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arxiv: 2605.19057 · v1 · pith:XKJBADCNnew · submitted 2026-05-18 · 🌌 astro-ph.HE · astro-ph.IM· astro-ph.SR· physics.flu-dyn· physics.plasm-ph

Magnetohydrodynamics Simulations

E. A. Huerta This is my paper

Pith reviewed 2026-05-20 07:39 UTC · model grok-4.3

classification 🌌 astro-ph.HE astro-ph.IMastro-ph.SRphysics.flu-dynphysics.plasm-ph
keywords magnetohydrodynamicsphysics-informed neural networksturbulence simulationastrophysical plasmasfusion plasmasnumerical methodsoperator learningAI surrogates
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The pith

Physics-informed AI integrated with conventional solvers offers a route to MHD regimes beyond classical discretization limits.

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

Magnetohydrodynamics simulations couple fluid and electromagnetic equations to model stellar interiors, fusion plasmas, and space weather. Classical numerical methods such as high-order discretizations and adaptive mesh refinement hit a hard wall because the number of degrees of freedom needed grows as O(Re to the 9/4) or faster. The paper reviews physics-informed neural networks, Fourier neural operators, and hybrid generative frameworks that learn solution operators from large-scale training data. A sympathetic reader would care because these approaches aim to reach the extreme Reynolds and Lundquist numbers that govern real astrophysical and fusion systems where direct simulation remains impossible.

Core claim

Magnetohydrodynamics couples the Navier-Stokes and Maxwell equations into a nonlinear system of partial differential equations governing stellar interiors, astrophysical jets, fusion plasmas, and space weather. Numerical advances have made MHD predictive for many phenomena, yet in three-dimensional turbulence the degrees of freedom scale as O(Re^{9/4}) or faster, rendering direct numerical simulation intractable when the Lundquist number exceeds 10^{10}. The central claim is that physics-informed AI, integrated with conventional solvers and trained on leadership-scale simulations, offers a credible route to regimes beyond the reach of classical discretisation alone.

What carries the argument

Physics-informed neural networks, Fourier neural operators, and hybrid operator-diffusion frameworks that learn solution operators across families of MHD problems while recovering broadband turbulent spectra.

If this is right

  • Predictive modeling of solar eruptions, tokamak confinement, and magnetized turbulence becomes feasible at previously inaccessible parameters.
  • Hybrid AI-conventional pipelines can incorporate kinetic closures, radiation transport, and uncertainty quantification at reduced cost.
  • Exascale high-order solvers, GPU acceleration, and task-based parallelism gain new sub-grid closure and operator-learning capabilities.
  • Prospective quantum algorithms for implicit linear systems in resistive MHD can be paired with learned surrogates.

Where Pith is reading between the lines

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

  • Operator-learning methods may naturally extend to coupled multi-physics problems such as MHD with radiation or kinetic effects.
  • Trained models could supply fast surrogates for ensemble forecasting in space-weather applications.
  • Systematic tests at the highest currently resolvable Re and S values would quantify the onset of generalization failure.

Load-bearing premise

AI models trained mainly on lower-resolution or lower-Reynolds data will generalize accurately to the extreme Reynolds and Lundquist numbers of astrophysics and fusion without introducing unphysical artifacts or violating conservation laws.

What would settle it

Side-by-side comparison of AI-augmented solutions against the highest-resolution conventional runs possible at moderately elevated Reynolds and Lundquist numbers, checking whether conservation laws hold and spectra remain free of spurious features.

Figures

Figures reproduced from arXiv: 2605.19057 by E. A. Huerta.

Figure 1.1
Figure 1.1. Figure 1.1: Schematic of the DINOs (Diffusion-Integrated Neural Operators) frame [PITH_FULL_IMAGE:figures/full_fig_p022_1_1.png] view at source ↗
read the original abstract

Magnetohydrodynamics (MHD) couples the Navier--Stokes and Maxwell equations into a nonlinear system of partial differential equations governing stellar interiors, astrophysical jets, fusion plasmas, and space weather. Numerical advances, including finite-volume Godunov schemes, constrained-transport algorithms, high-order spectral-element and discontinuous-Galerkin discretisations, and adaptive mesh refinement, have made MHD a predictive tool for solar eruptions, tokamak confinement, and magnetised turbulence. A fundamental barrier nevertheless remains. In three-dimensional MHD turbulence, the degrees of freedom required to resolve all active scales grow as $\mathcal{O}(\mathrm{Re}^{9/4})$ or faster, where $\mathrm{Re}$ is the Reynolds number. Direct numerical simulation is therefore intractable at astrophysical and fusion-relevant parameters, particularly when the Lundquist number $S$ exceeds $10^{10}$ and both viscous and resistive dissipation ranges must be resolved. Kinetic closures, radiation transport, and uncertainty quantification further increase the cost. This chapter examines how AI may help bridge this gap. We review physics-informed neural networks, Fourier neural operators and physics-informed neural operators, which learn solution operators across families of MHD problems; and hybrid operator-diffusion frameworks that combine deterministic surrogates with score-based generative models to recover broadband turbulent spectra. These developments are set within the wider landscape of exascale high-order solvers, GPU acceleration, task-based parallelism, data-driven sub-grid closures, and prospective quantum algorithms for implicit linear systems in resistive MHD. The central claim is that physics-informed AI, integrated with conventional solvers and trained on leadership-scale simulations, offers a credible route to regimes beyond the reach of classical discretisation alone.

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 / 1 minor

Summary. The manuscript reviews the fundamental computational barrier in three-dimensional MHD turbulence simulations, where the number of degrees of freedom scales as O(Re^{9/4}), rendering direct numerical simulation intractable at astrophysical and fusion-relevant Reynolds numbers (Re ≫ 10^6) and Lundquist numbers (S > 10^{10}). It surveys AI approaches including physics-informed neural networks (PINNs), Fourier neural operators (FNOs), physics-informed neural operators, and hybrid operator-diffusion frameworks that combine deterministic surrogates with score-based generative models, positioning these as integrable with conventional high-order solvers, exascale computing, and data-driven sub-grid closures to access otherwise unreachable regimes.

Significance. If the surveyed AI methods can be shown to generalize accurately from leadership-scale training data to extreme Re and S while preserving conservation laws and avoiding unphysical artifacts, the review would usefully frame a pathway toward predictive modeling of stellar interiors, astrophysical jets, and fusion plasmas. The manuscript provides a timely catalog of existing techniques set against the landscape of GPU-accelerated solvers and prospective quantum algorithms, though the absence of quantitative support for the extrapolation claim limits its immediate utility as a roadmap.

major comments (1)
  1. [Abstract] Abstract: The central claim that 'physics-informed AI, integrated with conventional solvers and trained on leadership-scale simulations, offers a credible route to regimes beyond the reach of classical discretisation alone' is not supported by any quantitative validation, error metrics, conservation diagnostics, or extrapolation tests at the target parameters (Re ≫ 10^6, S > 10^{10}). The text catalogs published methods (PINNs, FNOs, hybrid frameworks) but supplies no evidence or cited bounds demonstrating that operators trained on finite-Re data avoid spurious dissipation, inverse cascades, or violations of conservation when applied to unresolved scales.
minor comments (1)
  1. [Abstract] The abstract refers to 'this chapter'; adding a brief statement of the manuscript's standalone scope or its relation to a larger volume would improve clarity for readers encountering it as a journal article.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive review of our survey on AI surrogates for high-Re and high-S MHD regimes. We agree that the central claim would benefit from explicit quantitative grounding drawn from the cited literature and have revised the manuscript to address this while preserving its character as a review.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that 'physics-informed AI, integrated with conventional solvers and trained on leadership-scale simulations, offers a credible route to regimes beyond the reach of classical discretisation alone' is not supported by any quantitative validation, error metrics, conservation diagnostics, or extrapolation tests at the target parameters (Re ≫ 10^6, S > 10^{10}). The text catalogs published methods (PINNs, FNOs, hybrid frameworks) but supplies no evidence or cited bounds demonstrating that operators trained on finite-Re data avoid spurious dissipation, inverse cascades, or violations of conservation when applied to unresolved scales.

    Authors: We acknowledge that the original presentation of the central claim in the abstract and introduction did not include sufficient explicit references to quantitative results. As this is a review paper whose purpose is to catalog and contextualize existing techniques rather than report new simulations, the claim was framed as a perspective based on the body of work surveyed. To directly address the referee's valid concern, we have revised the abstract to qualify the claim, added a new subsection summarizing reported performance metrics (including L2 errors, conservation violations, and generalization tests to higher Re/S) from representative PINN, FNO, and hybrid operator-diffusion papers, and inserted a table compiling these diagnostics along with cited bounds on artifacts such as spurious dissipation. These additions draw on the existing literature without introducing original numerical results. revision: yes

Circularity Check

0 steps flagged

Literature survey with no internal derivation chain

full rationale

This manuscript functions as a literature review that catalogs existing AI methods for MHD (PINNs, FNOs, hybrid operator-diffusion models) without advancing any new derivation, operator, or prediction of its own. The central claim is an evaluative statement about the field rather than a mathematical result derived from equations or data within the text. The O(Re^{9/4}) scaling is used only to motivate the computational barrier and is not part of any load-bearing derivation that reduces to fitted parameters or self-citations. No self-definitional, fitted-input-as-prediction, or uniqueness-via-self-citation patterns appear.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a review and introduces no new free parameters, axioms, or invented entities beyond the standard MHD equations and established machine-learning concepts already present in the prior literature.

axioms (1)
  • standard math MHD is governed by the coupled Navier-Stokes and Maxwell equations
    Invoked in the opening paragraph as the foundational system.

pith-pipeline@v0.9.0 · 5833 in / 1152 out tokens · 32576 ms · 2026-05-20T07:39:00.788341+00:00 · methodology

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Works this paper leans on

75 extracted references · 75 canonical work pages

  1. [1]

    Existence of electromagnetic-hydrodynamic waves.Nature, 150:405–406, 1942

    Hannes Alfv ´en. Existence of electromagnetic-hydrodynamic waves.Nature, 150:405–406, 1942

    work page 1942

  2. [2]

    F. J. Artola, A. Loarte, M. Hoelzl, M. Lehnen, N. Schwarz, and Jorek Team. Non-axisymmetric MHD simulations of the current quench phase of ITER mitigated disruptions.Nuclear Fusion, 62(5):056023, May 2022

    work page 2022

  3. [3]

    Balsara and Chi-Wang Shu

    Dinshaw S. Balsara and Chi-Wang Shu. Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. Journal of Computational Physics, 160(2):405–452, 2000

    work page 2000

  4. [4]

    Deep neural networks for data-driven les closure models.Journal of Computational Physics, 398:108910, 2019

    Andrea Beck, David Flad, and Claus-Dieter Munz. Deep neural networks for data-driven les closure models.Journal of Computational Physics, 398:108910, 2019. 1 Magnetohydrodynamics Simulations 27

    work page 2019

  5. [5]

    Beresnyak and A

    A. Beresnyak and A. Lazarian. Strong imbalanced turbulence.The Astrophys- ical Journal, 682(2):1070, aug 2008

    work page 2008

  6. [6]

    MHD turbulence.Living Reviews in Computational Astro- physics, 5(1):2, September 2019

    Andrey Beresnyak. MHD turbulence.Living Reviews in Computational Astro- physics, 5(1):2, September 2019

    work page 2019

  7. [7]

    Spectrum of magnetohydrodynamic turbulence.Phys

    Stanislav Boldyrev. Spectrum of magnetohydrodynamic turbulence.Phys. Rev. Lett., 96:115002, Mar 2006

    work page 2006

  8. [8]

    S. I. Braginskii. Transport Processes in a Plasma.Reviews of Plasma Physics, 1:205, January 1965

    work page 1965

  9. [9]

    Clifford neural layers for pde modeling

    Johannes Brandstetter, Rianne van den Berg, Max Welling, and Jayesh K Gupta. Clifford neural layers for pde modeling. InThe Eleventh International Conference on Learning Representations, 2023

    work page 2023

  10. [10]

    Cerezo, Yigit Subasi, Lukasz Cincio, and Patrick J

    Carlos Bravo-Prieto, Ryan LaRose, M. Cerezo, Yigit Subasi, Lukasz Cincio, and Patrick J. Coles. Variational Quantum Linear Solver.Quantum, 7:1188, November 2023

    work page 2023

  11. [11]

    An upwind differencing scheme for the equations of ideal magnetohydrodynamics.Journal of Computational Physics, 75(2):400–422, 1988

    M Brio and C.C Wu. An upwind differencing scheme for the equations of ideal magnetohydrodynamics.Journal of Computational Physics, 75(2):400–422, 1988

    work page 1988

  12. [12]

    Burns, Geoffrey M

    Keaton J. Burns, Geoffrey M. Vasil, Jeffrey S. Oishi, Daniel Lecoanet, and Benjamin P. Brown. Dedalus: A flexible framework for numerical simulations with spectral methods.Phys. Rev. Res., 2:023068, Apr 2020

    work page 2020

  13. [13]

    Oxford University Press, Oxford, 1961

    Subrahmanyan Chandrasekhar.Hydrodynamic and Hydromagnetic Stability. Oxford University Press, Oxford, 1961

    work page 1961

  14. [14]

    Tianping Chen and Hong Chen. Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems.IEEE Transactions on Neural Networks, 6(4):911–917, 1995

    work page 1995

  15. [15]

    Gr-athena++: General-relativistic magnetohydrodynamics simulations of neutron star space- times.The Astrophysical Journal Supplement Series, 277(1):3, feb 2025

    William Cook, Boris Daszuta, Jacob Fields, Peter Hammond, Simone Albanesi, Francesco Zappa, Sebastiano Bernuzzi, and David Radice. Gr-athena++: General-relativistic magnetohydrodynamics simulations of neutron star space- times.The Astrophysical Journal Supplement Series, 277(1):3, feb 2025

    work page 2025

  16. [16]

    T. G. Cowling. The magnetic field of sunspots.Monthly Notices of the Royal Astronomical Society, 94(1):39–48, 11 1933

    work page 1933

  17. [17]

    Cowling.Magnetohydrodynamics

    Thomas G. Cowling.Magnetohydrodynamics. Interscience Publishers, New York, 1957

    work page 1957

  18. [18]

    Dedner, F

    A. Dedner, F. Kemm, D. Kr ¨oner, C.-D. Munz, T. Schnitzer, and M. Wesen- berg. Hyperbolic Divergence Cleaning for the MHD Equations.Journal of Computational Physics, 175(2):645–673, January 2002

    work page 2002

  19. [19]

    Magnetic 28 Eliu Huerta control of tokamak plasmas through deep reinforcement learning.Nature, 602(7897):414–419, February 2022

    Jonas Degrave, Federico Felici, Jonas Buchli, Michael Neunert, Brendan Tracey, Francesco Carpanese, Timo Ewalds, Roland Hafner, Abbas Abdol- maleki, Diego De Las Casas, Craig Donner, Leslie Fritz, Cristian Galperti, An- drea Huber, James Keeling, Maria Tsimpoukelli, Jackie Kay, Antoine Merle, Jean-Marc Moret, Seb Noury, Federico Pesamosca, David Pfau, Oli...

    work page 2022

  20. [20]

    Turbulence model- ing in the age of data.Annual Review of Fluid Mechanics, 51(Volume 51, 2019):357–377, 2019

    Karthik Duraisamy, Gianluca Iaccarino, and Heng Xiao. Turbulence model- ing in the age of data.Annual Review of Fluid Mechanics, 51(Volume 51, 2019):357–377, 2019

    work page 2019

  21. [21]

    Etienne, Vasileios Paschalidis, Roland Haas, Philipp M ¨osta, and Stuart L

    Zachariah B. Etienne, Vasileios Paschalidis, Roland Haas, Philipp M ¨osta, and Stuart L. Shapiro. IllinoisGRMHD: An Open-Source, User-Friendly GRMHD Code for Dynamical Spacetimes.Class. Quant. Grav., 32:175009, 2015

    work page 2015

  22. [22]

    Evans and John F

    Charles R. Evans and John F. Hawley. Simulation of Magnetohydrodynamic Flows: A Constrained Transport Model.The Astrophysical Journal, 332:659, September 1988

    work page 1988

  23. [23]

    Freidberg.Ideal Magnetohydrodynamics

    Jeffrey P. Freidberg.Ideal Magnetohydrodynamics. Plenum Press, New York, 1987

    work page 1987

  24. [24]

    Fryxell, K

    B. Fryxell, K. Olson, P. Ricker, F. X. Timmes, M. Zingale, D. Q. Lamb, P. MacNeice, R. Rosner, J. W. Truran, and H. Tufo. FLASH: An Adaptive Mesh Hydrodynamics Code for Modeling Astrophysical Thermonuclear Flashes.The Astrophysical Journal Supplement Series, 131(1):273–334, November 2000

    work page 2000

  25. [25]

    Gammie, Julian C

    Charles F. Gammie, Julian C. McKinney, and G ´abor T ´oth. Harm: A scheme for general relativistic magnetohydrodynamics.The Astrophysical Journal, 589:444–457, 2003

    work page 2003

  26. [26]

    Transformers for modeling physical systems.Neural Networks, 146:272–289, 2022

    Nicholas Geneva and Nicholas Zabaras. Transformers for modeling physical systems.Neural Networks, 146:272–289, 2022

    work page 2022

  27. [27]

    WhiskyMHD: A New numerical code for general relativistic magnetohydrodynamics.Class

    Bruno Giacomazzo and Luciano Rezzolla. WhiskyMHD: A New numerical code for general relativistic magnetohydrodynamics.Class. Quant. Grav., 24:S235–S258, 2007

    work page 2007

  28. [28]

    J. P. Goedbloed and S. Poedts.Principles of Magnetohydrodynamics: With Applications to Laboratory and Astrophysical Plasmas. Cambridge University Press, Cambridge, 2004

    work page 2004

  29. [29]

    Goldreich and S

    P. Goldreich and S. Sridhar. Toward a Theory of Interstellar Turbulence. II. Strong Alfvenic Turbulence.The Astrophysical Journal, 438:763, January 1995

    work page 1995

  30. [30]

    Vlaykov, Wolfram Schmidt, Dominik R

    Philipp Grete, Dimitar G. Vlaykov, Wolfram Schmidt, Dominik R. G. Schle- icher, and Christoph Federrath. Nonlinear closures for scale separation in supersonic magnetohydrodynamic turbulence.New Journal of Physics, 17(2):023070, February 2015

    work page 2015

  31. [31]

    Hamiltonian neural networks

    Samuel Greydanus, Misko Dzamba, and Jason Yosinski. Hamiltonian neural networks. InAdvances in Neural Information Processing Systems, volume 32, pages 15379–15389, 2019

    work page 2019

  32. [32]

    Dynamical diffusion: Learning temporal dynamics with diffusion models

    Xingzhuo Guo, Yu Zhang, Baixu Chen, Haoran Xu, Jianmin Wang, and Ming- sheng Long. Dynamical diffusion: Learning temporal dynamics with diffusion models. InThe Thirteenth International Conference on Learning Representa- tions, 2025

    work page 2025

  33. [33]

    GNOT: A general neural 1 Magnetohydrodynamics Simulations 29 operator transformer for operator learning

    Zhongkai Hao, Zhengyi Wang, Hang Su, Chengyang Ying, Yinpeng Dong, Songming Liu, Ze Cheng, Jian Song, and Jun Zhu. GNOT: A general neural 1 Magnetohydrodynamics Simulations 29 operator transformer for operator learning. InInternational conference on machine learning, pages 12556–12569. PMLR, 2023

    work page 2023

  34. [34]

    Harrow, Avinatan Hassidim, and Seth Lloyd

    Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum algorithm for linear systems of equations.Phys. Rev. Lett., 103:150502, Oct 2009

    work page 2009

  35. [35]

    Denoising diffusion probabilistic models

    Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. InProceedings of the 34th International Conference on Neural In- formation Processing Systems, NIPS ’20, Red Hook, NY, USA, 2020. Curran Associates Inc

    work page 2020

  36. [36]

    Hoelzl, G

    M. Hoelzl, G. T. A. Huijsmans, S. J. P. Pamela, M. B ´ecoulet, E. Nardon, F. J. Artola, B. Nkonga, C. V. Atanasiu, V. Bandaru, A. Bhole, D. Bonfiglio, A. Cathey, O. Czarny, A. Dvornova, T. Feh ´er, A. Fil, E. Franck, S. Futatani, M. Gruca, H. Guillard, J. W. Haverkort, I. Holod, D. Hu, S. K. Kim, S. Q. Korv- ing, L. Kos, I. Krebs, L. Kripner, G. Latu, F. ...

    work page 2021

  37. [37]

    Physics informed neural networks for magnetohydrodynamic equations

    Eva Jaillon and Pavlos Protopapas. Physics informed neural networks for magnetohydrodynamic equations. InAI&PDE: ICLR 2026 Workshop on AI and Partial Differential Equations, 2026

    work page 2026

  38. [38]

    Resolving turbulent magne- tohydrodynamics: a hybrid operator-diffusion framework.Machine Learning: Science and Technology, 6(3):035057, sep 2025

    Semih Kacmaz, E A Huerta, and Roland Haas. Resolving turbulent magne- tohydrodynamics: a hybrid operator-diffusion framework.Machine Learning: Science and Technology, 6(3):035057, sep 2025

    work page 2025

  39. [39]

    Physics-informed machine learning.Nature Reviews Physics, pages 1–19, 05 2021

    George Karniadakis, Yannis Kevrekidis, Lu Lu, Paris Perdikaris, Sifan Wang, and Liu Yang. Physics-informed machine learning.Nature Reviews Physics, pages 1–19, 05 2021

    work page 2021

  40. [40]

    Kolda and Brett W

    Tamara G. Kolda and Brett W. Bader. Tensor decompositions and applications. SIAM Review, 51(3):455–500, 2009

    work page 2009

  41. [41]

    Ten- sorly: Tensor learning in python.Journal of Machine Learning Research, 20(26):1–6, 2019

    Jean Kossaifi, Yannis Panagakis, Anima Anandkumar, and Maja Pantic. Ten- sorly: Tensor learning in python.Journal of Machine Learning Research, 20(26):1–6, 2019

    work page 2019

  42. [42]

    Krebs, S

    I. Krebs, S. C. Jardin, S. G¨ unter, K. Lackner, M. Hoelzl, E. Strumberger, and N. Ferraro. Magnetic flux pumping in 3D nonlinear magnetohydrodynamic simulations.Physics of Plasmas, 24(10):102511, October 2017

    work page 2017

  43. [43]

    Kulsrud.Plasma Physics for Astrophysics

    Russell M. Kulsrud.Plasma Physics for Astrophysics. Princeton University Press, Princeton, NJ, 2005

    work page 2005

  44. [44]

    Scalable transformer for PDE surrogate modeling.Advances in Neural Information Processing Systems, 36:28010–28039, 2023

    Zijie Li, Dule Shu, and Amir Barati Farimani. Scalable transformer for PDE surrogate modeling.Advances in Neural Information Processing Systems, 36:28010–28039, 2023

    work page 2023

  45. [45]

    Fourier 30 Eliu Huerta neural operator for parametric partial differential equations

    Zongyi Li, Nikola Borislavov Kovachki, Kamyar Azizzadenesheli, Burigede liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Fourier 30 Eliu Huerta neural operator for parametric partial differential equations. InInternational Conference on Learning Representations, 2021

    work page 2021

  46. [46]

    Physics- informed neural operator for learning partial differential equations.ACM / IMS J

    Zongyi Li, Hongkai Zheng, Nikola Kovachki, David Jin, Haoxuan Chen, Burigede Liu, Kamyar Azizzadenesheli, and Anima Anandkumar. Physics- informed neural operator for learning partial differential equations.ACM / IMS J. Data Sci., 1(3), May 2024

    work page 2024

  47. [47]

    Bharath, and Chris D

    Mario Lino, Stathi Fotiadis, Anil A. Bharath, and Chris D. Cantwell. Multi- scale rotation-equivariant graph neural networks for unsteady eulerian fluid dynamics.Physics of Fluids, 34(8):087110, 08 2022

    work page 2022

  48. [48]

    Lithwick, P

    Y. Lithwick, P. Goldreich, and S. Sridhar. Imbalanced strong mhd turbulence. The Astrophysical Journal, 655(1):269, jan 2007

    work page 2007

  49. [49]

    Lopez Armengol, Zachariah B

    Federico G. Lopez Armengol, Zachariah B. Etienne, Scott C. Noble, Bernard J. Kelly, Leonardo R. Werneck, Brendan Drachler, Manuela Campanelli, Federico Cipolletta, Yosef Zlochower, Ariadna Murguia-Berthier, Lorenzo Ennoggi, Mark Avara, Riccardo Ciolfi, Joshua Faber, Grace Fiacco, Bruno Giaco- mazzo, Tanmayee Gupte, Trung Ha, Julian H. Krolik, Vassilios Me...

    work page 2022

  50. [50]

    Lu Lu, Pengzhan Jin, and George Em Karniadakis. DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators.Nature Machine Intelligence, 3(3):218– 229, 2021

    work page 2021

  51. [51]

    A comprehensive and fair comparison of two neural operators (with practical extensions) based on fair data.Computer Methods in Applied Mechanics and Engineering, 393:114778, 2022

    Lu Lu, Xuhui Meng, Shengze Cai, Zhiping Mao, Somdatta Goswami, Zhongqiang Zhang, and George Em Karniadakis. A comprehensive and fair comparison of two neural operators (with practical extensions) based on fair data.Computer Methods in Applied Mechanics and Engineering, 393:114778, 2022

    work page 2022

  52. [52]

    Dynamic alignment in driven magnetohydrodynamic turbulence.Phys

    Joanne Mason, Fausto Cattaneo, and Stanislav Boldyrev. Dynamic alignment in driven magnetohydrodynamic turbulence.Phys. Rev. Lett., 97:255002, Dec 2006

    work page 2006

  53. [53]

    Mignone, G

    A. Mignone, G. Bodo, S. Massaglia, T. Matsakos, O. Tesileanu, C. Zanni, and A. Ferrari. Pluto: A numerical code for computational astrophysics.The Astrophysical Journal Supplement Series, 170:228–242, 2007

    work page 2007

  54. [54]

    Mistrangelo, L

    C. Mistrangelo, L. B¨ uhler, S. Smolentsev, V. Kl¨ uber, I. Maione, and J. Aubert. Mhd flow in liquid metal blankets: Major design issues, mhd guidelines and numerical analysis.Fusion Engineering and Design, 173:112795, 2021

    work page 2021

  55. [55]

    A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics.Journal of Computational Physics, 208(1):315–344, September 2005

    Takahiro Miyoshi and Kanya Kusano. A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics.Journal of Computational Physics, 208(1):315–344, September 2005

    work page 2005

  56. [56]

    Optimization of the dc magnetohydrodynamic pump for the dual fluid reactor.Annals of Nuclear Energy, 174:109142, 2022

    Mateusz Nowak, Micha l Spirzewski, and Konrad Czerski. Optimization of the dc magnetohydrodynamic pump for the dual fluid reactor.Annals of Nuclear Energy, 174:109142, 2022

    work page 2022

  57. [57]

    Vivek Oommen, Aniruddha Bora, Zhen Zhang, and George Em Karniadakis. Integrating neural operators with diffusion models improves spectral represen- 1 Magnetohydrodynamics Simulations 31 tation in turbulence modelling.Proceedings of the Royal Society A: Mathe- matical, Physical and Engineering Sciences, 481(2309):20240819, 03 2025

    work page 2025

  58. [58]

    E. N. Parker.Cosmical Magnetic Fields: Their Origin and Their Activity. Oxford University Press, Oxford, 1979

    work page 1979

  59. [59]

    Learning mesh-based simulation with graph networks

    Tobias Pfaff, Meire Fortunato, Alvaro Sanchez-Gonzalez, and Peter Battaglia. Learning mesh-based simulation with graph networks. InInternational Con- ference on Learning Representations, 2021

    work page 2021

  60. [60]

    The black hole accretion code.Comput

    Oliver Porth, Hector Olivares, Yosuke Mizuno, Ziri Younsi, Luciano Rezzolla, Monika Moscibrodzka, Heino Falcke, and Michael Kramer. The black hole accretion code.Comput. Astrophys. Cosmol., 4(1):1, 2017

    work page 2017

  61. [61]

    Raissi, P

    M. Raissi, P. Perdikaris, and G.E. Karniadakis. Physics-informed neural net- works: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations.Journal of Computational Physics, 378:686–707, 2019

    work page 2019

  62. [62]

    U-net: Convolutional networks for biomedical image segmentation

    Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomedical image segmentation. InInternational Conference on Medical image computing and computer-assisted intervention, pages 234–241. Springer, 2015

    work page 2015

  63. [63]

    Applications of physics informed neural operators.Machine Learning: Science and Technology, 4(2):025022, may 2023

    Shawn G Rosofsky, Hani Al Majed, and E A Huerta. Applications of physics informed neural operators.Machine Learning: Science and Technology, 4(2):025022, may 2023

    work page 2023

  64. [64]

    Magnetohydrodynamics with physics informed neural operators.Machine Learning: Science and Technology, 4(3):035002, jul 2023

    Shawn G Rosofsky and E A Huerta. Magnetohydrodynamics with physics informed neural operators.Machine Learning: Science and Technology, 4(3):035002, jul 2023

    work page 2023

  65. [65]

    Sadowski, R

    A. Sadowski, R. Narayan, J. C. McKinney, and A. Tchekhovskoy. Numerical simulations of super-critical black hole accretion flows in general relativity. Mon. Not. Roy. Astron. Soc., 439(1):503–520, 2014

    work page 2014

  66. [66]

    Score-based generative modeling through stochastic differential equations

    Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. InInternational Conference on Learning Representations, 2021

    work page 2021

  67. [67]

    Stone, Thomas A

    James M. Stone, Thomas A. Gardiner, Peter Teuben, John F. Hawley, and Ja- cob B. Simon. Athena: A New Code for Astrophysical MHD.The Astrophysical Journal Supplement Series, 178(1):137–177, September 2008

    work page 2008

  68. [68]

    Stone and Michael L

    James M. Stone and Michael L. Norman. ZEUS-2D: A Radiation Magnetohy- drodynamics Code for Astrophysical Flows in Two Space Dimensions. I. The Hydrodynamic Algorithms and Tests.The Astrophysical Journal Supplement Series, 80:753, June 1992

    work page 1992

  69. [69]

    Data-driven subgrid-scale modeling of forced Burgers turbulence using deep learning with generalization to higher Reynolds numbers via transfer learning

    Adam Subel, Ashesh Chattopadhyay, Yifei Guan, and Pedram Hassanzadeh. Data-driven subgrid-scale modeling of forced Burgers turbulence using deep learning with generalization to higher Reynolds numbers via transfer learning. Physics of Fluids, 33(3):031702, March 2021

    work page 2021

  70. [70]

    Z. Sun, J. Al Salami, A. Khodak, F. Saenz, B. Wynne, R. Maingi, K. Hanada, C.H. Hu, and E. Kolemen. Magnetohydrodynamics in free surface liquid metal 32 Eliu Huerta flow relevant to plasma-facing components.Nuclear Fusion, 63(7):076022, jun 2023

    work page 2023

  71. [71]

    The∇ ·𝐵=0 constraint in shock-capturing magnetohydrodynam- ics codes.Journal of Computational Physics, 161(2):605–652, 2000

    G ´abor T´oth. The∇ ·𝐵=0 constraint in shock-capturing magnetohydrodynam- ics codes.Journal of Computational Physics, 161(2):605–652, 2000

    work page 2000

  72. [72]

    A connection between score matching and denoising autoen- coders.Neural Computation, 23(7):1661–1674, July 2011

    Pascal Vincent. A connection between score matching and denoising autoen- coders.Neural Computation, 23(7):1661–1674, July 2011

    work page 2011

  73. [73]

    Understanding and mitigating gradient flow pathologies in physics-informed neural networks.SIAM Journal on Scientific Computing, 43(5):A3055–A3081, 2021

    Sifan Wang, Yujun Teng, and Paris Perdikaris. Understanding and mitigating gradient flow pathologies in physics-informed neural networks.SIAM Journal on Scientific Computing, 43(5):A3055–A3081, 2021

    work page 2021

  74. [74]

    Learning the solution op- erator of parametric partial differential equations with physics-informed deep- onets.Science Advances, 7(40):eabi8605, 2021

    Sifan Wang, Hanwen Wang, and Paris Perdikaris. Learning the solution op- erator of parametric partial differential equations with physics-informed deep- onets.Science Advances, 7(40):eabi8605, 2021

    work page 2021

  75. [75]

    Les simulation on heavy liquid metal flow in a bare rod bundle for assessment of turbulent prandtl number

    Yiqi Yu, Emily Shemon, and Elia Merzari. Les simulation on heavy liquid metal flow in a bare rod bundle for assessment of turbulent prandtl number. Nuclear Engineering and Design, 404:112175, 2023

    work page 2023