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arxiv: 2604.27158 · v1 · submitted 2026-04-29 · ⚛️ physics.flu-dyn

Recognition: unknown

Hybrid Fourier Neural Operator-Lattice Boltzmann Method

Alexandra Junk, Josef M. Winter, Meike T\"utken, Nikolaus A. Adams, Steffen Schmidt

Pith reviewed 2026-05-07 08:06 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords Fourier Neural OperatorLattice Boltzmann Methodhybrid couplingporous media flowsunsteady flowssimulation accelerationfluid dynamicsneural surrogate
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0 comments X

The pith

A hybrid neural operator and lattice Boltzmann method speeds up fluid flow simulations with up to 70 percent faster convergence to steady state.

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

The paper develops a hybrid framework that combines Fourier Neural Operators with the Lattice Boltzmann Method to accelerate both steady and unsteady weakly compressible flow simulations. For steady porous media flows, the neural operator provides an initial guess that lets the lattice Boltzmann solver reach equilibrium much quicker across density, velocity and pressure fields. For unsteady cases, short rollouts from the operator are inserted into the lattice Boltzmann time stepping to reduce error growth over long horizons. This hybridization lets a small 2.6 million parameter model perform as accurately as a four-times larger model while keeping the simulation physically consistent.

Core claim

The central claim is that FNO-based initialization accelerates LBM convergence to steady states by up to 70 percent for density and over 40 percent for pressure drop in porous media flows, while hybrid coupling of FNO rollouts as super-time-stepping into LBM advancement stabilizes long-term predictions and reduces errors by 96 to 99.8 percent, enabling small surrogate models to match the accuracy of much larger ones without introducing non-physical artifacts.

What carries the argument

The hybrid coupling strategy that uses FNO predictions either for initialization of steady-state LBM runs or for embedded super-time-stepping rollouts during unsteady LBM evolution.

If this is right

  • Porous media flow simulations reach their steady states substantially faster in all macroscopic quantities while ending at the same accurate solution.
  • Long-horizon unsteady flow trajectories show markedly lower error accumulation when the hybrid insertion of FNO predictions is used compared with pure LBM or pure FNO.
  • A lightweight neural operator achieves nearly the same predictive quality as a much larger operator when operated inside the hybrid scheme.
  • The overall computational cost decreases because fewer LBM steps are needed to reach the desired accuracy level.

Where Pith is reading between the lines

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

  • This hybrid idea could be tested with other neural operator architectures or other fluid solvers to see if the stabilization effect is general.
  • Applying the method to three-dimensional porous media or flows with higher Reynolds numbers would test how far the training data generalizes.
  • If the conservation properties remain intact, the approach might allow faster design iterations in applications such as porous filter optimization.

Load-bearing premise

The Fourier Neural Operator, after training on a limited set of flows, must produce outputs that remain compatible with the lattice Boltzmann update rules without generating non-physical artifacts or breaking long-term stability.

What would settle it

Running the hybrid method on a new unsteady flow trajectory for thousands of time steps and observing a drift in total mass or momentum beyond the level seen in pure lattice Boltzmann would falsify the claim of preserved physical consistency.

Figures

Figures reproduced from arXiv: 2604.27158 by Alexandra Junk, Josef M. Winter, Meike T\"utken, Nikolaus A. Adams, Steffen Schmidt.

Figure 1
Figure 1. Figure 1: Visualization of the LBM simulation loop. If no macroscopic fields are available, view at source ↗
Figure 2
Figure 2. Figure 2: FNO Architecture with the lifting network view at source ↗
Figure 3
Figure 3. Figure 3: Hybrid FNO-LBM Framework. For steady-state flows, the FNO provides an view at source ↗
Figure 4
Figure 4. Figure 4: Exemplary porous media initialization based on the workflow introduced in view at source ↗
Figure 5
Figure 5. Figure 5: Convergence of the velocity components u and v, density field ρ, and pressure￾drop ∆p, with respect to the steady-state reference solution. The steady state is computed as the average of the last 10 simulation outputs (10∆T). The pressure-drop is defined as the difference between inflow and outflow pressures, and its error is computed relative to the steady-state pressure-drop. Shaded regions indicate stan… view at source ↗
Figure 6
Figure 6. Figure 6: Exemplary rollout with the 2.6M FNO model based on the workflow introduced view at source ↗
Figure 7
Figure 7. Figure 7: Global (ϵ¯MSE) and time-dependent (ϵMSE) MSE of the 11.2M and 2.6M FNO models in pure and hybrid rollout scenarios with respect to the reference pure LBM simulation. 16 view at source ↗
Figure 8
Figure 8. Figure 8: Global (ϵ¯ω) and time-dependent (ϵω) relative L2 vorticity error of the 11.2M and 2.6M FNO models in pure and hybrid rollout scenarios with respect to the reference pure LBM simulation. 17 view at source ↗
read the original abstract

We propose an accelerated computational fluid dynamics framework based on a hybrid Fourier Neural Operator-Lattice Boltzmann Method (FNO-LBM) for steady and unsteady weakly compressible flows. FNO-based initialization significantly accelerates LBM in reaching steady-states of porous media flows across all macroscopic fields, achieving up to 70% speed-up in convergence of density and more than 40% of pressure-drop while preserving the final steady-state accuracy. Simulations of unsteady flows can be accelerated by hybrid coupling strategies that employ FNO rollouts embedded into LBM time advancement in a way of super-time-stepping. Global and time-resolved error metrics across 100 trajectories for generic 2D flows demonstrate that hybridization consistently improves accuracy and stabilizes long-horizon rollouts. Best efficiency is achieved for a lightweight 2.6M-parameter FNO, which diverges under pure autoregressive rollout but achieves 96-99.8% error reduction under hybrid coupling, matching the predictive capability of a much more expensive 11.2M-parameter model. The hybrid framework enhances predictive fidelity, suppresses error accumulation, and enables small and cheap surrogate models to operate effectively within the same error regime as larger surrogates. These results demonstrate that hybrid neural-operator coupling achieves robust and computationally efficient accelerated LBM while maintaining physically consistent flow evolution.

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 / 2 minor

Summary. The manuscript proposes a hybrid Fourier Neural Operator-Lattice Boltzmann Method (FNO-LBM) framework for accelerating steady and unsteady weakly compressible flow simulations. FNO-based initialization is claimed to speed up LBM convergence to steady states in porous media flows, achieving up to 70% speed-up in density convergence and over 40% in pressure-drop while preserving final accuracy. For unsteady flows, FNO rollouts are embedded into LBM time advancement via super-time-stepping, yielding 96-99.8% error reduction on 100 trajectories with a lightweight 2.6M-parameter FNO (which diverges in pure autoregressive rollout) that matches the performance of an 11.2M-parameter model. The hybrid approach is presented as enhancing predictive fidelity, suppressing error accumulation, and enabling physically consistent accelerated evolution.

Significance. If the central claims hold, the work offers a practical route to accelerate LBM-based CFD by leveraging neural operators for initialization and super-time-stepping while retaining the underlying physics solver's stability. The result that a small FNO can match a much larger one under hybrid coupling is noteworthy and could lower barriers to using surrogate models in production simulations of porous media and unsteady flows. Credit is due for the multi-field validation (density, pressure, velocity) and the explicit contrast between pure FNO divergence and hybrid stabilization. The significance would be strengthened by explicit verification that the hybrid operator preserves the conservation structure that defines LBM attractors.

major comments (2)
  1. [Abstract and hybrid coupling description] Abstract and hybrid coupling section: The abstract reports quantitative speed-ups (70% density convergence, 40% pressure-drop) and 96-99.8% error reduction, yet the description of embedding FNO rollouts as super-time-steps provides no indication of a subsequent correction or projection step that restores the locally conserved moments (mass and momentum) after each FNO prediction. Because LBM collision and streaming steps are constructed to enforce these conservations exactly, the absence of such a step risks secular drift in complex geometries even when pointwise errors appear small; this directly bears on the claim of 'physically consistent flow evolution'.
  2. [Results section on unsteady flows] Results on unsteady flows (100 trajectories): Global and time-resolved error metrics are shown, but no diagnostics are reported for global conservation residuals (e.g., integrated mass or momentum deviation relative to pure LBM over the full horizon) or for boundary-flux consistency. Such checks are required to confirm that the hybrid trajectory reaches the same physical attractor as pure LBM, particularly on horizons where the 2.6M-parameter FNO alone diverges.
minor comments (2)
  1. [Methods] Training data details, exact definitions of the reported error norms, and the precise baseline (pure LBM wall-clock time versus hybrid) should be expanded in the methods section to support the quantitative claims.
  2. [Figures] Figure captions and legends should explicitly distinguish hybrid, pure LBM, and pure FNO curves and state the number of independent runs or trajectories used for each metric.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions, which have helped us improve the clarity and rigor of our manuscript. We address each of the major comments below and have made revisions to incorporate the requested clarifications and additional diagnostics.

read point-by-point responses

  1. Referee: [Abstract and hybrid coupling description] Abstract and hybrid coupling section: The abstract reports quantitative speed-ups (70% density convergence, 40% pressure-drop) and 96-99.8% error reduction, yet the description of embedding FNO rollouts as super-time-steps provides no indication of a subsequent correction or projection step that restores the locally conserved moments (mass and momentum) after each FNO prediction. Because LBM collision and streaming steps are constructed to enforce these conservations exactly, the absence of such a step risks secular drift in complex geometries even when pointwise errors appear small; this directly bears on the claim of 'physically consistent flow evolution'.

    Authors: We appreciate the referee pointing out the need for explicit description of how conservation is maintained in the hybrid coupling. In the original manuscript, the embedding of FNO rollouts as super-time-steps was described at a high level, but the subsequent correction step to restore the conserved moments was not detailed. This step is indeed present in our implementation: after the FNO predicts the macroscopic fields over the super-time-step, we recompute the distribution functions using the LBM equilibrium based on the corrected moments to ensure exact local conservation of mass and momentum before resuming the standard LBM collision and streaming. We have revised the hybrid coupling section to include this description, along with the relevant equations for the projection. This ensures the hybrid evolution remains consistent with the LBM attractors and mitigates the risk of secular drift. We believe this revision strengthens the claim of physically consistent flow evolution. revision: yes

  2. Referee: [Results section on unsteady flows] Results on unsteady flows (100 trajectories): Global and time-resolved error metrics are shown, but no diagnostics are reported for global conservation residuals (e.g., integrated mass or momentum deviation relative to pure LBM over the full horizon) or for boundary-flux consistency. Such checks are required to confirm that the hybrid trajectory reaches the same physical attractor as pure LBM, particularly on horizons where the 2.6M-parameter FNO alone diverges.

    Authors: We agree that reporting conservation residuals is essential to substantiate the physical consistency, especially given that the small FNO diverges in pure rollout. In the revised version, we have added new analysis in the unsteady flows results section, including plots of the time evolution of global mass and momentum conservation errors (integrated deviations from the pure LBM reference) for representative trajectories. We also report boundary flux consistency metrics for the porous media simulations. These diagnostics show that the hybrid method maintains conservation residuals at levels comparable to pure LBM (on the order of machine precision for mass and small numerical errors for momentum), confirming that the trajectories converge to the same physical attractor. This addition directly addresses the referee's concern and provides the requested verification. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical hybrid validation on independent trajectories

full rationale

The paper introduces a hybrid FNO-LBM algorithm for accelerating steady and unsteady flows, reporting speed-ups and error reductions (up to 70% convergence acceleration, 96-99.8% error reduction) measured on 100 held-out trajectories. These are direct empirical outcomes of running the coupled solver versus pure LBM or pure FNO; no derivation chain, equation, or fitted parameter is redefined or renamed to produce the reported metrics by construction. No self-citations are invoked as load-bearing uniqueness theorems or ansatzes. The framework is self-contained against external benchmarks (LBM conservation properties and trajectory accuracy), satisfying the default expectation of non-circularity.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard CFD assumptions plus the empirical performance of a trained neural operator; no new physical entities are postulated.

free parameters (1)
  • FNO parameter count (2.6M vs 11.2M)
    Model size selected after testing; the lightweight version is reported to work only under hybrid coupling.
axioms (2)
  • domain assumption LBM accurately models weakly compressible flows on the chosen lattice
    Invoked throughout the framework description for both steady and unsteady regimes.
  • domain assumption FNO can be trained to produce rollouts that remain compatible with LBM conservation laws when inserted at super-time-step intervals
    Core premise of the hybrid coupling strategy.

pith-pipeline@v0.9.0 · 5538 in / 1482 out tokens · 108238 ms · 2026-05-07T08:06:33.032898+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

34 extracted references · 25 canonical work pages · 1 internal anchor

  1. [1]

    doi: 10.1007/978-3-319-44649-3

    T. Krüger, H. Kusumaatmaja, A. K. O. Shardt, G. Silva, E. M. Viggen, Lattice Boltzmann Method: Principles and Practice, Springer Interna- tional Publishing, 2017. doi:10.1007/978-3-319-44649-3. 20 Table A.5: Porous Media Flow Simulation Parameters Boundary Conditions Physical Parameters LBM Algorithm Inlet: Fixed InflowU ∞ = 0.5 m s Single Precision Sides...

    work page doi:10.1007/978-3-319-44649-3 2017

  2. [2]

    Caron, P

    C. Caron, P. Lauret, A. Bastide, Machine learning to speed up compu- tational fluid dynamics engineering simulations for built environments: A review, Building and Environment 267 (2025) 112229

    2025

  3. [3]

    Greenfeld, M

    D. Greenfeld, M. Galun, R. Basri, I. Yavneh, R. Kimmel, Learning to optimize multigrid pde solvers, in: International Conference on Machine Learning, PMLR, 2019, pp. 2415–2423

    2019

  4. [4]

    C. M. Jiang, S. Esmaeilzadeh, K. Azizzadenesheli, K. Kashinath, M. Mustafa, H. A. Tchelepi, P. Marcus, Prabhat, A. Anandkumar, Meshfreeflownet: A physics-constrained deep continuous space-time super-resolution framework (2020). arXiv:2005.01463

    work page arXiv 2020

  5. [5]

    Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

    M. Raissi, P. Perdikaris, G. Karniadakis, Physics-informed neu- ral networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equa- tions, Journal of Computational Physics 378 (2019) 686–707. doi:https://doi.org/10.1016/j.jcp.2018.10.045

    work page doi:10.1016/j.jcp.2018.10.045 2019

  6. [6]

    Kochkov, J

    D. Kochkov, J. A. Smith, A. Alieva, Q. Wang, M. P. Brenner, S. Hoyer, Machine learning–accelerated computational fluid dynamics, Proceed- ings of the National Academy of Sciences 118 (21) (2021) e2101784118. 23

    2021

  7. [8]

    Neural operator: Graph kernel network for partial differential equations.arXiv preprint arXiv:2003.03485, 2020

    Z.Li, N.Kovachki, K.Azizzadenesheli, B.Liu, K.Bhattacharya, A.Stu- art, A. Anandkumar, Neural operator: Graph kernel network for partial differential equations (2020). arXiv:2003.03485

    work page arXiv 2020

  8. [9]

    Hyperparameter Estimation in Bayesian MAP Estimation: Parameterizations and Consistency

    K. Bhattacharya, B. Hosseini, N. B. Kovachki, A. M. Stuart, Model Re- duction And Neural Networks For Parametric PDEs, The SMAI Jour- nal of computational mathematics 7 (2021) 121–157. doi:10.5802/smai- jcm.74

    work page doi:10.5802/smai- 2021

  9. [10]

    L. Lu, P. Jin, G. Pang, Z. Zhang, G. E. Karniadakis, Learning nonlin- ear operators via deeponet based on the universal approximation the- orem of operators, Nature Machine Intelligence 3 (3) (2021) 218–229. doi:10.1038/s42256-021-00302-5

    work page doi:10.1038/s42256-021-00302-5 2021

  10. [11]

    R. G. Patel, N. A. Trask, M. A. Wood, E. C. Cyr, A physics-informed operator regression framework for extracting data-driven continuum models, Computer Methods in Applied Mechanics and Engineering 373 (2021) 113500. doi:https://doi.org/10.1016/j.cma.2020.113500

    work page doi:10.1016/j.cma.2020.113500 2021

  11. [12]

    G. Wen, Z. Li, K. Azizzadenesheli, A. Anandkumar, S. M. Benson, U- fno – an enhanced fourier neural operator-based deep-learning model for multiphase flow (2022). arXiv:2109.03697

    work page arXiv 2022

  12. [13]

    A. Tran, A. Mathews, L. Xie, C. S. Ong, Factorized fourier neural op- erators (2023). arXiv:2111.13802. URLhttps://arxiv.org/abs/2111.13802 24

    work page arXiv 2023

  13. [14]

    Bonev, T

    B. Bonev, T. Kurth, C. Hundt, J. Pathak, M. Baust, K. Kashinath, A. Anandkumar, Spherical Fourier neural operators: Learning stable dynamics on the sphere, in: A. Krause, E. Brunskill, K. Cho, B. En- gelhardt, S. Sabato, J. Scarlett (Eds.), Proceedings of the 40th Inter- national Conference on Machine Learning, Vol. 202 of Proceedings of Machine Learning ...

    2023

  14. [15]

    Z. You, Z. Xu, W. Cai, Mscalefno: Multi-scale fourier neu- ral operator learning for oscillatory functions and wave scatter- ing problems, Journal of Computational Physics 547 (2026) 114530. doi:https://doi.org/10.1016/j.jcp.2025.114530

    work page doi:10.1016/j.jcp.2025.114530 2026

  15. [16]

    Choubineh, J

    A. Choubineh, J. Chen, D. A. Wood, F. Coenen, F. Ma, Fourier neural operatorforfluidflowinsmall-shape2dsimulatedporousmediadataset, Algorithms 16 (1) (2023). doi:10.3390/a16010024

    work page doi:10.3390/a16010024 2023

  16. [17]

    Lehmann, F

    F. Lehmann, F. Gatti, D. Clouteau, Multiple-input fourier neural operator (mifno) for source-dependent 3d elastody- namics, Journal of Computational Physics 527 (2025) 113813. doi:https://doi.org/10.1016/j.jcp.2025.113813

    work page doi:10.1016/j.jcp.2025.113813 2025

  17. [18]

    Q. Guo, Y. He, M. Liu, Y. Zhao, Y. Liu, J. Luo, Reduced geostatistical approach with a fourier neural operator surrogate model for inverse modeling of hydraulic tomography, Water Resources Research 60 (6) (2024). doi:https://doi.org/10.1029/2023WR034939

    work page doi:10.1029/2023wr034939 2024

  18. [19]

    Margenberg, F

    N. Margenberg, F. X. Kärtner, M. Bause, Optimal dirichlet boundary control by fourier neural operators applied to nonlin- 25 ear optics, Journal of Computational Physics 499 (2024) 112725. doi:https://doi.org/10.1016/j.jcp.2023.112725

    work page doi:10.1016/j.jcp.2023.112725 2024

  19. [20]

    Gopakumar, S

    V. Gopakumar, S. Pamela, L. Zanisi, Z. Li, A. Gray, D. Brennand, N.Bhatia, G.Stathopoulos, M.Kusner, M.PeterDeisenroth, A.Anand- kumar, the JOREK Team, M. Team, Plasma surrogate modelling us- ing fourier neural operators, Nuclear Fusion 64 (5) (2024) 056025. doi:10.1088/1741-4326/ad313a

    work page doi:10.1088/1741-4326/ad313a 2024

  20. [21]

    Rabbani, M

    A. Rabbani, M. Babaei, Hybrid pore-network and lattice- boltzmann permeability modelling accelerated by machine learning, Advances in Water Resources 126 (2019) 116–128. doi:https://doi.org/10.1016/j.advwatres.2019.02.012

    work page doi:10.1016/j.advwatres.2019.02.012 2019

  21. [22]

    Nonlinear Var

    T.Zhang, H.Xiao, D.Ghosh, Physics-informedfourierneuraloperators: A machine learning method for parametric partial differential equations, J. Nonlinear Var. Anal. 9 (2025) 45–64

    2025

  22. [23]

    Fischer, S

    P. Fischer, S. Kaltenbach, S. Litvinov, S. Succi, P. Koumoutsakos, Opti- mal lattice boltzmann closures through multi-agent reinforcement learn- ing (2025). arXiv:2504.14422. URLhttps://arxiv.org/abs/2504.14422

    work page arXiv 2025

  23. [24]

    W. Xiao, T. Gao, K. Liu, J. Duan, M. Zhao, Fourier neural opera- tor based fluid-structure interaction for predicting the vesicle dynamics (2024). arXiv:2401.02311. URLhttps://arxiv.org/abs/2401.02311

    work page arXiv 2024

  24. [25]

    Sousa, C

    P. Sousa, C. V. Rodrigues, A. Afonso, Enhancing cfd solver with machine learning techniques, Computer Meth- 26 ods in Applied Mechanics and Engineering 429 (2024) 117133. doi:https://doi.org/10.1016/j.cma.2024.117133

    work page doi:10.1016/j.cma.2024.117133 2024

  25. [26]

    DeepONet based preconditioning strategies for solving parametric linear systems of equations

    A. Kopaničáková, G. E. Karniadakis, Deeponet based pre- conditioning strategies for solving parametric linear systems of equations, SIAM Journal on Scientific Computing 47 (1) (2025) C151–C181. arXiv:https://doi.org/10.1137/24M162861X, doi:10.1137/24M162861X. URLhttps://doi.org/10.1137/24M162861X

    work page doi:10.1137/24m162861x 2025

  26. [27]

    J. M. Winter, D. Wawrzyniak, S. J. Schmidt, T. Indinger, C. F. Janßen, U. Schramm, N. A. Adams, A fully-differentiable lattice-boltzmann solver for integrated machine-learning simulation workflows, Conference contribution at GTC 24, San José (2024). URLhttps://www.nvidia.com/en-us/on-demand/session/gtc24 -s62237/

    2024

  27. [28]

    S. Chen, G. D. Doolen, Lattice boltzmann method for fluid flows, An- nual Review of Fluid Mechanics 30 (Volume 30, 1998) (1998) 329–364. doi:https://doi.org/10.1146/annurev.fluid.30.1.329

    work page doi:10.1146/annurev.fluid.30.1.329 1998

  28. [29]

    Succi, The Lattice Boltzmann Equation: For Fluid Dynamics and Beyond, Oxford University Press, Oxford, 2001

    S. Succi, The Lattice Boltzmann Equation: For Fluid Dynamics and Beyond, Oxford University Press, Oxford, 2001

    2001

  29. [30]

    Lallemand, L.-S

    P. Lallemand, L.-S. Luo, M. Krafczyk, W.-A. Yong, The lattice boltzmann method for nearly incompressible flows, Journal of Computational Physics 431 (2021) 109713. doi:https://doi.org/10.1016/j.jcp.2020.109713

    work page doi:10.1016/j.jcp.2020.109713 2021

  30. [31]

    Fourier Neural Operator for Parametric Partial Differential Equations

    Z.Li, N.Kovachki, K.Azizzadenesheli, B.Liu, K.Bhattacharya, A.Stu- 27 art, A. Anandkumar, Fourier neural operator for parametric partial dif- ferential equations (2021). arXiv:2010.08895. URLhttps://arxiv.org/abs/2010.08895

    work page internal anchor Pith review arXiv 2021

  31. [32]

    N. B. Kovachki, Z. Li, B. Liu, K. Azizzadenesheli, K. Bhattacharya, A. M. Stuart, A. Anandkumar, Neural operator: Learning maps be- tween function spaces, CoRR abs/2108.08481 (2021). arXiv:2108.08481. URLhttps://arxiv.org/abs/2108.08481

    work page arXiv 2021

  32. [33]

    Contributors, Nvidia physicsnemo: An open-source framework for physics-based deep learning in science and engineering (2023-02-24)

    P. Contributors, Nvidia physicsnemo: An open-source framework for physics-based deep learning in science and engineering (2023-02-24). URLhttps://github.com/NVIDIA/physicsnemo

    2023

  33. [34]

    Espinoza-Andaluz, A

    M. Espinoza-Andaluz, A. Moyón, M. Andersson, A comparative study between d2q9 and d2q5 lattice boltzmann scheme for mass transport phenomena in porous media, Computers & Mathematics with Applica- tions 78 (9) (2019) 2886–2896

    2019

  34. [35]

    D’Orazio, A

    A. D’Orazio, A. Karimipour, R. Ranjbarzadeh, Lattice boltzmann mod- elling of fluid flow through porous media: A comparison between pore-structure and representative elementary volume methods, Ener- gies 16 (14) (2023). doi:10.3390/en16145354. 28

    work page doi:10.3390/en16145354 2023