Pith. sign in

REVIEW 3 major objections 5 minor 41 references

Accurate and scalable deep Maxwell solvers using multilevel iterative methods

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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

Pith's one-line read A neural operator trained on subdomains solves Maxwell problems on grids up to 3000 by 3000.

desk verdict A genuinely new integration of neural preconditioners with spectral coarse spaces for two-level Schwarz, worth a serious referee despite some overclaimed headlining. read the letter →

arxiv 2509.03622 v1 pith:KFX4WNUL submitted 2025-09-03 physics.comp-ph cs.CEcs.LGphysics.app-ph

Chenkai Mao , Jonathan A. Fan This is my paper
classification physics.comp-phcs.CEcs.LGphysics.app-ph
keywords neuraloperatorMaxwell'sequationsdomaindecompositionoverlappingSchwarzcoarsespaceF-GMRESpreconditioninginversedesignFDFD
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 argues that neural PDE surrogates can be both accurate and scalable if they are embedded in iterative multilevel solvers instead of being asked to produce global solutions directly. It trains a single subdomain neural operator on small 64x64 or 256x256 boundary-value problems with arbitrary Robin-type boundary conditions, then reuses that network in two ways: as a preconditioner inside an F-GMRES loop to solve each subdomain to a chosen residual tolerance, and as an engine for building the coarse space of a two-level overlapping Schwarz method. With this setup, the authors report solving 2D frequency-domain Maxwell problems on global grids up to about 3000 by 3000 and physical sizes up to about 200 wavelengths, with iteration counts that stay nearly constant on smoothly varying dielectric distributions. The same solver drives adjoint-based inverse design of photonic devices larger than 100 wavelengths, producing layouts and efficiencies nearly identical to a ground-truth FDFD solver. If the claims hold, neural surrogates gain a credible path to large, practical electromagnetic simulation with controllable accuracy.

What carries the argument

The central object is a modified Fourier neural operator that maps a residual field, together with auxiliary permittivity, source, and PML maps, to an approximate error field. It is trained inside a recurrent F-GMRES loop, so the network sees the same residual-error distribution it will encounter at inference. Its two load-bearing uses are (1) as a flexible preconditioner generating Krylov directions for inner subdomain solves, and (2) as a fast evaluator of Robin-to-Robin maps: pseudo-plane-wave probe boundary values are fed through the network, neighboring boundary values are extracted, and Rayleigh-Ritz on this compact linear map produces the coarse-space eigenvectors. These eigenvectors

What would settle it

Run the same 256x256 subdomain network and two-level Schwarz method on a global domain built from binary random blocks of permittivity 1 and 8 with sharp edges, at 4x4, 8x8, and 12x12 subdomains with a fixed kcoarse, and compare the number of iterations to the grayscale cases reported in Table 2. If the iteration counts grow with the number of subdomains at a rate close to the one-level method, the near-optimal scaling claim is falsified for that material class.

Watch

Extended reading notes

Core claim

In its own terms, the paper's central claim is that a subdomain neural operator can be turned into an accurate global Maxwell solver by nesting two iterative loops: an inner network-preconditioned F-GMRES that solves subdomain problems with bounded residual, and an outer overlapping Schwarz iteration that updates subdomain boundary conditions until the global residual is small. The same network accelerates coarse-space construction: probe Robin boundary conditions are pushed through the network, the resulting Robin-to-Robin map is assembled, and Rayleigh-Ritz on that map yields slowly decaying modes used as global basis functions. The paper shows one-level overlapping Schwarz can reach machi

Load-bearing premise

The claim that iteration counts stay almost constant as the domain grows rests on coarse spaces that work well for smoothly varying grayscale dielectrics; if the coarse space loses effectiveness on sharp, high-contrast, resonant material distributions, the advertised 200-wavelength scalability does not carry over.

Editorial extensions

If this is right

  • One trained subdomain model transfers across global domain sizes, resolutions, wavelengths, and dielectric maps, because the network only ever receives subdomain-sized inputs.
  • Global accuracy is set by residual thresholds rather than by the network's fixed approximation error, giving users a direct accuracy-versus-cost dial.
  • The coarse space is built once per dielectric layout and then reused for many source configurations, which is the case that arises in optimization and design sweeps.
  • Because training and inference share the same F-GMRES loop, the network is optimized for how it is actually deployed, and the training recurrence itself expands the effective dataset from residual-error pairs produced along the iteration path.

Reading between the lines

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

  • The scaling tables use smoothly varying grayscale permittivity; the paper's own observation that resonant subdomains with discrete material boundaries degrade the coarse space implies the near-constant iteration counts should be re-tested on binary, high-contrast, sharply edged layouts before claiming general nanophotonic scalability.
  • The network's implicit learning of wavelength and resolution from the residual encoding suggests the same recurrent-training recipe may carry over to other parameter-dependent PDE families, such as Helmholtz or elastodynamics, without explicit frequency inputs; the paper does not claim this transfer.
  • If the coarse space is the bottleneck, then a more direct test of the method's ceiling is how many coarse modes kcoarse must grow as subdomains become more resonant; measuring that growth rate would separate network-precision limits from coarse-space limits.
  • The reported speed comparison is at 2D sizes where direct sparse solvers still fit in memory; the decisive regime the paper anticipates is larger 2D and 3D problems, so a battery of memory-bound 3D Maxwell benchmarks would test whether the batching and coarse-space setup retains its advantage.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper presents a hybrid neural-operator/iterative solver for 2D frequency-domain Maxwell equations. A Fourier neural operator is trained to map residuals to error fields and is used as a flexible preconditioner inside F-GMRES; the same model is used to build approximate Robin-to-Robin maps for coarse-space construction in a two-level overlapping Schwarz method. The authors claim machine-precision global solutions, near-optimal iteration scaling up to roughly 3000x3000 grids (about 200 wavelengths), and inverse design of devices larger than 100 wavelengths that matches an FDFD ground truth. The manuscript reports one-level Schwarz results (Figure 4), scaling tables for 64x64 and 256x256 subdomain models (Tables 1-2), and three inverse-design demonstrations (Figure 6).

Significance. The central design choice is sound: convergence is checked against the true FDFD operator residual, so the neural network acts as a preconditioner rather than a standalone approximate solver. This is a meaningful step toward using neural surrogates in a way that preserves a user-controlled accuracy guarantee. The recurrent F-GMRES training scheme, the modulation-based FNO architecture, and the use of the subdomain model for coarse-space eigenproblems are interesting contributions, and the ground-truth FDFD comparison in the inverse-design section is a genuine strength. However, the headline claims outrun the evidence: the 'machine precision' statement is unsupported by the reported losses, and the scaling results are restricted to smoothly varying grayscale media, while the paper itself reports degradation for discrete-material/resonant subdomains and increasing sensitivity to hyperparameters as the problem grows. The significance is therefore conditional on strengthening and honestly scoping the empirical claims.

major comments (3)
  1. [§2.2, Figure 4] The claim that the subdomain preconditioner 'can combine with one-level overlapping Schwarz methods to solve global problems with machine precision' is not supported by the data. Figure 4 reports L1 losses down to 3.6e-6 in the best case and 9.9e-5 in others, with final-error plots drawn on a ±1% scale and simulations in single precision. These are residual-bounded accuracies, not machine precision. Since this accuracy statement is central to the paper's promise, either replace 'machine precision' with 'prescribed residual accuracy' or provide evidence of convergence to roughly 1e-7 in the reported metric.
  2. [§2.2, Tables 1-2 and §3] The near-optimal scaling claim is established only for dielectric maps that the authors themselves describe as 'grayscale and varies smoothly.' The same paragraph concedes that 'highly resonant subdomains with discrete material boundaries worsen the effectiveness of the coarse space,' and the Discussion adds that convergence became increasingly sensitive to overlap and momentum and that coarse-space effectiveness 'slowly degraded' as problem size increased. The inverse-design devices in Figure 6 are explicitly binary, so the 200-wavelength scalability claim is not supported in the regime the paper advertises. Moreover, Tables 1-2 do not state the global residual tolerance, number of random realizations, error bars, or a rule for choosing kcoarse; these omissions make the iteration counts difficult to interpret. Please add scaling experiments on binary/high-contrast media and report conve
  3. [§2.2, Figure 6] The inverse-design evidence for 'accurate' large-scale optimization is limited. As written, the ground-truth FDFD comparison is explicitly made for the multiplexer optimization trajectory; the coupler and metalens are presented without an independent FDFD comparison, even though these are the binary/large-scale cases where the coarse space is most at risk. If the ground-truth comparison is meant to cover all three devices, the manuscript should state this and show the quantitative convergence/efficiency comparisons. Otherwise, a second ground-truth comparison is needed to support the generalization claimed in the text.
minor comments (5)
  1. [Table 2] Typo in the table title: 'Schwawrz' should be 'Schwarz.' In the Discussion, 'course grid construction' should be 'coarse grid construction.'
  2. [Figure 4] The final-error plots are described as spanning -1% to 1% of the maximum ground-truth field, but no colorbar or normalization formula is provided. Please add a colorbar and state whether the L1 losses are normalized by the L1 norm of the ground truth.
  3. [Algorithms 1-2] The training recurrence N, clipping threshold th_clip, and residual thresholds are load-bearing hyperparameters, but no default values or sensitivity analysis are reported. Please provide the values used in the experiments for reproducibility.
  4. [§4.1] Training details for the 256x256 physics-only model are sparse, and the manuscript repeatedly refers to the Supplementary Information for details. If the arXiv version is the review version, include the SI or summarize the omitted training hyperparameters and schedules.
  5. [§3] The Discussion notes that the solver is about 2x slower than PARDISO for the tested problems and would only become advantageous for much larger problems. This is important context for the scalability claims and should be mentioned in the abstract or introduction as well.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the solver is validated against true PDE residuals and an independent FDFD ground truth, not against its own outputs.

full rationale

The paper's derivation chain is not circular. The subdomain neural operator is trained to approximate A^{-1} acting on residuals (Sec. 2.1, Fig. 2), but it never defines accuracy in terms of its own outputs: F-GMRES uses the true FDFD operator A, and inference stops only when the true residual falls below a threshold (Algorithm 1, lines 22-24). Global solutions are likewise evaluated against ground-truth FDFD solutions (Fig. 4) and, in inverse design, against an independent FDFD adjoint solver (Fig. 6: optimization trajectories and final devices are 'nearly identical'). The coarse space is constructed via Rayleigh-Ritz approximations to eigenproblems of the Robin-to-Robin map (Sec. 4.3, Algorithm 3), a standard spectral coarse-space framework; the neural operator only accelerates the B-matrix-vector products, and the final coarse-space correction is still checked by the global residual. The one self-citation ([24]) is contextual and not load-bearing. The Discussion's caveats about discrete material boundaries degrading coarse-space effectiveness and about growing hyperparameter sensitivity are limitations on test coverage and robustness, not circularity: they do not make the advertised scaling a tautology or a fitted prediction. The reported comparisons to PETSc solvers and PARDISO further anchor the claims externally. Score 0 reflects the absence of any step where a prediction reduces by construction to a fitted input or to a self-citation chain.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central claims rest on standard linear algebra (GMRES convergence via true residual), the two-level Schwarz framework, and on heuristic assumptions about the network's generalization and the coarse-space quality. The numerical values of most hand-chosen hyperparameters (clipping threshold, overlap, momentum, exact k_coarse choices, residual thresholds) are not reported, which hampers replication. No new physical entities are introduced.

free parameters (5)
  • training data recurrence N (F-GMRES iterations in training) = 1 to 16, models compared; specifics in SI
    Chosen by hand; larger N improves preconditioner but training time held constant by adjusting epochs. Affects convergence performance.
  • fixed-point clipping threshold th_clip = not given; 'problem-specific'
    Algorithm 2 states clipping threshold depends on N and the PDE; required for stable training. This is a hand-tuned stabilization parameter.
  • coarse space size k_coarse = 16-36 for 64x64 model; 100-160 for 256x256 model, depending on grids per lambda
    Chosen per resolution; larger k_coarse costs more but accelerates convergence (Tables 1, 2).
  • domain decomposition hyperparameters (overlap, momentum m) = not reported
    Authors state convergence is sensitive to overlap and momentum as problem size grows; values are not given in text.
  • subdomain model size = approximately 1.5M weights selected as optimal
    Chosen by parametric sweep of Fourier modes and hidden channels (Figure 3d). Architecture hyperparameters.
assumptions (4)
  • domain assumption The FDFD-discretized Maxwell operator A (Eq. 1-2) is an accurate discretization of the target boundary value problem
    All residual computations, training losses, and convergence checks use this discrete operator; the ground-truth comparisons use the same FDFD family (Methods 4.1, ref [41]).
  • domain assumption The training distribution of subdomains (permittivity 1-8, wavelengths 2-4 cm, resolutions 0.2-0.8 mm, Gaussian-random/Voronoi dielectrics) is representative of the test and inverse-design problems
    Section 4.1 defines the training set; out-of-distribution results (Fig. 3e) show degraded performance for spatial frequencies outside the trained range.
  • domain assumption Rayleigh-Ritz with pseudo-plane-wave probes yields coarse basis functions that approximate the dominant modes of the Robin-to-Robin map
    Methods 4.3; the paper notes random/Fourier probes give worse results and discrete resonant subdomains degrade coarse-space quality, so this is a heuristic assumption.
  • standard math GMRES/Schwarz iterative convergence in residual implies convergence to the true PDE solution
    Standard numerical linear algebra; relies on A being nonsingular for the indefinite Maxwell system, a known subtlety acknowledged in Discussion.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Accurate and scalable deep Maxwell solvers using multilevel iterative methods." pith.science (2026). https://pith.science/paper/KFX4WNUL

@misc{pith2026250903622,
  author       = {Pith},
  title        = {Pith review of: Accurate and scalable deep Maxwell solvers using multilevel iterative methods},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KFX4WNUL}},
  note         = {Machine review of arXiv:2509.03622}
}
read the original abstract

Neural networks have promise as surrogate partial differential equation (PDE) solvers, but it remains a challenge to use these concepts to solve problems with high accuracy and scalability. In this work, we show that neural network surrogates can combine with iterative algorithms to accurately solve PDE problems featuring different scales, resolutions, and boundary conditions. We develop a subdomain neural operator model that supports arbitrary Robin-type boundary condition inputs, and we show that it can be utilized as a flexible preconditioner to iteratively solve subdomain problems with bounded accuracy. We further show that our subdomain models can facilitate the construction of global coarse spaces to enable accelerated, large scale PDE problem solving based on iterative multilevel domain decomposition. With two-dimensional Maxwell's equations as a model system, we train a single network to simulate large scale problems with different sizes, resolutions, wavelengths, and dielectric media distribution. We further demonstrate the utility of our platform in performing the accurate inverse design of multi-wavelength nanophotonic devices. Our work presents a promising path to building accurate and scalable multi-physics surrogate solvers for large practical problems.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

41 extracted references · 35 canonical work pages

  1. [1]

    Journal of Heat Transfer 143(6), 060801 (2021)

    Cai, S., Wang, Z., Wang, S., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks for heat transfer problems. Journal of Heat Transfer 143(6), 060801 (2021)

    work page 2021

  2. [2]

    npj Computational Materials 5(1), 56 (2019) 19

    Wu, Y.-J., Fang, L., Xu, Y.: Predicting interfacial thermal resistance by machine learning. npj Computational Materials 5(1), 56 (2019) 19

    work page 2019

  3. [3]

    Communications Engineering 3(1), 151 (2024)

    He, W., Li, J., Kong, X., Deng, L.: Multi-level physics informed deep learning for solving partial differential equations in computational structural mechanics. Communications Engineering 3(1), 151 (2024)

    work page 2024

  4. [4]

    Nature Communications 15(1), 9229 (2024)

    Li, X., Bolandi, H., Masmoudi, M., Salem, T., Jha, A., Lajnef, N., Boddeti, V.N.: Mechanics-informed autoencoder enables automated detection and localization of unforeseen structural damage. Nature Communications 15(1), 9229 (2024)

    work page 2024

  5. [5]

    Nature communications 16(1), 1422 (2025)

    Font, B., Alc´ antara-´Avila, F., Rabault, J., Vinuesa, R., Lehmkuhl, O.: Deep rein- forcement learning for active flow control in a turbulent separation bubble. Nature communications 16(1), 1422 (2025)

    work page 2025

  6. [6]

    Nature Machine Intelligence 6(4), 393–403 (2024)

    Li, T., Biferale, L., Bonaccorso, F., Scarpolini, M.A., Buzzicotti, M.: Synthetic lagrangian turbulence by generative diffusion models. Nature Machine Intelligence 6(4), 393–403 (2024)

    work page 2024

  7. [7]

    Nature Communications 16(1), 2251 (2025)

    Zhu, E., Zong, Z., Li, E., Lu, Y., Zhang, J., Xie, H., Li, Y., Yin, W.-Y., Wei, Z.: Frequency transfer and inverse design for metasurface under multi-physics coupling by euler latent dynamic and data-analytical regularizations. Nature Communications 16(1), 2251 (2025)

    work page 2025

  8. [8]

    Scientific reports 9(1), 19728 (2019)

    Trivedi, R., Su, L., Lu, J., Schubert, M.F., Vuckovic, J.: Data-driven acceleration of photonic simulations. Scientific reports 9(1), 19728 (2019)

    work page 2019

Show all 41 references
  1. [9]

    arXiv preprint arXiv:1910.03193 (2019)

    Lu, L., Jin, P., Karniadakis, G.E.: Deeponet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators. arXiv preprint arXiv:1910.03193 (2019)

    1910 arXiv

  2. [10]

    arXiv preprint arXiv:1906.01200 (2019)

    Hsieh, J.-T., Zhao, S., Eismann, S., Mirabella, L., Ermon, S.: Learning neural pde solvers with convergence guarantees. arXiv preprint arXiv:1906.01200 (2019)

    1906 arXiv

  3. [11]

    SIAM Journal on Scientific Computing 47(1), 151–181 (2025)

    Kopaniˇ c´ akov´ a, A., Karniadakis, G.E.: Deeponet based preconditioning strategies for solving parametric linear systems of equations. SIAM Journal on Scientific Computing 47(1), 151–181 (2025)

    2025

  4. [12]

    Advances in Neural Information Processing Systems 37, 128636–128667 (2024)

    Luo, J., Wang, J., Wang, H., Geng, Z., Chen, H., Kuang, Y., et al.: Neural krylov iteration for accelerating linear system solving. Advances in Neural Information Processing Systems 37, 128636–128667 (2024)

    2024

  5. [13]

    arXiv preprint arXiv:2405.15557 (2024)

    Trifonov, V., Rudikov, A., Iliev, O., Laevsky, Y.M., Oseledets, I., Muravleva, E.: Learning from linear algebra: A graph neural network approach to preconditioner design for conjugate gradient solvers. arXiv preprint arXiv:2405.15557 (2024)

    2024 arXiv

  6. [14]

    Journal of Computational physics 378, 686–707 (2019) 20

    Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378, 686–707 (2019) 20

    2019

  7. [15]

    arXiv preprint arXiv:2010.08895 (2020)

    Li, Z., Kovachki, N., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A., Anandkumar, A.: Fourier neural operator for parametric partial differential equations. arXiv preprint arXiv:2010.08895 (2020)

    2010 arXiv

  8. [16]

    Nature Machine Intelligence 6(6), 631–640 (2024)

    Cao, Q., Goswami, S., Karniadakis, G.E.: Laplace neural operator for solving differential equations. Nature Machine Intelligence 6(6), 631–640 (2024)

    2024

  9. [17]

    In: International Conference on Machine Learning, pp

    Sanchez-Gonzalez, A., Godwin, J., Pfaff, T., Ying, R., Leskovec, J., Battaglia, P.: Learning to simulate complex physics with graph networks. In: International Conference on Machine Learning, pp. 8459–8468 (2020). PMLR

    2020

  10. [18]

    arXiv preprint arXiv:2402.03541 (2024)

    Bryutkin, A., Huang, J., Deng, Z., Yang, G., Sch¨ onlieb, C.-B., Aviles-Rivero, A.: Hamlet: Graph transformer neural operator for partial differential equations. arXiv preprint arXiv:2402.03541 (2024)

    2024 arXiv

  11. [19]

    arXiv preprint arXiv:2410.05744 (2024)

    Dong, D., Suo, W., Kou, J., Zhang, W.: Pinn-mg: A multigrid-inspired hybrid framework combining iterative method and physics-informed neural networks. arXiv preprint arXiv:2410.05744 (2024)

    2024 arXiv

  12. [20]

    Nature Machine Intelligence, 1–11 (2024)

    Zhang, E., Kahana, A., Kopaniˇ c´ akov´ a, A., Turkel, E., Ranade, R., Pathak, J., Karniadakis, G.E.: Blending neural operators and relaxation methods in pde numerical solvers. Nature Machine Intelligence, 1–11 (2024)

    2024

  13. [21]

    Communications in Computational Physics 28(5) (2020)

    Jagtap, A.D., Karniadakis, G.E.: Extended physics-informed neural networks (xpinns): A generalized space-time domain decomposition based deep learn- ing framework for nonlinear partial differential equations. Communications in Computational Physics 28(5) (2020)

    2020

  14. [22]

    arXiv preprint arXiv:2307.12435 (2023)

    Basir, S., Senocak, I.: A generalized schwarz-type non-overlapping domain decomposition method using physics-constrained neural networks. arXiv preprint arXiv:2307.12435 (2023)

    2023 arXiv

  15. [23]

    Com- puter Methods in Applied Mechanics and Engineering 429, 117116 (2024)

    Dolean, V., Heinlein, A., Mishra, S., Moseley, B.: Multilevel domain decomposition-based architectures for physics-informed neural networks. Com- puter Methods in Applied Mechanics and Engineering 429, 117116 (2024)

    2024

  16. [24]

    In: Salakhutdinov, R., Kolter, Z., Heller, K., Weller, A., Oliver, N., Scarlett, J., Berkenkamp, F

    Mao, C., Lupoiu, R., Dai, T., Chen, M., Fan, J.: Towards general neural surro- gate solvers with specialized neural accelerators. In: Salakhutdinov, R., Kolter, Z., Heller, K., Weller, A., Oliver, N., Scarlett, J., Berkenkamp, F. (eds.) Pro- ceedings of the 41st International ...

    2024

  17. [25]

    In: International Conference on Machine Learning, pp

    Luz, I., Galun, M., Maron, H., Basri, R., Yavneh, I.: Learning algebraic multigrid using graph neural networks. In: International Conference on Machine Learning, pp. 6489–6499 (2020). PMLR 21

    2020

  18. [26]

    SIAM Journal on Scientific Computing 46(5), 123–144 (2024)

    Lerer, B., Ben-Yair, I., Treister, E.: Multigrid-augmented deep learning precondi- tioners for the helmholtz equation using compact implicit layers. SIAM Journal on Scientific Computing 46(5), 123–144 (2024)

    2024

  19. [27]

    SIAM Journal on Scientific Computing 43(5), 816–838 (2021)

    Heinlein, A., Klawonn, A., Lanser, M., Weber, J.: Combining machine learning and adaptive coarse spaces—a hybrid approach for robust feti-dp methods in three dimensions. SIAM Journal on Scientific Computing 43(5), 816–838 (2021)

    2021

  20. [28]

    SIAM Journal on Scientific Computing 41(6), 3887–3912 (2019)

    Heinlein, A., Klawonn, A., Lanser, M., Weber, J.: Machine learning in adaptive domain decomposition methods—predicting the geometric location of constraints. SIAM Journal on Scientific Computing 41(6), 3887–3912 (2019)

    2019

  21. [29]

    In: Domain Decomposition Methods in Science and Engineering XXVI, pp

    Heinlein, A., Klawonn, A., Lanser, M., Weber, J.: Predicting the geometric location of critical edges in adaptive gdsw overlapping domain decomposition methods using deep learning. In: Domain Decomposition Methods in Science and Engineering XXVI, pp. 307–315. Springer, ??? (2023)

    2023

  22. [30]

    SIAM Journal on Scientific Computing 45(3), 127–151 (2022)

    Azulay, Y., Treister, E.: Multigrid-augmented deep learning preconditioners for the helmholtz equation. SIAM Journal on Scientific Computing 45(3), 127–151 (2022)

    2022

  23. [31]

    arXiv preprint arXiv:2404.02493 (2024)

    Cui, C., Jiang, K., Shu, S.: A neural multigrid solver for helmholtz equations with high wavenumber and heterogeneous media. arXiv preprint arXiv:2404.02493 (2024)

    2024 arXiv

  24. [32]

    In: The Thirteenth International Conference on Learning Representations (2025)

    Xie, Y., Lv, M., Zhang, C.-S.: Mgcfnn: A neural multigrid solver with novel fourier neural network for high wave number helmholtz equations. In: The Thirteenth International Conference on Learning Representations (2025)

    2025

  25. [33]

    arXiv preprint arXiv:2303.02590 (2023)

    Knoke, T., Kinnewig, S., Beuchler, S., Demircan, A., Morgner, U., Wick, T.: Domain decomposition with neural network interface approximations for time- harmonic maxwell’s equations with different wave numbers. arXiv preprint arXiv:2303.02590 (2023)

    2023 arXiv

  26. [34]

    IEEE Antennas and Wireless Propagation Letters 23(10), 2905–2909 (2024)

    Piao, S., Gu, H., Wang, A., Qin, P.: A domain-adaptive physics-informed neu- ral network for inverse problems of maxwell’s equations in heterogeneous media. IEEE Antennas and Wireless Propagation Letters 23(10), 2905–2909 (2024)

    2024

  27. [35]

    Advances in Water Resources 34(9), 1124–1139 (2011)

    Dalcin, L.D., Paz, R.R., Kler, P.A., Cosimo, A.: Parallel distributed computing using python. Advances in Water Resources 34(9), 1124–1139 (2011)

    2011

  28. [36]

    SIAM, ??? (2015)

    Dolean, V., Jolivet, P., Nataf, F.: An Introduction to Domain Decomposition Methods: Algorithms, Theory, and Parallel Implementation. SIAM, ??? (2015)

    2015

  29. [37]

    Comptes Rendus

    Nataf, F., Xiang, H., Dolean, V.: A two level domain decomposition pre- conditioner based on local dirichlet-to-neumann maps. Comptes Rendus. Math´ ematique348(21-22), 1163–1167 (2010) 22

    2010

  30. [38]

    Comptes Rendus Mathematique 353(10), 959–963 (2015)

    Haferssas, R., Jolivet, P., Nataf, F.: A robust coarse space for optimized schwarz methods: Soras-geneo-2. Comptes Rendus Mathematique 353(10), 959–963 (2015)

    2015

  31. [39]

    Numerische Mathematik 85(2), 283–308 (2000)

    Farhat, C., Macedo, A., Lesoinne, M.: A two-level domain decomposition method for the iterative solution of high frequency exterior helmholtz problems. Numerische Mathematik 85(2), 283–308 (2000)

    2000

  32. [40]

    Future Generation Computer Systems 18(1), 69–78 (2001)

    Schenk, O., G¨ artner, K., Fichtner, W., Stricker, A.: Pardiso: a high-performance serial and parallel sparse linear solver in semiconductor device simulation. Future Generation Computer Systems 18(1), 69–78 (2001)

    2001

  33. [41]

    ACS Photonics 6(11), 3010–3016 (2019) 23

    Hughes, T.W., Williamson, I.A., Minkov, M., Fan, S.: Forward-mode differentia- tion of maxwell’s equations. ACS Photonics 6(11), 3010–3016 (2019) 23

    2019

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.