Pith. sign in

REVIEW 4 major objections 5 minor 2 cited by

NeuralMag: an open-source nodal finite-difference code for inverse micromagnetics

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read NeuralMag's central claim is that a nodal finite-difference scheme can handle material interfaces with finite-element accuracy while keeping finite-difference speed.

desk verdict A genuinely useful open-source micromagnetic code with a real but incremental numerical idea, whose headline accuracy claim over conventional finite difference is asserted rather than demonstrated. read the letter →

arxiv 2411.11725 v1 pith:XKKY4WAI submitted 2024-11-18 physics.comp-ph

Claas Abert , Florian Bruckner , Andrey Voronov , Martin Lang , Swapneel Amit Pathak , Samuel Holt , Robert Kraft , Ruslan Allayarov
show 6 more authors
Peter Flauger Sabri Koraltan Thomas Schrefl Andrii Chumak Hans Fangohr Dieter Suess
This is my paper · ORCID
classification physics.comp-ph
keywords micromagneticsnodalfinite-differencefinite-elementdiscretizationinverseproblemsautomaticdifferentiationadjoint-statemethodLandau-Lifshitz-Gilbertequationopen-sourcesoftware
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

NeuralMag is an open-source micromagnetic simulation library built on high-performance tensor backends with automatic differentiation. The paper's central claim is a nodal finite-difference discretization that stores the magnetization at grid vertices instead of cell centers, so interface conditions like the exchange jump condition are handled variationally as in finite elements, while the demagnetization field is still computed with the fast FFT convolution of standard finite differences. The authors argue this improves accuracy at material interfaces without increasing computational complexity, and that time-dependent inverse problems—finding fields or material parameters that produce a desired magnetic trajectory—become practical through automatic differentiation and an adjoint-state solver. Benchmarks in the paper show the code is less than a factor of two slower than the fastest established GPU code when using its just-in-time-compiling backend. If these claims hold, NeuralMag gives researchers a flexible Python tool that combines finite-element-style interface treatment with finite-difference speed for both forward simulation and inverse design.

What carries the argument

The load-bearing object is the nodal finite-difference basis: a nodal basis $\phi_n$ of piecewise-linear hat functions on a Cartesian mesh, with support spanning the eight cells around each vertex, paired with piecewise-constant cell basis functions $\vartheta_c = \mathbb{1}_{\Omega_c}$ for material parameters. This choice makes the discretized weak form (17) local and matrix-free: each cell contributes a fixed $24 \times 24$ element matrix, and assembly is a shift-and-add over the local vertex index $i$, so every local field term costs $O(N)$. Mass lumping diagonalizes the mass matrix (each row summed onto the diagonal), avoiding a linear solve for the effective field, and the FFT demagnetization routine is kept by averaging between nodal and cell-centered values.

What would settle it

Compute switching fields for the two-phase domain-wall pinning problem on a sequence of refined grids and compare the error slopes against a high-resolution finite-element reference; if the nodal scheme's error does not decline faster than the cell-centered scheme's, the claimed interface-accuracy advantage is not realized.

Watch

Extended reading notes

Core claim

The central contribution is a discretization scheme that merges finite-element variational handling with finite-difference efficiency on a cuboid grid. The magnetization is expanded in piecewise-linear, globally continuous nodal basis functions $\phi_n$, while material parameters such as $M_s$ are piecewise-constant per cell. Local field terms are assembled from the discretized weak form with an element matrix per cell and summed over the eight cells sharing each node, giving $\delta E = \sum_i \delta E^{*i}$; the effective field is obtained by mass lumping, $H_{n,j} = -(\int_\Omega \mu_0 M_s^h \phi_n\,dx)^{-1} \delta E_{n,j}$. The demagnetization field reuses the standard cell-centered FFT convolution through an averaging pre- and post-processing step. The authors validate the scheme against a standard dynamic problem and a two-phase domain-wall pinning problem with discontinuous material parameters, reporting close agreement with reference and analytical switching fields.

Load-bearing premise

The accuracy gain depends on the assumption that the simple averaging step between node-based and cell-based grids for the demagnetization field, and the mass-lumping shortcut in the local field update, do not introduce errors as large as the interface errors the new scheme removes.

Editorial extensions

If this is right

  • Material-interface problems such as two-phase domain-wall pinning can be simulated on a regular grid with finite-element-like accuracy at the same asymptotic cost as standard finite differences.
  • Thin-film simulations can use 2D basis functions with full 3D integration, cutting the number of degrees of freedom by about half compared with a full 3D nodal grid.
  • Time-dependent inverse problems become tractable with a single framework: gradients are computed by one backward adjoint pass whose complexity matches the forward solve.
  • The form compiler converts symbolic weak forms into backend tensor operations with no loops or conditionals, so new energy terms can be added without hand-written stencils while remaining just-in-time compilable.
  • The just-in-time-compiling backend keeps Python overhead low even for small systems, making the code competitive with optimized GPU implementations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the interface-accuracy claim survives a careful convergence study, the same nodal approach could be applied to interfacial Dzyaloshinskii-Moriya or RKKY contributions, where cell-centered finite differences need elaborate boundary corrections.
  • The paper's accuracy argument would be strengthened by an error decomposition separating the nodal discretization error from the demagnetization-averaging error; such a decomposition is not reported, but a high-resolution finite-element reference on the two-phase problem would provide it.
  • A natural testable extension is to benchmark the adjoint-state gradient against finite differences of the full forward solve on a small time-dependent problem, quantifying the reduced accuracy the paper mentions for the backward reconstruction.
  • Because the gradient computation is 'discretize first', the same architecture could be reused for other regular-grid variational PDEs beyond micromagnetics, where interface conditions are also delicate.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. NeuralMag is an open-source Python micromagnetic simulation library built on PyTorch and JAX. The paper proposes a nodal finite-difference discretization in which the magnetization is represented by continuous piecewise-linear nodal basis functions on a regular cuboid grid, while material parameters such as Ms are represented as piecewise-constant cell values. Local field terms are assembled from a finite-element weak form with mass lumping, while the demagnetization field is computed with the standard cell-centered FFT convolution plus an averaging pre- and post-processing step between nodal and cell-centered representations. The manuscript also describes a symbolic form compiler, dynamic state attributes, automatic differentiation and adjoint-state methods for inverse problems, two validation cases (MuMag Standard Problem #4 and a domain-wall pinning problem), and a performance benchmark against mumax3 and magnum.np. The central claims are that the nodal scheme provides improved accuracy over traditional finite-difference methods at material interfaces at the same computational complexity, and that NeuralMag is competitive with state-of-the-art micromagnetic codes while enabling convenient inverse-problem workflows.

Significance. If the accuracy claim is substantiated, NeuralMag would be a useful open-source contribution that combines finite-element-like treatment of local interfacial terms with the FFT-based demagnetization computation, and its differentiable-programming design is well suited for inverse magnonics and related optimization problems. The manuscript has clear strengths: the code is publicly available under LGPL, the repository includes CI tests for both backends, the form compiler generates backend-specific tensor code, and the paper states that scripts for the numerical problems and benchmarks are included. However, the central accuracy advantage over conventional finite-difference methods is not currently demonstrated: the demagnetization averaging step is unquantified, the validation cases do not provide a head-to-head error-versus-cell-size comparison with a standard finite-difference code, and the interface test reports only scalar depinning fields. The performance claim is plausible but under-specified, and the inverse-problem demonstration is too simple to validate the adjoint implementation.

major comments (4)
  1. [Section III B and Section VI.] The abstract and Section III claim improved accuracy over traditional finite-difference methods, but the validation does not establish this claim. The demagnetization field is computed with the standard cell-centered FFT convolution and a 'straightforward pre- and post-processing step' that averages between nodal and cell-centered discretizations (Section III B), and no error analysis or convergence study is provided for this step. Standard Problem #4 (Fig. 5) contains no material interfaces, and the domain-wall pinning test (Table I) compares only scalar depinning fields without a cell-size dependence or a comparison with a conventional cell-centered finite-difference code. Please provide a head-to-head error-versus-cell-size experiment on an interface problem, such as the Heistracher pinning problem, that separates the interface handling from the demagnetization averaging error.
  2. [Section III A, Eq. (22).] The mass-lumping formula is load-bearing for the local-field accuracy claim, but the manuscript does not quantify how lumping interacts with cell-wise discontinuous Ms at material interfaces. Replacing the consistent mass matrix by the diagonal cell-wise integral changes the effective nodal weight, and when a node support straddles an interface with a jump in Ms, the discretized interface condition depends on this choice. Please provide either an error analysis or a numerical test that isolates the lumping error from the discretization error in a two-phase system.
  3. [Section VI, Fig. 6.] The performance benchmark reports only the right-hand-side evaluation time for the exchange and demagnetization fields and omits hardware details, GPU model, software versions, and repetition counts. The claim that NeuralMag with the JAX backend is less than a factor of two slower than mumax3 is therefore not reproducible as reported. Please add the missing benchmark specifications and consider reporting end-to-end LLG time-stepping performance.
  4. [Section VII.] The time-dependent inverse-problem demonstration is a two-parameter single-domain problem and does not validate the adjoint-state gradient implementation. Since Section IV itself notes that the backwards pass reconstructs the magnetization trajectory with reduced accuracy, a gradient check against backpropagation or finite differences, together with a nontrivial example such as distributed material parameters, would substantiate the claim that NeuralMag is well suited for time-dependent inverse problems.
minor comments (5)
  1. [Section III and Section IX.] The word 'rigoros' appears in the introduction to the nodal finite-difference scheme and in the conclusion; it should be 'rigorous'.
  2. [Section IV.] The text contains 'minization problem'; this should be 'minimization problem'.
  3. [Section VI.] The reference to 'Tab. 1 of the original paper' should specify the table and, ideally, include the exact material parameters and field-rate values used for the NeuralMag runs, so the comparison in Table I can be reproduced.
  4. [Section V A and Listings 1-2.] The manuscript abbreviates the generated code with ellipses; please state explicitly that the listings are shortened excerpts and point to the repository for the complete generated kernels.
  5. [Section VIII.] The documentation link is given as a bare URL; a versioned citation or a software-archive DOI would make the released version more durable and citable.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the nodal finite-difference scheme is derived from the weak form with no fitted parameters, and validation uses external and analytical references.

full rationale

NeuralMag's central discretization is derived explicitly from the weak form (Eqs. 7-22), with basis functions defined in Eqs. 11-12 and no adjustable parameters fitted to the validation targets. The accuracy claims are checked against two references: MuMag Standard Problem #4, an external community standard, and the analytical depinning-field values of Heistracher et al. (2022). The latter is authored by overlapping researchers, but it is an analytical reference rather than a fitted value, and Table I reports errors against it rather than using it to define the method. The demagnetization-field averaging in Section III B is not derived in detail, but it is an implementation choice, not an input that predetermines the reported predictions; the paper does not fit an averaging parameter to benchmark outcomes. The only notable self-citations (mass lumping in Abert 2019, magnum.np, and the pinning standard problem) are used as standard-technique or benchmark references, not as a uniqueness argument that forces the nodal scheme. Thus no step reduces by construction to its own inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No parameters were fitted to data in the paper. Material constants in the validation cases are taken from the problem definitions (MuMag SP4 and Heistracher et al.). The optimized field angles in Section VII are design variables of the inverse problem, not free parameters of the method. No new physical entities are introduced; the nodal finite-difference scheme is a numerical discretization, not a new physical mechanism.

assumptions (5)
  • domain assumption Micromagnetic dynamics follow the Landau-Lifshitz-Gilbert equation with Brown's boundary conditions and the exchange jump condition A1 dm1/dn = A2 dm2/dn.
    Standard micromagnetics model invoked in Section II, Eqs. (1) and (6); the nodal scheme is built to satisfy these conditions implicitly.
  • domain assumption Magnetization is discretized with continuous piecewise-linear nodal basis functions, while material parameters are piecewise constant per cell.
    Section III A, Eqs. (11), (14), and (15). This choice assumes magnetization continuity across interfaces and is the basis of the method.
  • domain assumption Mass lumping of the left-hand side of the weak form is valid and does not significantly degrade accuracy.
    Eq. (22) replaces the consistent mass matrix with a diagonal approximation; no error estimate is provided in this paper.
  • domain assumption The demagnetization field can be computed with the standard FFT convolution on cell-centered values, using averaged nodal values as input and output.
    Section III B introduces averaging between nodal and cell-centered discretizations without error analysis; the accuracy claim rests partly on this step.
  • domain assumption For thin films, 2D basis functions that are constant along the thickness, combined with full 3D integration in the weak form, give accurate results.
    Section III C, Eq. (23). This assumes negligible magnetization variation across the film thickness.

how reviews work

0 comments
Cite this review

Pith. "Pith review of NeuralMag: an open-source nodal finite-difference code for inverse micromagnetics." pith.science (2026). https://pith.science/paper/XKKY4WAI

@misc{pith2026241111725,
  author       = {Pith},
  title        = {Pith review of: NeuralMag: an open-source nodal finite-difference code for inverse micromagnetics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XKKY4WAI}},
  note         = {Machine review of arXiv:2411.11725}
}
read the original abstract

We present NeuralMag, a flexible and high-performance open-source Python library for micromagnetic simulations. NeuralMag leverages modern machine learning frameworks, such as PyTorch and JAX, to perform efficient tensor operations on various parallel hardware, including CPUs, GPUs, and TPUs. The library implements a novel nodal finite-difference discretization scheme that provides improved accuracy over traditional finite-difference methods without increasing computational complexity. NeuralMag is particularly well-suited for solving inverse problems, especially those with time-dependent objectives, thanks to its automatic differentiation capabilities. Performance benchmarks show that NeuralMag is competitive with state-of-the-art simulation codes while offering enhanced flexibility through its Python interface and integration with high-level computational backends.

Figures

Figures reproduced from arXiv: 2411.11725 by the authors.

Figure 1
Figure 1. FIG. 1. Illustration of the discretization of the magnetization [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Cell and vertex numbering using multiindices for nodal finite-differences. (a) Two [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Two-dimensional representation of a basis function [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Representation of the degrees of freedom for a square-shaped rod using the following [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Results for MuMag Standard Problem #4 are presented using both 2D and 3D discretiza [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Comparison of the computation time for evaluating the right-hand side of the Landau [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Simple inverse micromagnetic problem for the optimization of the external field direction [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Inverse-design topology optimization of magnonic devices using level-set method

    cond-mat.other 2024-11 conditional novelty 5.0 of 10

    A level-set plus adjoint-state optimization framework, implemented in the NeuralMag micromagnetic solver, designs magnonic devices such as a frequency-selective YIG demultiplexer with constant memory cost.

  2. Review on spin-wave RF applications

    physics.app-ph 2024-11 conditional novelty 3.0 of 10

    A comprehensive review of spin-wave RF components that concludes YIG-based magnonic devices show strong promise for 5G and 6G communication systems.

Reference graph

Works this paper leans on

34 extracted references · 28 canonical work pages · cited by 2 Pith papers

  1. [1]

    author author M. J. \ Donahue \ and\ author D. G. \ Porter ,\ @noop title OOMMF user's guide, version 1.0 , \ ( year 1999 ) NoStop

    work page 1999

  2. [2]

    Fidimag -- a finite difference atomistic and micromagnetic simulation package

    author author M.-A. \ Bisotti , author D. Cort \'e s-Ortu \ n o , author R. A. \ Pepper , author W. Wang , author M. Beg , author T. Kluyver ,\ and\ author H. Fangohr ,\ title title Fidimag--a finite difference atomistic and micromagnetic simulation package , \ @noop journal journal arXiv preprint arXiv:2002.04318 \ ( year 2020 ) NoStop

    work page Pith review arXiv 2002

  3. [3]

    Vansteenkiste , author J

    author author A. Vansteenkiste , author J. Leliaert , author M. Dvornik , author M. Helsen , author F. Garcia-Sanchez ,\ and\ author B. Van Waeyenberge ,\ title title The design and verification of MuMax3 , \ https://doi.org/10.1063/1.4899186 journal journal AIP Adv. \ volume 4 ( year 2014 ),\ 10.1063/1.4899186 NoStop

    doi:10.1063/1.4899186

  4. [4]

    Bruckner , author S

    author author F. Bruckner , author S. Koraltan , author C. Abert ,\ and\ author D. Suess ,\ title title magnum. np: a pytorch based gpu enhanced finite difference micromagnetic simulation framework for high level development and inverse design , \ @noop journal journal Scientific Reports \ volume 13 ,\ pages 12054 ( year 2023 ) NoStop

    work page 2023

  5. [5]

    Fischbacher , author M

    author author T. Fischbacher , author M. Franchin , author G. Bordignon ,\ and\ author H. Fangohr ,\ title title A systematic approach to multiphysics extensions of finite-element-based micromagnetic simulations: Nmag , \ @noop journal journal IEEE Transactions on Magnetics \ volume 43 ,\ pages 2896--2898 ( year 2007 ) NoStop

    work page 2007

  6. [6]

    Kakay , author E

    author author A. Kakay , author E. Westphal ,\ and\ author R. Hertel ,\ title title Speedup of fem micromagnetic simulations with graphical processing units , \ @noop journal journal IEEE transactions on magnetics \ volume 46 ,\ pages 2303--2306 ( year 2010 ) NoStop

    work page 2010

  7. [7]

    Chang , author S

    author author R. Chang , author S. Li , author M. Lubarda , author B. Livshitz ,\ and\ author V. Lomakin ,\ title title Fastmag: Fast micromagnetic simulator for complex magnetic structures , \ @noop journal journal Journal of Applied Physics \ volume 109 ( year 2011 ) NoStop

    work page 2011

  8. [8]

    \ Bisotti , author M

    author author M.-A. \ Bisotti , author M. Beg , author W. Wang , author M. Albert , author D. Chernyshenko , author D. Cort \'e s-Ortu \ n o , author R. A. \ Pepper , author M. Vousden , author R. Carey , author H. Fuchs , et al. ,\ title title Finmag: finite-element micromagnetic simulation tool (version 1.0) , \ @noop journal journal Zenodo \ ( year 201...

    work page 2018

Show all 34 references
  1. [9]

    Abert , author L

    author author C. Abert , author L. Exl , author F. Bruckner , author A. Drews ,\ and\ author D. Suess ,\ title title magnum. fe: A micromagnetic finite-element simulation code based on fenics , \ @noop journal journal Journal of Magnetism and Magnetic Materials \ volume 345 ,\...

    2013

  2. [10]

    Bruckner , author C

    author author F. Bruckner , author C. Abert , author G. Wautischer , author C. Huber , author C. Vogler , author M. Hinze ,\ and\ author D. Suess ,\ title title Solving large-scale inverse magnetostatic problems using the adjoint method , \ @noop journal journal Scientific rep...

    2017

  3. [11]

    Abert , author C

    author author C. Abert , author C. Huber , author F. Bruckner , author C. Vogler , author G. Wautischer ,\ and\ author D. Suess ,\ title title A fast finite-difference algorithm for topology optimization of permanent magnets , \ @noop journal journal Journal of Applied Physics...

    2017

  4. [12]

    Huber , author C

    author author C. Huber , author C. Abert , author F. Bruckner , author C. Pfaff , author J. Kriwet , author M. Groenefeld , author I. Teliban , author C. Vogler ,\ and\ author D. Suess ,\ title title Topology optimized and 3d printed polymer-bonded permanent magnets for a pred...

    2017

  5. [13]

    Papp , author W

    author author \'A . Papp , author W. Porod ,\ and\ author G. Csaba ,\ title title Nanoscale neural network using non-linear spin-wave interference , \ https://doi.org/10.1038/s41467-021-26711-z journal journal Nat Commun \ volume 12 ,\ pages 6422 ( year 2021 ) NoStop

    doi

  6. [14]

    Wang , author A

    author author Q. Wang , author A. V. \ Chumak ,\ and\ author P. Pirro ,\ title title Inverse-design magnonic devices , \ https://doi.org/10.1038/s41467-021-22897-4 journal journal Nat Commun \ volume 12 ,\ pages 2636 ( year 2021 ) NoStop

    doi

  7. [15]

    Yan , author Y

    author author Z. Yan , author Y. Xing ,\ and\ author X. Han ,\ title title Inverse design of magnonic filter , \ @noop journal journal Journal of Magnetism and Magnetic Materials \ volume 563 ,\ pages 169976 ( year 2022 ) NoStop

    2022

  8. [16]

    Zenbaa , author C

    author author N. Zenbaa , author C. Abert , author F. Majcen , author M. Kerber , author R. Serha , author S. Knauer , author Q. Wang , author T. Schrefl , author D. Suess ,\ and\ author A. Chumak ,\ title title Magnonic inverse-design processor , \ @noop journal journal arXiv...

    2024 arXiv

  9. [17]

    author author W. F. \ Brown Jr ,\ title title Micromagnetics, domains, and resonance , \ @noop journal journal Journal of Applied Physics \ volume 30 ,\ pages S62--S69 ( year 1959 ) NoStop

    1959

  10. [18]

    Abert ,\ title title Micromagnetics and spintronics: Models and numerical methods , \ https://doi.org/10.1140/epjb/e2019-90599-6 journal journal Eur

    author author C. Abert ,\ title title Micromagnetics and spintronics: Models and numerical methods , \ https://doi.org/10.1140/epjb/e2019-90599-6 journal journal Eur. Phys. J. B \ volume 92 ,\ pages 1--45 ( year 2019 ) NoStop

    2019 doi

  11. [19]

    author author J. E. \ Miltat \ and\ author M. J. \ Donahue ,\ title title Numerical micromagnetics: Finite difference methods , \ @noop journal journal Handbook of magnetism and advanced magnetic materials \ volume 2 ,\ pages 742--764 ( year 2007 ) NoStop

    2007

  12. [20]

    Schrefl , author G

    author author T. Schrefl , author G. Hrkac , author S. Bance , author D. Suess , author O. Ertl ,\ and\ author J. Fidler ,\ title title Numerical micromagnetics (finite element method) , \ @noop journal journal Handbook of magnetism and advanced magnetic materials \ volume 2 ,...

    2007

  13. [21]

    Heistracher , author C

    author author P. Heistracher , author C. Abert , author F. Bruckner , author T. Schrefl ,\ and\ author D. Suess ,\ title title Proposal for a micromagnetic standard problem: domain wall pinning at phase boundaries , \ @noop journal journal Journal of Magnetism and Magnetic Mat...

    2022

  14. [22]

    Suess , author S

    author author D. Suess , author S. Koraltan , author F. Slanovc , author F. Bruckner ,\ and\ author C. Abert ,\ title title Accurate finite-difference micromagnetics of magnets including rkky interaction: Analytical solution and comparison to standard micromagnetic codes , \ @...

    2023

  15. [23]

    Berkov , author K

    author author D. Berkov , author K. Ramst \"o cck ,\ and\ author A. Hubert ,\ title title Solving micromagnetic problems. towards an optimal numerical method , \ @noop journal journal physica status solidi (a) \ volume 137 ,\ pages 207--225 ( year 1993 ) NoStop

    1993

  16. [24]

    Ramst \"o ck , author T

    author author K. Ramst \"o ck , author T. Leibl ,\ and\ author A. Hubert ,\ title title Optimizing stray field computations in finite-element micromagnetics , \ @noop journal journal Journal of magnetism and magnetic materials \ volume 135 ,\ pages 97--110 ( year 1994 ) NoStop

    1994

  17. [25]

    Paszke , author S

    author author A. Paszke , author S. Gross , author S. Chintala , author G. Chanan , author E. Yang , author Z. DeVito , author Z. Lin , author A. Desmaison , author L. Antiga ,\ and\ author A. Lerer ,\ title title Automatic differentiation in pytorch , \ @noop \ ( year 2017 ) NoStop

    2017

  18. [26]

    author author R. T. \ Chen , author Y. Rubanova , author J. Bettencourt ,\ and\ author D. K. \ Duvenaud ,\ title title Neural ordinary differential equations , \ @noop journal journal Advances in neural information processing systems \ volume 31 ( year 2018 ) NoStop

    2018

  19. [27]

    Paszke , author S

    author author A. Paszke , author S. Gross , author F. Massa , author A. Lerer , author J. Bradbury , author G. Chanan , author T. Killeen , author Z. Lin , author N. Gimelshein , author L. Antiga , et al. ,\ title title Pytorch: An imperative style, high-performance deep learn...

    2019

  20. [28]

    Frostig , author M

    author author R. Frostig , author M. J. \ Johnson ,\ and\ author C. Leary ,\ title title Compiling machine learning programs via high-level tracing , \ @noop journal journal Systems for Machine Learning \ volume 4 ( year 2018 ) NoStop

    2018

  21. [29]

    Bradbury , author R

    author author J. Bradbury , author R. Frostig , author P. Hawkins , author M. J. \ Johnson , author C. Leary , author D. Maclaurin , author G. Necula , author A. Paszke , author J. VanderPlas , author S. Wanderman-Milne , et al. ,\ title title Jax: Autograd and xla , \ @noop j...

    2021

  22. [30]

    Meurer , author C

    author author A. Meurer , author C. P. \ Smith , author M. Paprocki , author O. C ert \' k , author S. B. \ Kirpichev , author M. Rocklin , author A. Kumar , author S. Ivanov , author J. K. \ Moore , author S. Singh , et al. ,\ title title Sympy: symbolic computing in python ,...

    2017

  23. [31]

    hey, that's not an ode

    author author P. Kidger , author R. T. \ Chen ,\ and\ author T. J. \ Lyons ,\ title title " hey, that's not an ode": Faster ode adjoints via seminorms. \ in\ @noop booktitle ICML \ ( year 2021 )\ pp.\ pages 5443--5452 NoStop

    2021

  24. [32]

    Kidger ,\ title O n N eural D ifferential E quations ,\ @noop Ph.D

    author author P. Kidger ,\ title O n N eural D ifferential E quations ,\ @noop Ph.D. thesis ,\ school University of Oxford ( year 2021 ) NoStop

    2021

  25. [33]

    @noop title MAG standard problem \#4 , \ howpublished https://www.ctcms.nist.gov/ rdm/std4/spec4.html ( year 2024 ) NoStop

    2024

  26. [34]

    @noop title Neuralmag: A machine learning framework for micromagnetic simulations , \ howpublished https://gitlab.com/neuralmag/neuralmag ( year 2024 ) NoStop

    2024

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.