from the generated pairs; the ledger records the training process and the domain assumptions of the two flow models.

free parameters (1)
  • Fourier neural operator weights
    The network parameters are fitted to the paired simulation dataset to minimize the difference between corrected Boussinesq and compressible outputs.
axioms (2)
  • domain assumption The Boussinesq approximation provides a usable base model whose errors are learnable from paired data
    The entire surrogate approach starts from running the Boussinesq model and learning its deviation from the compressible reference.
  • standard math Implicit monolithic mixed finite element discretization accurately solves both flow models
    Both training data and reference solutions rely on this numerical method implemented in FEniCSx.

pith-pipeline@v0.9.1-grok · 5741 in / 1405 out tokens · 26121 ms · 2026-06-25T20:38:10.764716+00:00 · methodology

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Reference graph

Works this paper leans on

97 extracted references · 66 canonical work pages

  1. [1]

    Chapter 1 - introduction

    Jiyuan Tu, Guan-Heng Yeoh, and Chaoqun Liu. Chapter 1 - introduction. In Jiyuan Tu, Guan- Heng Yeoh, and Chaoqun Liu, editors,Computational Fluid Dynamics (Third Edition), pages 1–31. Butterworth-Heinemann, third edition edition, 2018. ISBN 978-0-08-101127-0. doi: https://doi.org/10. 1016/B978-0-08-101127-0.00001-5. URLhttps://www.sciencedirect.com/scienc...

    2018

  2. [2]

    Spalart and V

    Philippe R. Spalart and V. Venkatakrishnan. On the role and challenges of CFD in the aerospace industry.The Aeronautical Journal, 120:209 – 232, 2016. URLhttps://api.semanticscholar.org/ CorpusID:7731130

    2016

  3. [3]

    Progress and future prospects of CFD in aerospace—wind tunnel and beyond.Progress in Aerospace Sciences, 41(6):455–470, 2005

    Kozo Fujii. Progress and future prospects of CFD in aerospace—wind tunnel and beyond.Progress in Aerospace Sciences, 41(6):455–470, 2005. ISSN 0376-0421. doi: https://doi.org/10.1016/j.paerosci. 2005.09.001. URLhttps://www.sciencedirect.com/science/article/pii/S0376042105001016

    work page doi:10.1016/j.paerosci 2005

  4. [4]

    Subramanian

    Venkateswarlu Chintala and Kallipatti A. Subramanian. A CFD (computational fluid dynamics) study for optimization of gas injector orientation for performance improvement of a dual-fuel diesel engine. Energy, 57:709–721, 2013. ISSN 0360-5442. doi: https://doi.org/10.1016/j.energy.2013.06.009. URL https://www.sciencedirect.com/science/article/pii/S0360544213005070

    work page doi:10.1016/j.energy.2013.06.009 2013

  5. [5]

    Hoekstra

    Dongwei Ye, Valeria Krzhizhanovskaya, and Alfons G. Hoekstra. Data-driven reduced-order modelling for blood flow simulations with geometry-informed snapshots.Journal of Computational Physics, 497: 112639, 2024. ISSN 0021-9991. doi: https://doi.org/10.1016/j.jcp.2023.112639. URLhttps://www. sciencedirect.com/science/article/pii/S0021999123007349

    work page doi:10.1016/j.jcp.2023.112639 2024

  6. [7]

    Springer International Publishing, Cham, 2021

    Luke Reid.An Introduction to Biomedical Computational Fluid Dynamics, pages 205–222. Springer International Publishing, Cham, 2021. ISBN 978-3-030-76951-2. doi: 10.1007/978-3-030-76951-2_10. URLhttps://doi.org/10.1007/978-3-030-76951-2_10

    work page doi:10.1007/978-3-030-76951-2_10 2021

  7. [8]

    High-performance hybrid modeling chemical reactors using differential evolution based fuzzy inference system.Scientific Reports, 10(1):21304, 2020

    Meisam Babanezhad, Iman Behroyan, Ali Taghvaie Nakhjiri, Azam Marjani, Mashallah Rezakazemi, and Saeed Shirazian. High-performance hybrid modeling chemical reactors using differential evolution based fuzzy inference system.Scientific Reports, 10(1):21304, 2020. doi: 10.1038/s41598-020-78277-3. URLhttps://doi.org/10.1038/s41598-020-78277-3

    work page doi:10.1038/s41598-020-78277-3 2020

  8. [9]

    Saleha, A

    Mohd Hafiz Zawawi, A. Saleha, A. Salwa, N.H. Hassan, Nazirul Mubin Zahari, Mohd Zakwan Ramli, and Zakaria Che Muda. A review: Fundamentals of computational fluid dynamics (CFD). InAIP conference proceedings, volume 2030, page 020252. AIP Publishing LLC, 2018. doi: 10.1063/1.5066893. URLhttps://doi.org/10.1063/1.5066893

    work page doi:10.1063/1.5066893 2030

  9. [10]

    Application of CFD to environmental flows.Journal of Wind Engineering and Industrial Aerodynamics, 81(1):145–158, 1999

    Sung-Eun Kim and Ferit Boysan. Application of CFD to environmental flows.Journal of Wind Engineering and Industrial Aerodynamics, 81(1):145–158, 1999. ISSN 0167-6105. doi: https://doi. org/10.1016/S0167-6105(99)00013-6. URLhttps://www.sciencedirect.com/science/article/pii/ S0167610599000136

    work page doi:10.1016/s0167-6105(99)00013-6 1999

  10. [11]

    CFD simulations of turbulent flow and dispersion in built environment: A per- spective review.Journal of Wind Engineering and Industrial Aerodynamics, 249:105741, 2024

    Yoshihide Tominaga. CFD simulations of turbulent flow and dispersion in built environment: A per- spective review.Journal of Wind Engineering and Industrial Aerodynamics, 249:105741, 2024. ISSN 0167-6105. doi: https://doi.org/10.1016/j.jweia.2024.105741. URLhttps://www.sciencedirect.com/ science/article/pii/S0167610524001041

    work page doi:10.1016/j.jweia.2024.105741 2024

  11. [12]

    Gauthier-Villars, 1903

    Joseph Boussinesq.Théorie analytique de la chaleur mise en harmonic avec la thermodynamique et avec la théorie mécanique de la lumière: Refroidissement et échauffement par rayonnement, conductibilité des tiges, lames et masses cristallines, courants de convection, théorie mécanique de la lumière, volume 2. Gauthier-Villars, 1903. 40

    1903

  12. [13]

    Incompressible flows and the Boussinesq approximation: 50 years of CFD.Comptes Rendus

    Marcello Lappa. Incompressible flows and the Boussinesq approximation: 50 years of CFD.Comptes Rendus. Mécanique, 350(S1):75–96, 2022. doi: 10.5802/crmeca.134. URLhttps://doi.org/10.5802/ crmeca.134

    work page doi:10.5802/crmeca.134 2022

  13. [14]

    Gray and Aldo Giorgini

    Donald D. Gray and Aldo Giorgini. The validity of the Boussinesq approximation for liquids and gases. International Journal of Heat and Mass Transfer, 19(5):545–551, 1976. ISSN 0017-9310. doi: https:// doi.org/10.1016/0017-9310(76)90168-X. URLhttps://www.sciencedirect.com/science/article/ pii/001793107690168X

    work page doi:10.1016/0017-9310(76)90168-x 1976

  14. [15]

    Toward improving Boussinesq flow simulations by learning with compressible flow

    Nurshat Mangnike and David Hyde. Toward improving Boussinesq flow simulations by learning with compressible flow. InProceedings of the Platform for Advanced Scientific Computing Conference, PASC ’24, New York, NY, USA, 2024. Association for Computing Machinery. ISBN 9798400706394. doi: 10.1145/3659914.3659919. URLhttps://doi.org/10.1145/3659914.3659919

    work page doi:10.1145/3659914.3659919 2024

  15. [16]

    Integrating physics- based modeling with machine learning: A survey.arXiv preprint arXiv:2003.04919, 1(1):1–34, 2020

    Jared Willard, Xiaowei Jia, Shaoming Xu, Michael Steinbach, and Vipin Kumar. Integrating physics- based modeling with machine learning: A survey.arXiv preprint arXiv:2003.04919, 1(1):1–34, 2020. URLhttps://arxiv.org/pdf/2003.04919.pdf

    arXiv 2003

  16. [17]

    Sharp interface approaches and deep learning techniques for multiphase flows.Journal of Computational Physics, 380:442–463, 2019

    Frederic Gibou, David Hyde, and Ron Fedkiw. Sharp interface approaches and deep learning techniques for multiphase flows.Journal of Computational Physics, 380:442–463, 2019. doi: 10.1016/j.jcp.2018.05

    work page doi:10.1016/j.jcp.2018.05 2019

  17. [18]

    URLhttps://www.sciencedirect.com/science/article/pii/S0021999118303371

  18. [19]

    Baratta, Joseph P

    Igor A. Baratta, Joseph P. Dean, Jørgen S. Dokken, Michal Habera, Jack S. Hale, Chris N. Richardson, Marie E. Rognes, Matthew W. Scroggs, Nathan Sime, and Garth N. Wells. DOLFINx: the next generation FEniCS problem solving environment. preprint, 2023. URLhttps://doi.org/10.5281/ zenodo.10447666

    2023

  19. [20]

    Wells.Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book, volume 84

    Anders Logg, Kent-Andre Mardal, and Garth N. Wells.Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book, volume 84. Springer Science & Business Media, 2012. URLhttps://api.semanticscholar.org/CorpusID:59832704

    2012

  20. [21]

    Taylor and P

    Cedric M. Taylor and P. Hood. A numerical solution of the navier-stokes equations using the fi- nite element technique.Computers & Fluids, 1(1):73–100, 1973. ISSN 0045-7930. doi: https: //doi.org/10.1016/0045-7930(73)90027-3. URLhttps://www.sciencedirect.com/science/article/ pii/0045793073900273

    work page doi:10.1016/0045-7930(73)90027-3 1973

  21. [22]

    David Hyde, Chennakesava Kadapa, Abdallah Alalem Albustami, and Ahmad F. Taha. Progress and perils of pinns: Perspectives on applying physics-informed neural networks to science and engineering problems.Computing in Science & Engineering, pages 1–9, 2026. doi: 10.1109/MCSE.2026.3675465. URLhttps://ieeexplore.ieee.org/document/11447375

    work page doi:10.1109/mcse.2026.3675465 2026

  22. [23]

    Ueckermann and P.F.J

    M.P. Ueckermann and P.F.J. Lermusiaux. Hybridizable discontinuous Galerkin projection methods for Navier–Stokes and Boussinesq equations.Journal of Computational Physics, 306:390–421, 2016. ISSN 0021-9991. doi: https://doi.org/10.1016/j.jcp.2015.11.028. URLhttps://www.sciencedirect.com/ science/article/pii/S0021999115007688

    work page doi:10.1016/j.jcp.2015.11.028 2016

  23. [24]

    Unified hybridization of dis- continuous galerkin, mixed, and continuous galerkin methods for second order elliptic problems

    Bernardo Cockburn, Jayadeep Gopalakrishnan, and Raytcho Lazarov. Unified hybridization of dis- continuous galerkin, mixed, and continuous galerkin methods for second order elliptic problems. SIAM Journal on Numerical Analysis, 47(2):1319–1365, 2009. doi: 10.1137/070706616. URLttps: //doi.org/10.1137/070706616

    work page doi:10.1137/070706616 2009

  24. [25]

    An implicit high-order hybridizable discontinuous Galerkin method for linear convection–diffusion equations.Journal of Computational Physics, 228(9):3232–3254, 2009

    Ngoc Cuong Nguyen, Jaume Peraire, and Bernardo Cockburn. An implicit high-order hybridizable discontinuous Galerkin method for linear convection–diffusion equations.Journal of Computational Physics, 228(9):3232–3254, 2009. doi: https://doi.org/10.1016/j.jcp.2009.01.030. URLhttps://www. sciencedirect.com/science/article/pii/S0021999109000308

    work page doi:10.1016/j.jcp.2009.01.030 2009

  25. [26]

    Schroeder and Gert Lube

    Philipp W. Schroeder and Gert Lube. Stabilised dG-FEM for incompressible natural convection flows with boundary and moving interior layers on non-adapted meshes.Journal of Computational Physics, 41 335:760–779, 2017. ISSN 0021-9991. doi: https://doi.org/10.1016/j.jcp.2017.01.055. URLhttps: //www.sciencedirect.com/science/article/pii/S0021999117300712

    work page doi:10.1016/j.jcp.2017.01.055 2017

  26. [27]

    A fictitious domain method for particulate flows withheattransfer.Journal of Computational Physics, 217(2):424–452, 2006

    Zhaosheng Yu, Xueming Shao, and Anthony Wachs. A fictitious domain method for particulate flows withheattransfer.Journal of Computational Physics, 217(2):424–452, 2006. ISSN0021-9991. doi: https: //doi.org/10.1016/j.jcp.2006.01.016. URLhttps://www.sciencedirect.com/science/article/pii/ S0021999106000167

    work page doi:10.1016/j.jcp.2006.01.016 2006

  27. [28]

    Lower, slower, louder: V ocal cues of sarcasm,

    Jian-Guo Liu, Cheng Wang, and Hans Johnston. A fourth order scheme for incompressible Boussinesq equations.Journal of Scientific Computing, 18(2):253–285, 2003. doi: https://doi.org/10.1023/A: 1021168924020. URLhttps://link.springer.com/article/10.1023/A:1021168924020

    work page doi:10.1023/a: 2003

  28. [29]

    Damanik, J

    H. Damanik, J. Hron, A. Ouazzi, and S. Turek. A monolithic FEM-multigrid solver for non-isothermal incompressible flow on general meshes.Journal of Computational Physics, 228(10):3869–3881, 2009. ISSN 0021-9991. doi: https://doi.org/10.1016/j.jcp.2009.02.024. URLhttps://www.sciencedirect. com/science/article/pii/S0021999109000941

    work page doi:10.1016/j.jcp.2009.02.024 2009

  29. [30]

    Smethurst, David J

    Christopher A. Smethurst, David J. Silvester, and Milan D. Mihajlovic. Unstructured finite element method for the solution of the Boussinesq problem in three dimensions.International Journal for Nu- merical Methods in Fluids, 73, 2013. URLhttps://api.semanticscholar.org/CorpusID:119789681

    2013

  30. [31]

    Numerical study of three-dimensional oscillatory natural convection at low prandtl number in rectangular enclosures.Journal of Heat Transfer, 123(1):77–83, 09 2000

    Shunichi Wakitani. Numerical study of three-dimensional oscillatory natural convection at low prandtl number in rectangular enclosures.Journal of Heat Transfer, 123(1):77–83, 09 2000. ISSN 0022-1481. doi: 10.1115/1.1336508. URLhttps://doi.org/10.1115/1.1336508

    work page doi:10.1115/1.1336508 2000

  31. [32]

    Miller, Xi Chen, and David M

    Edward A. Miller, Xi Chen, and David M. Williams. Versatile mixed methods for non-isothermal incompressible flows.Computers & Mathematics with Applications, 125:150–175, 2022. ISSN 0898-1221. doi: https://doi.org/10.1016/j.camwa.2022.08.044. URLhttps://www.sciencedirect.com/science/ article/pii/S0898122122003704

    work page doi:10.1016/j.camwa.2022.08.044 2022

  32. [33]

    Fast iterative solvers for buoyancy driven flow problems.Journal of Computational Physics, 230(10):3900–3914, 2011

    Howard Elman, Milan Mihajlović, and David Silvester. Fast iterative solvers for buoyancy driven flow problems.Journal of Computational Physics, 230(10):3900–3914, 2011. ISSN 0021-9991. doi: https: //doi.org/10.1016/j.jcp.2011.02.014. URLhttps://www.sciencedirect.com/science/article/pii/ S0021999111001033

    work page doi:10.1016/j.jcp.2011.02.014 2011

  33. [34]

    A numerical approach to the optimal control of thermally convective flows.Journal of Computational Physics, 494:112458, 2023

    Yongcun Song, Xiaoming Yuan, and Hangrui Yue. A numerical approach to the optimal control of thermally convective flows.Journal of Computational Physics, 494:112458, 2023. ISSN 0021-9991. doi: https://doi.org/10.1016/j.jcp.2023.112458. URLhttps://www.sciencedirect.com/science/ article/pii/S0021999123005533

    work page doi:10.1016/j.jcp.2023.112458 2023

  34. [35]

    A finite-element toolbox for the simulation of solid–liquid phase-change systems with natural convection.Computer Physics Communications, 253: 107188, 2020

    Aina Rakotondrandisa, Georges Sadaka, and Ionut Danaila. A finite-element toolbox for the simulation of solid–liquid phase-change systems with natural convection.Computer Physics Communications, 253: 107188, 2020. ISSN 0010-4655. doi: https://doi.org/10.1016/j.cpc.2020.107188. URLhttps://www. sciencedirect.com/science/article/pii/S0010465520300151

    work page doi:10.1016/j.cpc.2020.107188 2020

  35. [36]

    Georges Sadaka, Aina Rakotondrandisa, Pierre-Henri Tournier, Francky Luddens, Corentin Lothodé, and Ionut Danaila. Parallel finite-element codes for the simulation of two-dimensional and three- dimensional solid–liquid phase-change systems with natural convection.Computer Physics Commu- nications, 257:107492, 2020. ISSN 0010-4655. doi: https://doi.org/10....

    work page doi:10.1016/j.cpc.2020.107492 2020

  36. [37]

    Peyman Mayeli and Gregory J. Sheard. Buoyancy-driven flows beyond the Boussinesq approximation: A brief review.International Communications in Heat and Mass Transfer, 125:105316, 2021. doi: https:// doi.org/10.1016/j.icheatmasstransfer.2021.105316. URLhttps://www.sciencedirect.com/science/ article/pii/S0735193321002098

    work page doi:10.1016/j.icheatmasstransfer.2021.105316 2021

  37. [38]

    Zeytounian

    Radyadour Kh. Zeytounian. Joseph Boussinesq and his approximation: a contemporary view.Comptes Rendus Mecanique, 331(8):575–586, 2003. doi: https://doi.org/10.1016/S1631-0721(03)00120-7. URL https://www.sciencedirect.com/science/article/pii/S1631072103001207. 42

    work page doi:10.1016/s1631-0721(03)00120-7 2003

  38. [39]

    Patankar and Dudley B

    Suhas V. Patankar and Dudley B. Spalding. A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows.International Journal of Heat and Mass Transfer, 15 (10):1787–1806, 1972. ISSN 0017-9310. doi: https://doi.org/10.1016/0017-9310(72)90054-3. URL https://www.sciencedirect.com/science/article/pii/0017931072900543

    work page doi:10.1016/0017-9310(72)90054-3 1972

  39. [40]

    Jeffrey Peter van Doormaal and George D. Raithby. Enhancements of the simple method for pre- dicting incompressible fluid flows.Numerical Heat Transfer, 7(2):147–163, 1984. doi: 10.1080/ 01495728408961817. URLhttps://doi.org/10.1080/01495728408961817

    work page doi:10.1080/01495728408961817 1984

  40. [41]

    Raad I. Issa. Solution of the implicitly discretised fluid flow equations by operator-splitting.Journal of Computational Physics, 62(1):40–65, 1986. ISSN0021-9991. doi: https://doi.org/10.1016/0021-9991(86) 90099-9. URLhttps://www.sciencedirect.com/science/article/pii/0021999186900999

    work page doi:10.1016/0021-9991(86 1986

  41. [42]

    Chae M. Rhie. Pressure-based navier-stokes solver using the multigrid method.AIAA journal, 27 (8):1017–1018, 1989. doi: 10.2514/6.1986-207. URLhttps://arc.aiaa.org/doi/pdf/10.2514/6. 1986-207

    work page doi:10.2514/6.1986-207 1989

  42. [43]

    Karki and Suhas V

    Kailash C. Karki and Suhas V. Patankar. Pressure based calculation procedure for viscous flows at all speedsin arbitrary configurations.AIAA journal, 27(9):1167–1174, 1989. URLhttps://doi.org/10. 2514/3.10242

    1989

  43. [44]

    Rehm and Howard R

    Ronald G. Rehm and Howard R. Baum. The equations of motion for thermally driven, buoyant flows. Journal of Research of the National Bureau of Standards, 83 3:297–308, 1978. URLhttps://api. semanticscholar.org/CorpusID:13971757

    1978

  44. [45]

    Pierce, and Parviz Moin

    Clifton Wall, Charles D. Pierce, and Parviz Moin. A semi-implicit method for resolution of acoustic waves in low mach number flows.Journal of Computational Physics, 181(2):545–563, 2002. ISSN 0021-

    2002

  45. [46]

    URLhttps://www.sciencedirect.com/science/ article/pii/S002199910297141X

    doi: https://doi.org/10.1006/jcph.2002.7141. URLhttps://www.sciencedirect.com/science/ article/pii/S002199910297141X

    work page doi:10.1006/jcph.2002.7141 2002

  46. [47]

    Le Quéré, R Masson, and Pierre Perrot

    Patrick L. Le Quéré, R Masson, and Pierre Perrot. A Chebyshev collocation algorithm for 2D non-Boussinesq convection.Journal of Computational Physics, 103(2):320–335, 1992. ISSN 0021-

    1992

  47. [48]

    URLhttps://www.sciencedirect.com/ science/article/pii/002199919290404M

    doi: https://doi.org/10.1016/0021-9991(92)90404-M. URLhttps://www.sciencedirect.com/ science/article/pii/002199919290404M

    work page doi:10.1016/0021-9991(92)90404-m

  48. [49]

    A pressure-based solver for low-Mach number flow using a discontinuous Galerkin method.Journal of Computational Physics, 425:109877, 2021

    Aldo Hennink, Marco Tiberga, and Danny Lathouwers. A pressure-based solver for low-Mach number flow using a discontinuous Galerkin method.Journal of Computational Physics, 425:109877, 2021. ISSN 0021-9991. doi: https://doi.org/10.1016/j.jcp.2020.109877. URLhttps://www.sciencedirect.com/ science/article/pii/S0021999120306513

    work page doi:10.1016/j.jcp.2020.109877 2021

  49. [50]

    A numerical method for low Mach number compressible flows by simultaneous relaxation of dependent variables.arXiv preprint arXiv:2502.08116, 2025

    Hideki Yanaoka and Yuji Sato. A numerical method for low Mach number compressible flows by simultaneous relaxation of dependent variables.arXiv preprint arXiv:2502.08116, 2025. doi: 10.48550/ arXiv.2502.08116. URLhttps://doi.org/10.48550/arXiv.2502.08116

    work page doi:10.48550/arxiv.2502.08116 2025

  50. [51]

    Preconditioned methods for solving the incompressible and low speed compressible equations.Journal of Computational Physics, 72(2):277–298, 1987

    Eli Turkel. Preconditioned methods for solving the incompressible and low speed compressible equations.Journal of Computational Physics, 72(2):277–298, 1987. ISSN 0021-9991. doi: https: //doi.org/10.1016/0021-9991(87)90084-2. URLhttps://www.sciencedirect.com/science/article/ pii/0021999187900842

    work page doi:10.1016/0021-9991(87)90084-2 1987

  51. [52]

    Preconditioning techniques in computational fluid dynamics.Annual Review of Fluid Me- chanics, 31(1):385–416, 1999

    Eli Turkel. Preconditioning techniques in computational fluid dynamics.Annual Review of Fluid Me- chanics, 31(1):385–416, 1999. doi: 10.1146/annurev.fluid.31.1.385. URLhttps://doi.org/10.1146/ annurev.fluid.31.1.385

    work page doi:10.1146/annurev.fluid.31.1.385 1999

  52. [53]

    Yunho Choi and Charles L. Merkle. The application of preconditioning in viscous flows.Journal of Computational Physics, 105(2):207–223, 1993. ISSN 0021-9991. doi: https://doi.org/10.1006/jcph.1993

    work page doi:10.1006/jcph.1993 1993

  53. [54]

    URLhttps://www.sciencedirect.com/science/article/pii/S0021999183710697

  54. [55]

    Weiss and Wayne Smith

    Jonathan M. Weiss and Wayne Smith. Preconditioning applied to variable and constant density flows. AIAA Journal, 33:2050–2057, 1995. URLhttps://api.semanticscholar.org/CorpusID:119737915. 43

    2050

  55. [56]

    On the behaviour of upwind schemes in the low mach number limit

    Hervé Guillard and Cécile Viozat. On the behaviour of upwind schemes in the low mach number limit. Computers & fluids, 28(1):63–86, 1999. doi: https://doi.org/10.1016/S0045-7930(98)00017-6. URL https://www.sciencedirect.com/science/article/pii/S0045793098000176

    work page doi:10.1016/s0045-7930(98)00017-6 1999

  56. [57]

    Brooks and Thomas J.R

    Alexander N. Brooks and Thomas J.R. Hughes. Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equa- tions.Computer Methods in Applied Mechanics and Engineering, 32(1):199–259, 1982. ISSN 0045-

    1982

  57. [58]

    URLhttps://www.sciencedirect.com/ science/article/pii/0045782582900718

    doi: https://doi.org/10.1016/0045-7825(82)90071-8. URLhttps://www.sciencedirect.com/ science/article/pii/0045782582900718

    work page doi:10.1016/0045-7825(82)90071-8

  58. [59]

    Franca, and Marc Balestra

    Thomas Joseph Robert Hughes, Leopoldo P. Franca, and Marc Balestra. A new finite element formula- tion for computational fluid dynamics: V. circumventing the babuška-brezzi condition: A stable Petrov- Galerkin formulation of the Stokes problem accommodating equal-order interpolations.Applied Mechan- ics and Engineering, 59:85–99, 1986. URLhttps://api.sema...

    1986

  59. [60]

    Hong Luo, Hidehiro Segawa, and Miguel R. Visbal. An implicit discontinuous Galerkin method for the unsteady compressible Navier–Stokes equations.Computers & Fluids, 53:133–144, 2012. ISSN 0045-

    2012

  60. [61]

    URLhttps://www.sciencedirect.com/ science/article/pii/S0045793011003082

    doi: https://doi.org/10.1016/j.compfluid.2011.10.009. URLhttps://www.sciencedirect.com/ science/article/pii/S0045793011003082

    work page doi:10.1016/j.compfluid.2011.10.009 2011

  61. [62]

    Satish Balay, William Gropp, Lois Curfman McInnes, and Barry F. Smith. Efficient management of parallelism in object-oriented numerical software libraries. InModern Software Tools in Scientific Computing, 1997. URLhttps://api.semanticscholar.org/CorpusID:56520908

    1997

  62. [63]

    Mikael Mortensen, Hans Petter Langtangen, and Garth N. Wells. A FEniCS-based programming frame- work for modeling turbulent flow by the Reynolds-averaged Navier–Stokes equations.Advances in Water Resources, 34(9):1082–1101, 2011. ISSN 0309-1708. doi: https://doi.org/10.1016/j.advwatres.2011.02

    work page doi:10.1016/j.advwatres.2011.02 2011

  63. [64]

    URLhttps://www.sciencedirect.com/science/article/pii/S030917081100039X

  64. [65]

    Rognes, and Stuart R

    Lyudmyla Vynnytska, Marie E. Rognes, and Stuart R. Clark. Benchmarking FEniCS for mantle convection simulations.Computers & Geosciences, 50:95–105, 2013. ISSN 0098-3004. doi: https: //doi.org/10.1016/j.cageo.2012.05.012. URLhttps://www.sciencedirect.com/science/article/ pii/S0098300412001689

    work page doi:10.1016/j.cageo.2012.05.012 2013

  65. [66]

    Zarrouk, and Rosalind Archer

    Chao Zhang, Sadiq J. Zarrouk, and Rosalind Archer. A mixed finite element solver for natural convection in porous media using automated solution techniques.Computers & Geosciences, 96: 181–192, 2016. ISSN 0098-3004. doi: https://doi.org/10.1016/j.cageo.2016.08.012. URLhttps: //www.sciencedirect.com/science/article/pii/S0098300416302746

    work page doi:10.1016/j.cageo.2016.08.012 2016

  66. [67]

    Physics-informed machine learning,

    George Em Karniadakis, Ioannis G. Kevrekidis, Lu Lu, Paris Perdikaris, Sifan Wang, and Liu Yang. Physics-informed machine learning.Nature Reviews Physics, 3(6):422–440, 2021. doi: https://doi.org/ 10.1038/s42254-021-00314-5. URLhttps://www.nature.com/articles/s42254-021-00314-5

    work page doi:10.1038/s42254-021-00314-5 2021

  67. [69]

    Lagaris, A

    I.E. Lagaris, A. Likas, and D.I. Fotiadis. Artificial neural networks for solving ordinary and partial differential equations.IEEE Transactions on Neural Networks, 9(5):987–1000, 1998. doi: 10.1109/72. 712178. URLhttps://ieeexplore.ieee.org/document/712178

    work page doi:10.1109/72 1998

  68. [70]

    Gamini Dissanayake and Nhan Phan-Thien

    M.W.M. Gamini Dissanayake and Nhan Phan-Thien. Neural-network-based approximations for solving partial differential equations.Communications in Numerical Methods in Engineering, 10(3):195–201,

  69. [71]

    URLhttps://onlinelibrary.wiley.com/doi/ abs/10.1002/cnm.1640100303

    doi: https://doi.org/10.1002/cnm.1640100303. URLhttps://onlinelibrary.wiley.com/doi/ abs/10.1002/cnm.1640100303. 44

    work page doi:10.1002/cnm.1640100303

  70. [72]

    A deep conjugate direction method for iteratively solving linear systems

    AyanoKaneda, OsmanAkar, JingyuChen, VictoriaAliciaTrevinoKala, DavidHyde, andJosephTeran. A deep conjugate direction method for iteratively solving linear systems. In Andreas Krause, Emma Brunskill, Kyunghyun Cho, Barbara Engelhardt, Sivan Sabato, and Jonathan Scarlett, editors,Proceed- ings of the 40th International Conference on Machine Learning, volume...

    2023

  71. [73]

    Reinforcement learning for adaptive mesh re- finement

    Jiachen Yang, Tarik Dzanic, Brenden Petersen, Jun Kudo, Ketan Mittal, Vladimir Tomov, Jean-Sylvain Camier, Tuo Zhao, Hongyuan Zha, Tzanio Kolev, et al. Reinforcement learning for adaptive mesh re- finement. InInternational Conference on Artificial Intelligence and Statistics, pages 5997–6014. PMLR,

  72. [74]

    URLhttps://proceedings.mlr.press/v206/yang23e.html

  73. [75]

    A supervised neural network for drag prediction of arbitrary 2d shapes in laminar flows at low reynolds number.Computers & Fluids, 210:104645, 2020

    Jonathan Viquerat and Elie Hachem. A supervised neural network for drag prediction of arbitrary 2d shapes in laminar flows at low reynolds number.Computers & Fluids, 210:104645, 2020. doi: https://doi.org/10.1016/j.compfluid.2020.104645. URLhttps://www.sciencedirect.com/science/ article/pii/S0045793020302164

    work page doi:10.1016/j.compfluid.2020.104645 2020

  74. [76]

    Norman, David M

    Muralikrishnan Gopalakrishnan Meena, Matthew R. Norman, David M. Hall, and Michael S. Pritchard. Spatially local surrogate modeling of subgrid-scale effects in idealized atmospheric flows: A deep learned approach using high-resolution simulation data.Artificial Intelligence for the Earth Systems, 3(4):e230043, 2024. doi: 10.1175/AIES-D-23-0043.1. URLhttps...

    work page doi:10.1175/aies-d-23-0043.1 2024

  75. [77]

    On deep-learning-based closures for algebraic surrogate models of turbulent flows.Journal of Fluid Mechanics, 1020:A36, 2025

    Benet Eiximeno, Marcial Sanchis-Agudo, Arnau Miró, Ivette Rodriguez, Ricardo Vinuesa, and Oriol Lehmkuhl. On deep-learning-based closures for algebraic surrogate models of tur- bulent flows.Journal of Fluid Mechanics, 1020:A36, 2025. doi: 10.1017/jfm.2025.10610. URLhttps://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/ on-deeplearning...

    work page doi:10.1017/jfm.2025.10610 2025

  76. [78]

    Hatashita, and Suhas S

    Anirban Bhattacharjee, Luis H. Hatashita, and Suhas S. Jain. A machine learning model for the prediction of sub-grid interfacial area in two-phase turbulent flows. InAIAA SCITECH 2026 Forum, page 1731, 2026. doi: https://doi.org/10.2514/6.2026-1731. URLhttps://arc.aiaa.org/doi/10. 2514/6.2026-1731

    work page doi:10.2514/6.2026-1731 2026

  77. [79]

    Arnold, and Trenton M

    Joshua Stuckner, Matthew Piekenbrock, Steven M. Arnold, and Trenton M. Ricks. Optimal experimen- tal design with fast neural network surrogate models.Computational Materials Science, 200:110747,

  78. [80]

    URLhttps://www.sciencedirect.com/ science/article/pii/S0927025621004742

    doi: https://doi.org/10.1016/j.commatsci.2021.110747. URLhttps://www.sciencedirect.com/ science/article/pii/S0927025621004742

    work page doi:10.1016/j.commatsci.2021.110747 2021

  79. [81]

    An artificial neural network for surrogate modeling of stress fields in viscoplastic polycrystalline materials,

    Mohammad S. Khorrami, Jaber R. Mianroodi, Nima H. Siboni, Pawan Goyal, Bob Svendsen, Peter Benner, and Dierk Raabe. An artificial neural network for surrogate modeling of stress fields in vis- coplastic polycrystalline materials.npj Computational Materials, 9(1):37, 2023. doi: https://doi.org/ 10.1038/s41524-023-00991-z. URLhttps://www.nature.com/articles...

    work page doi:10.1038/s41524-023-00991-z 2023

  80. [82]

    Latent representation learning based model correction and uncertainty quantification for PDEs.arXiv preprint arXiv:2603.24948, 2026

    Wenwen Zhou, Xiaodong Feng, Ling Guo, and Hao Wu. Latent representation learning based model correction and uncertainty quantification for PDEs.arXiv preprint arXiv:2603.24948, 2026. doi: http: //dx.doi.org/10.2139/ssrn.6629548. URLhttps://api.semanticscholar.org/CorpusID:286790070

    work page doi:10.2139/ssrn.6629548 2026

Showing first 80 references.