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REVIEW 3 major objections 4 minor 51 references

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

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper demonstrates a level-set and adjoint-state optimization framework that designs nanoscale magnonic devices—including a spin-wave demultiplexer separating 2.6 and 2.8 GHz signals—with memory cost independent of simulation time.

desk verdict A useful integration of level-set topology optimization with adjoint gradients for magnonics, but the printed adjoint equations don't match the time-distributed objectives—needs a gradient check and better documentation. read the letter →

arxiv 2411.19109 v2 pith:UFDGM2MR submitted 2024-11-28 cond-mat.other physics.comp-ph

Andrey A. Voronov , Marcos Cuervo Santos , Florian Bruckner , Dieter Suess , Andrii V. Chumak , Claas Abert This is my paper · ORCID
classification cond-mat.otherphysics.comp-ph
keywords inversedesignlevel-setmethodadjoint-statetopologyoptimizationmagnonicsspin-wavedemultiplexermicromagneticsimulationyttriumirongarnet
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

The paper aims to make inverse design practical for magnonic devices by combining a level-set description of material boundaries with an adjoint-state method for computing gradients, and it claims this combination removes the main memory bottleneck that limited earlier approaches. The boundary between magnetic and non-magnetic regions is written as the zero contour of a smooth function built from radial basis functions, so the geometry can change topology freely while remaining differentiable. Gradients of a device-performance objective are obtained by solving an adjoint equation backward in time, which keeps memory use constant as simulation time grows. The method is demonstrated on two tasks: shaping a magnetic particle to follow a target hysteresis curve, and carving air holes in a yttrium-iron-garnet strip so that 2.6 GHz and 2.8 GHz spin waves separate into different output conduits. The paper concludes that the framework is a versatile and universal tool for inverse design in magnonics.

What carries the argument

The load-bearing machinery is the pair formed by the level-set parameterization and the adjoint-state gradient formula. The level-set function $\Phi(x,y)$ is built from $n$ radial basis functions $g_i(x,y)$ such that $\Phi = (\sum_i g_i^p + \Delta\phi)^{1/p} - \Delta\phi$ with $p=90$; its zero contour defines the boundary, and a sigmoid $\psi = 1/(1+\exp(-a\Phi))$ with $a=50$ converts it into smooth material parameters. The adjoint-state method solves the system $\partial m/\partial t = L(t,m,s)$ and $\partial a/\partial t = -a^T \partial L/\partial m$ backward in time, then forms $\partial J/\partial s = -\int_T^{t_0} a(t)^T \partial L/\partial s\,dt$; this replaces step-by-step backpropagation with a second time integration, so intermediate magnetization states need not be stored. That integral and the backward integration are evaluated by automatic differentiation, and the amplitudes are updated by a gradient-descent optimizer with adaptive moments. Together these pieces let the algorithm nucleate, merge, and remove holes while keeping memory independent of simulation time.

What would settle it

Take the 30×30 particle design task, leave the discretization and objective unchanged, and compare the adjoint gradient against a central finite-difference gradient for a handful of radial-basis amplitudes at the first optimization step; any relative discrepancy well above the solver's own tolerance would falsify the gradient premise. A second, independent check: fix the design region, double the simulated time for the demultiplexer, and monitor peak GPU memory—a noticeable rise with time would falsify the constant-memory claim.

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Extended reading notes

Core claim

The central claim is that level-set topology optimization combined with the adjoint-state method can design functional nanoscale magnonic devices with a memory cost that does not grow with simulation time. The level-set function is assembled from radial basis functions through a differentiable p-norm approximation of the maximum, and a sigmoid maps it to material parameters, making the geometry a smooth function of optimizable amplitudes. The adjoint-state approach computes the gradient of the objective by integrating a second dynamics equation backward in time and an integral formula, rather than backpropagating through every step of the forward simulation. On the two test problems, the framework converges smoothly: the particle optimization reaches a stable objective value within 100 steps and can split a single shape into multiple particles, and the demultiplexer optimization separates spin waves at the two target frequencies with output amplitudes differing by an order of magnitude and remains functional over a roughly 70 MHz window. The authors state that these results validate the software as a versatile and universal tool for future inverse-design studies in magnonics.

Load-bearing premise

The load-bearing premise is that the adjoint-state equations, solved backward in time, return the exact gradient of the objective with respect to the level-set parameters for the damped magnetization dynamics and the smoothed sigmoid mapping; the paper provides no comparison with a brute-force finite-difference gradient, so if the adjoint gradient is biased, the reported convergence and geometries could be optimizer artifacts.

Editorial extensions

If this is right

  • Because only the objective function is device-specific, the same optimizer can be pointed at other magnonic functions—mode multiplexing, directional coupling, or logic gates—without changing the machinery.
  • Devices designed this way tolerate fabrication errors: the demultiplexer keeps separating spin waves when frequencies deviate by up to roughly 70 MHz from the design values.
  • Longer or more detailed spin-wave simulations become affordable, since the adjoint method holds memory use constant in simulation time.
  • Topology changes such as hole nucleation and merging are handled automatically, so the optimizer can start from a minimal design and grow only the features that are needed.
  • The resulting geometries have smoother, rounder edges than binary-search designs, which is advantageous for lithographic fabrication.

Reading between the lines

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

  • The adjoint-gradient premise can be tested cheaply on the 30×30 particle case by comparing against central finite differences; if it holds, the method should transfer to any objective built from local magnetization measurements.
  • Because the optimizer finds distinct local minima from different initial hole patterns, pairing the level-set descent with a global search (e.g., multi-start or restart with random initial amplitudes) could systematically find better-performing geometries.
  • The paper's extension list—3D structures, chiral exchange interactions, and finite-temperature stochastic dynamics—suggests the same machinery could be applied to design devices based on topological spin textures or thermally stable memory elements, though these are not demonstrated.
  • A direct benchmark against the earlier binary-search demultiplexer on identical geometry, tracking number of simulations to convergence and final output contrast, would quantify the claimed efficiency advantage.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper presents a level-set method for inverse topology optimization of magnonic devices, parameterizing the level-set function by radial basis functions and mapping it to magnetic/nonmagnetic regions through a smooth sigmoid. The design parameters (RBF amplitudes) are updated by gradient descent, with gradients computed via an adjoint-state solution of the Landau-Lifshitz-Gilbert equation, implemented in the GPU-accelerated NeuralMag solver with torchdiffeq. The method is validated on two tasks: optimizing a nanoparticle's hysteresis curve to match a hard-axis linear response, and designing a 300-nm-wide YIG demultiplexer that separates 2.6 GHz and 2.8 GHz spin waves. The paper reports smooth convergence, topology-splitting behavior, frequency-selective output separation, robustness to different initial hole configurations, and a frequency operating window of about 70 MHz.

Significance. If the adjoint-gradient computation is correct, the work is a valuable contribution to inverse design in magnonics: it introduces a level-set parameterization that handles topology changes, combines it with a micromagnetic solver, and demonstrates application to a realistic nanoscale device. The multiple-initialization study and the frequency-sweep robustness check strengthen the demultiplexer claim. The demultiplexer separation is, of course, a direct consequence of the chosen objective and thus validates the optimizer rather than predicting new physics; this is normal for inverse-design papers and is not itself a flaw. The main value would be as a reusable computational framework, but the paper as written does not make the code available and, more importantly, contains an inconsistency in the adjoint equations relative to the stated objectives.

major comments (3)
  1. [Optimisation with adjoint method (Eqs. 6-7)] The adjoint system in Eq. (6) is written for a terminal-cost objective: the backward equation has no source term, and the terminal condition is given as a(T)=∂J/∂m(T). The objectives actually used in the paper are distributed in time: Eq. (8) sums over field-ramp steps j=0...N, and Eq. (10) sums FFT amplitudes over frequency windows, so ∂J/∂m(t) is nonzero throughout the integration interval. The correct continuous adjoint for an objective J=∫g(t,m,s)dt contains a source term -(∂g/∂m)^T in the adjoint equation and has a(T)=0 (plus any terminal contribution), and the gradient formula must include the corresponding source contribution. As printed, Eq. (7) omits the running-cost contribution to ∂J/∂s. This is an internal inconsistency: if the implementation literally follows Eqs. (6)-(7), the optimizer is not descending the stated objectives, and the demultiplexer results would not validate the claimed method. The authors should either correct the adjoint equations to include the source terms, or document the actual augmented-state/checkpointing/automatic-differentiation procedure used, and provide a finite-difference gradient check for at least one small problem.
  2. [Demultiplexer example (Eq. (10), Discussion)] The memory-efficiency claim that the adjoint approach ensures constant memory as a function of simulation time is not demonstrated for the FFT-based objective of Eq. (10). Computing the FFT of the magnetization at the output cells requires either storing the time series over the simulation window or checkpointing the state, so the memory footprint is not trivially constant in simulation time unless a specific streaming or partial-FFT strategy is used. The paper should quantify the memory scaling with the number of time steps and describe how the FFT objective is incorporated in the adjoint computation. This point is load-bearing for the 'versatile and universal tool' claim in the Discussion.
  3. [Methods / all examples] No numerical verification of the adjoint gradient against brute-force finite differences or forward-mode automatic differentiation is reported. Because the paper introduces an optimization framework and the gradient formula is at issue (see previous comment), a gradient check for a small problem (e.g., the particle case with a reduced number of RBFs) should be included. Such a check would resolve whether the smooth convergence and optimized geometries in Figs. 3-7 are genuine minima of the stated objectives or artifacts of the optimizer operating on a different functional.
minor comments (4)
  1. [Results, Demultiplexer example] In the paragraph after Eq. (10), the sentence 'These frequency windows are illustrated in Fig. 6(a, b)' should presumably refer to Fig. 6(f, g), since the colored frequency windows appear in the FFT spectra panels; Fig. 6(a, b) show the objective evolution and propagation maps.
  2. [Eq. (8)] The summation is written from j=0 to N, which gives N+1 terms, while the text says the summation is performed over N simulation steps; please clarify the indexing.
  3. [Eqs. (3)-(4)] The statements that p=90 and a=50 introduce negligible error are not quantified. A brief sensitivity study or convergence check with respect to p and a would support this assertion.
  4. [Data availability] The statement 'No datasets were generated or analysed during the current study' is inconsistent with the reported simulation results; please clarify what simulation data, parameter files, and post-processing scripts are available, especially given that the code is not public.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the optimized geometries are independent outputs of the objective, and no claim reduces to its own input by construction.

full rationale

The paper's chain is: define a level-set parameterization (Eqs. 1-4), solve LLG forward, use the adjoint equations (Eqs. 6-7) to obtain ∂J/∂s, and update s by gradient descent (Eq. 5). The objective functions in Eqs. (8) and (10) are user-specified targets (a linear hard-axis loop and frequency-selective output amplitudes). The quantities reported as results—the wire-like particle shapes and the hole distributions that separate f1 and f2—are not contained in those objectives; they are the geometries found by optimization. Thus the demultiplexer separation is indeed the objective being minimized, but the paper does not present it as a physical prediction; the novel output is the geometry, and the convergence from different initial designs and the ~70 MHz bandwidth are additional results not forced by J. The Stoner-Wohlfarth benchmark is also a genuine external check: the target curve comes from the known theory, and the optimizer must discover a shape that realizes it. Self-citations to NeuralMag (ref. 33) and prior magnonic inverse design (ref. 17) provide the solver and problem context; no uniqueness theorem or ansatz is imported from same-author work to forbid alternatives. The one substantive concern is that the printed adjoint equation (Eq. 6) has terminal-cost form while the objectives in Eqs. (8) and (10) are distributed in time, which would require a source term; this is an implementation/correctness issue that could affect the validity of the gradients, but it is not an input-output equivalence and does not make the derivation circular. Overall, no claim reduces by construction to its own input.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central claim rests on a small set of hand-chosen numerical parameters (p, a, learning-rate schedule, FFT windows, initial topology) and on standard domain assumptions about LLG, the adjoint method, and the 2D film approximation. No new physical entities are introduced. The most fragile item is the unverified correctness of the discrete adjoint gradient, which is assumed from the NeuralMag and torchdiffeq implementations.

free parameters (6)
  • p-norm exponent p = 90
    Used in Eq. (3) to approximate the max of RBFs; chosen by hand; controls the smoothness of the geometry representation.
  • sigmoid sharpness a = 50
    Used in Eq. (4) to map the LSF to material parameters; chosen by hand; controls boundary width and convergence rate.
  • learning rate schedule = not stated (decreases every 10 steps)
    Adam optimizer learning rate is not specified; the schedule is a hand-chosen hyperparameter that affects convergence, especially in the regularized particle task.
  • FFT frequency window widths = not stated (around f1 and f2)
    Objective function Eq. (10) sums FFT values within windows around 2.6 and 2.8 GHz; the widths are not given and are chosen by the authors, directly affecting the optimized design.
  • initial RBF configuration = 4x4 grid of holes (demux)
    The optimization starts from a 4x4 array of holes; the 2x2 and single-hole cases show the result depends on initialization, so the initial design is a choice affecting the outcome.
  • constraint weights (zeta, xi, nu) = zeta=10, xi=0.05, nu=1
    Weights in Eq. (9) for the particle task; chosen by hand to balance curve matching, size and center constraints.
assumptions (5)
  • domain assumption LLG equation with Zeeman, demagnetizing, and exchange fields accurately models magnetization dynamics in the simulated thin films
    Used as the forward model throughout; stated in Methods, Eq. (11).
  • domain assumption The adjoint-state equations (Eq. 6) provide the exact gradient of the discrete objective for the LLG solver
    Invoked in 'Optimisation with adjoint method', Eqs. (6)-(7); correctness is cited to refs. 32,33, not proven or checked here.
  • ad hoc to paper The p-norm max approximation with p=90 and the sigmoid mapping (a=50) introduce negligible error in optimized topologies
    Eqs. (3)-(4) define the mapping; the authors do not analyze sensitivity to p or a.
  • domain assumption Spin-wave dynamics remain linear for sub-mT excitation, so FFT amplitudes at f1/f2 characterize the device
    Assumed in the demultiplexer example; stated in Results: 'magnetization oscillation angles below 1 degree'.
  • domain assumption A single 100 nm-thick layer treated as 2D with 20 nm cells captures the physics of the YIG film
    The simulation mesh is 512x64x1 with 20x20x100 nm^3 cells; no convergence study vs cell size is shown.

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Pith. "Pith review of Inverse-design topology optimization of magnonic devices using level-set method." pith.science (2026). https://pith.science/paper/UFDGM2MR

@misc{pith2026241119109,
  author       = {Pith},
  title        = {Pith review of: Inverse-design topology optimization of magnonic devices using level-set method},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UFDGM2MR}},
  note         = {Machine review of arXiv:2411.19109}
}
read the original abstract

The inverse design approach in magnonics exploits the wave nature of magnons and machine learning to develop logical devices with functionalities that exceed the capabilities of analytical methods. While promising for analog, Boolean, and neuromorphic computing, current implementations face memory limitations that hinder the design of complex systems. This study presents a level-set parameterization method for topology optimization, combined with an adjoint-state approach for memory-efficient simulation of magnetization dynamics. The framework is implemented in NeuralMag, a GPU-accelerated micromagnetic solver featuring a nodal finite-difference scheme and automatic differentiation tools. To validate the method, we optimized the shape of a magnetic nanoparticle by applying constraints to the objective function, and designed a 300-nm-wide yttrium iron garnet demultiplexer achieving frequency-selective spin-wave separation. These results highlight the algorithm's efficiency in exploring local minima across various initial configurations, establishing its utility as a versatile tool for the inverse design of magnonic logic devices.

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Reference graph

Works this paper leans on

51 extracted references · 49 canonical work pages

  1. [1]

    Chumak, A. V. et al. Advances in magnetics roadmap on spin-wave computing. IEEE Trans. Magnetics 58,1 – 72 (2022)

    work page 2022

  2. [2]

    V., Serga, A

    Chumak, A. V., Serga, A. A. & Hillebrands, B. Magnon transistor for all- magnon data processing. Nat. Commun. 5, 4700 (2014)

    work page 2014

  3. [3]

    & Grundler, D

    Kruglyak, V., Demokritov, S. & Grundler, D. Magnonics. J. Phys. D: Appl. Phys. 43, 264001 (2010)

    work page 2010

  4. [4]

    Mahmoud, A. et al. Introduction to spin wave computing. J. Appl. Phys. 128, 161101 (2020)

    work page 2020

  5. [5]

    & Wang, K

    Khitun, A., Bao, M. & Wang, K. L. Magnonic logic circuits. J. Phys. D: Appl. Phys. 43, 264005 (2010)

    work page 2010

  6. [6]

    Schneider, T. et al. Realization of spin-wave logic gates. Appl. Phys. Lett. 92, 022505 (2008)

    work page 2008

  7. [7]

    & Hillebrands, B

    Kostylev, M., Serga, A., Schneider, T., Leven, B. & Hillebrands, B. Spin-wave logical gates. Appl. Phys. Lett. 87, 153501 (2005)

    work page 2005

  8. [8]

    Wang, Q. et al. Recon figurable nanoscale spin-wave directional coupler. Sci. Adv. 4, e1701517 (2018)

    work page 2018

Show all 51 references
  1. [9]

    Wang, Q. et al. A magnonic directional coupler for integrated magnonic half-adders. Nat. Electron. 3, 765– 774 (2020)

    2020

  2. [10]

    Barman, A. et al. The 2021 magnonics roadmap. J. Phys.: Condens. Matter 33, 413001 (2021)

    2021

  3. [11]

    Wang, Q., Csaba, G., Verba, R., Chumak, A. V. & Pirro, P. Nanoscale magnonic networks. Phys. Rev. Appl. 21, 040503 (2024)

    2024

  4. [12]

    Molesky, S. et al. Inverse design in nanophotonics. Nat. Photonics 12, 659– 670 (2018)

    2018

  5. [13]

    F., Ding, Y., Sigmund, O

    Frellsen, L. F., Ding, Y., Sigmund, O. & Frandsen, L. H. Topology optimized mode multiplexing in silicon-on-insulator photonic wire waveguides. Opt. express 24, 16866– 16873 (2016)

    2016

  6. [14]

    & Fan, J

    Sell, D., Yang, J., Doshay, S., Yang, R. & Fan, J. A. Large-angle, multifunctional metagratings based on freeform multimode geometries. Nano Lett. 17, 3752– 3757 (2017)

    2017

  7. [15]

    & Aydin, K

    Callewaert, F., Velev, V., Kumar, P., Sahakian, A. & Aydin, K. Inverse- designed broadband all-dielectric electromagnetic metadevices. Sci. Rep. 8, 1358 (2018)

    2018

  8. [16]

    & Menon, R

    Shen, B., Wang, P., Polson, R. & Menon, R. Ultra-high-ef ficiency metamaterial polarizer. Optica 1, 356– 360 (2014)

    2014

  9. [17]

    Wang, Q., Chumak, A. V. & Pirro, P. Inverse-design magnonic devices. Nat. Commun. 12, 2636 (2021)

    2021

  10. [18]

    & Csaba, G

    Papp, Á., Porod, W. & Csaba, G. Nanoscale neural network using non- linear spin-wave interference. Nat. Commun. 12, 6422 (2021)

    2021

  11. [19]

    Kiechle, M. et al. Spin-wave optics in yig realized by ion-beam irradiation. Small 19, 2207293 (2023)

    2023

  12. [20]

    Zenbaa, N. et al. A universal inverse-design magnonic device. Nat. Electron 8, 106– 115 (2025)

    2025

  13. [21]

    Sethian, J. A. Evolution, implementation, and application of level set and fast marching methods for advancing fronts. J. Comput. Phys. 169, 503– 555 (2001)

    2001

  14. [22]

    & Sethian, J

    Osher, S. & Sethian, J. A. Fronts propagating with curvature- dependent speed: Algorithms based on hamilton-jacobi formulations. J. Comput. Phys. 79,1 2– 49 (1988)

    1988

  15. [23]

    P., Maute, K., Langelaar, M

    Van Dijk, N. P., Maute, K., Langelaar, M. & Van Keulen, F. Level-set methods for structural topology optimization: a review. Struct. Multidiscip. Optim. 48, 437– 472 (2013)

    2013

  16. [24]

    & Van Keulen, F

    Van Dijk, N., Langelaar, M. & Van Keulen, F. Explicit level-set-based topology optimization using an exact heaviside function and consistent sensitivity analysis. Int. J. Numer. Methods Eng. 91,6 7– 97 (2012)

    2012

  17. [25]

    & Stancil, D

    Prabhakar, A. & Stancil, D. D. Spin waves: Theory and applications , vol. 5 (Springer, 2009)

    2009

  18. [26]

    A tutorial on the adjoint method for inverse problems

    Givoli, D. A tutorial on the adjoint method for inverse problems. Comput. Methods Appl. Mech. Eng. 380, 113810 (2021)

    2021

  19. [27]

    & Stam, J

    McNamara, A., Treuille, A., Popovi ć, Z. & Stam, J. Fluid control using the adjoint method. ACM Trans. Graph. (TOG) 23, 449– 456 (2004)

    2004

  20. [28]

    K., Mader, C

    Kenway, G. K., Mader, C. A., He, P. & Martins, J. R. Effective adjoint approaches for computationalfluid dynamics. Prog. Aerosp. Sci. 110, 100542 (2019)

    2019

  21. [29]

    Aerodynamic shape optimization using the adjoint method

    Jameson, A. Aerodynamic shape optimization using the adjoint method. Lectures at the Von Karman Institute, Brussels (2003)

    2003

  22. [30]

    & Igel, H

    Fichtner, A., Bunge, H.-P. & Igel, H. The adjoint method in seismology: I. theory. Phys. Earth Planet. Inter. 157,8 6– 104 (2006)

    2006

  23. [31]

    A review of the adjoint-state method for computing the gradient of a functional with geophysical applications.Geophys

    Plessix, R.-E. A review of the adjoint-state method for computing the gradient of a functional with geophysical applications.Geophys. J. Int. 167, 495– 503 (2006). https://doi.org/10.1038/s44306-025-00082-3 Article npj Spintronics | (2025) 3:19 7

    2006 doi

  24. [32]

    T., Rubanova, Y., Bettencourt, J

    Chen, R. T., Rubanova, Y., Bettencourt, J. & Duvenaud, D. K. Neural ordinary differential equations. Adv. Neural Inform. Proces. Sys. 31, (2018)

    2018

  25. [33]

    Abert, C. et al. NeuralMag: an open-source nodal finite-difference code for inverse micromagnetics. https://doi.org/10.48550/arXiv. 2411.11725 (2024)

    2024 doi arXiv

  26. [34]

    & Fedkiw, R

    Osher, S. & Fedkiw, R. P. Level set methods: an overview and some recent results. J. Comput. Phys. 169, 463– 502 (2001)

    2001

  27. [35]

    Li, H., Li, P., Gao, L., Zhang, L. & Wu, T. A level set method for topological shape optimization of 3d structures with extrusion constraints. Comput. Methods Appl. Mech. Eng.283, 615– 635 (2015)

    2015

  28. [36]

    & Wang, M

    Wang, S. & Wang, M. Y. Radial basis functions and level set method for structural topology optimization. Int. J. Numer. methods Eng. 65, 2060– 2090 (2006)

    2006

  29. [37]

    & Ling, L

    Kansa, E., Power, H., Fasshauer, G. & Ling, L. A volumetric integral radial basis function method for time-dependent partial differential equations. i. formulation.Eng. Anal. Bound. Elem.28, 1191–1206 (2004)

    2004

  30. [38]

    & Maute, K

    Pingen, G., Waidmann, M., Evgrafov, A. & Maute, K. A parametric level-set approach for topology optimization of flow domains. Struct. Multidiscip. Optim. 41, 117– 131 (2010)

    2010

  31. [39]

    Neurocomputing 5, 185– 196 (1993)

    Amari, S.-i Backpropagation and stochastic gradient descent method. Neurocomputing 5, 185– 196 (1993)

    1993

  32. [40]

    Natural gradient works ef ficiently in learning

    Amari, S.-I. Natural gradient works ef ficiently in learning. Neural Comput. 10, 251– 276 (1998)

    1998

  33. [41]

    Pontryagin, L. S. Mathematical theory of optimal processes (Routledge, 2018)

    2018

  34. [42]

    & Gieraltowski, J

    Tannous, C. & Gieraltowski, J. The stoner – wohlfarth model of ferromagnetism. Eur. J. Phys. 29, 475 (2008)

    2008

  35. [43]

    Introduction to the Theory of Ferromagnetism , vol

    Aharoni, A. Introduction to the Theory of Ferromagnetism , vol. 109 (Clarendon Press, 2000)

    2000

  36. [44]

    Zheng, S. et al. Tutorial: nonlinear magnonics. J. Appl. Phys. 134, 151101 (2023)

    2023

  37. [45]

    Han, D.-S. et al. Asymmetric hysteresis for probing dzyaloshinskii–moriya interaction.Nano Lett.16, 4438–4446 (2016)

    2016

  38. [46]

    L., Chen, R

    Li, X., Wong, T.-K. L., Chen, R. T. & Duvenaud, D. Scalable gradients for stochastic differential equations. In International Conference on Artifi cial Intelligence and Statistics , 3870– 3882 (PMLR, 2020)

    2020

  39. [47]

    On Neural Differential Equations

    Kidger, P. On Neural Differential Equations. Ph.D. thesis, University of Oxford (2021)

    2021

  40. [48]

    Garanin, D. A. Fokker-planck and landau-lifshitz-bloch equations for classical ferromagnets. Phys. Rev. B 55, 3050 (1997)

    1997

  41. [49]

    Lopez-Diaz, L. et al. Micromagnetic simulations using graphics processing units. J. Phys. D: Appl. Phys. 45, 323001 (2012)

    2012

  42. [50]

    & Liu, C

    Wei, D., Song, J. & Liu, C. Micromagnetics at finite temperature. IEEE Trans. Magn. 52,1 – 8 (2016)

    2016

  43. [51]

    Kingma, D. P. & Ba, J. Adam: A method for stochastic optimization. https://doi.org/10.48550/arXiv.1412.6980 (2014). Acknowledgements This research was funded in whole or in part by the Austrian Science Fund (FWF) projects: 10.55776/P34671, 10.55776/I6068, MagFunc [10.55776/I49...

    2014 doi arXiv

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