Pith. sign in

REVIEW 5 major objections 6 minor 45 references

Momentum-Accelerated Richardson(m) and Their Multilevel Neural Solvers

T0 review · 5 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read A meta-network learns the weights of long-step Richardson iteration, and momentum, SSOR preconditioning, and multilevel embedding make the resulting solvers converge faster than optimal-weight Chebyshev iteration without eigenvalue…

desk verdict The single-level learned Richardson weights are a genuine, well-tested contribution, but the abstract overclaims multilevel parameter-independence and computational complexity that the paper's own tables contradict. read the letter →

arxiv 2412.08076 v1 pith:S4UBJEZY submitted 2024-12-11 math.NA cs.NA

Zhen Wang , Yun Liu , Chen Cui , Shi Shu This is my paper · ORCID
classification math.NAcs.NA MSC 65F1065N5568T07
keywords Richardson(m)iterationneuralsolvermomentumaccelerationSSORpreconditioningmultilevelFourieranisotropicdiffusionHelmholtzequation
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's aim is to replace hand-tuned or eigenvalue-derived weights in the long-step Richardson iteration with weights predicted by a small neural network, and to show that the resulting solvers beat classical Chebyshev acceleration. It introduces Richardson(m)-NS, where one outer iteration bundles m weighted inner iterations and a meta-network maps PDE coefficients to the weights. Adding momentum (especially a Nesterov-style correction), SSOR preconditioning, and then embedding the solver inside two multilevel neural solvers yields, in the paper's experiments, faster convergence and lower computational cost than optimal-weight Chebyshev iteration and Chebyshev semi-iteration. The multilevel versions, Richardson(m)-FNS for anisotropic diffusion and NAG-Richardson(m)-WANS for Helmholtz, are the paper's answer to the single-level solvers' sensitivity to PDE parameters and grid size. If true, the practical consequence is that expensive eigenvalue estimation can be replaced by a cheap learned map from PDE parameters to iteration weights.

What carries the argument

The central object is the Richardson(m) iteration, which treats m consecutive weighted updates $u_k = (I-\omega_i A)u_{k-1}+\omega_i f$ as one outer iteration with polynomial operator $T_m(A,\omega)=\prod_{i=1}^m (I-\omega_i A)$. The load-bearing mechanism is a meta-network $\mathrm{Meta}(\theta;\mu)$ that maps PDE parameters $\mu$ (anisotropy strength and angle, transformed to $\lg\varepsilon$) directly to the weight vector $\omega$, replacing eigenvalue-based weight formulas. The paper then layers on momentum variables ($\alpha_i$), a Nesterov-style lookahead (with an extension $\tilde{\alpha}_i$), and an SSOR preconditioner for the single-level variants, and an alternating-optimization training algorithm when the solver is embedded as a smoother in the FNS and WANS multilevel architectures.

What would settle it

A direct test: train Richardson(3)-NS on epsilon drawn uniformly from 1e-6 to 1 and theta from 0 to pi at N=64, as in the paper, then freeze the network and measure the relative residual after a fixed number of iterations on N=512 anisotropic systems with the same parameter ranges. If the iteration count to reach 1e-6 grows at the same rate as the paper's own Table 4 (27 to 977 iterations), the claimed transfer across grid sizes fails, and the multilevel results cannot be credited to the learned weights.

Watch

Extended reading notes

Core claim

The paper's central claim is that the weights of the m-step Richardson iteration do not need to come from the spectrum of A: a meta-network trained on random PDE parameters can predict weights that make Richardson(m) converge nearly as fast as the Chebyshev iteration with optimal weights, and faster than Chebyshev semi-iteration, while avoiding eigenvalue estimates. The claim is made in two stages. At the single level, momentum and preconditioning variants, MOM-Richardson(m), NAG-Richardson(m), and NAGex-Richardson(m), are proposed, with the SSOR-preconditioned NAGex version reported to be nearly 20 times faster than plain Richardson(3)-NS on the test problem. At the multilevel level, replacing the fixed smoother in FNS with Richardson(m)-NS, and the Chebyshev semi-iterative smoother in WANS with NAG-Richardson(m), is claimed to overcome the strong dependence on PDE parameters and grid size: Richardson(10)-FNS converges in 4 to 10 iterations for anisotropy strengths from 1 down to 1e-8 and all tested angles, and NAG-Richardson(5)-WANS preconditioning FGMRES gives lower iteration counts than WANS with Chebyshev semi-iteration up to N=4096. The paper presents these as numerical findings, with the theoretical backbone being the known faster convergence of long-step Richardson and the Chebyshev optimal weights as benchmark.

Load-bearing premise

The assumption that carries the argument is that the meta-network, trained only on 64x64 grids with anisotropy parameters drawn uniformly, predicts Richardson(m) weights that remain effective on finer grids (up to N=512 or N=4096 in the multilevel tests) and at Helmholtz wavenumbers; if the learned weights do not transfer across discretization scale, the claimed multilevel robustness and parameter-independent convergence rate collapse.

Editorial extensions

If this is right

  • If the learned weights generalize as reported, solving a parametric family of linear systems no longer requires per-matrix eigenvalue estimates: the same small network supplies the Richardson weights for new parameter values.
  • The reported single-level speedups imply that momentum and SSOR preconditioning can be composed with learned long-step weights: in the m=3 anisotropic test, NAGex-Richardson(3)-NS with SSOR needs about 20 iterations where Richardson(3)-NS needs hundreds.
  • Embedding learned Richardson smoothers in FNS gives a concrete recipe for anisotropic problems: Richardson(10)-FNS converges in 4 to 10 iterations across the tested anisotropy strengths and directions, whereas the original FNS can diverge or need many more.
  • For Helmholtz systems, using NAG-Richardson(m) as a WANS smoother removes the power-method eigenvalue estimation and keeps FGMRES iteration counts competitive or better than Chebyshev semi-iteration at all tested grid sizes up to N=4096.

Reading between the lines

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

  • The paper's tables show that the single-level learned weights degrade sharply as N grows (e.g., 27 iterations at N=64 versus 977 at N=512 for SSOR-preconditioned NAGex); a natural inference is that the multilevel wrapping, not the learned weights alone, is what earns grid robustness, and that testing the meta-network on unstructured meshes would likely require retraining.
  • Because the FNS and WANS gains are demonstrated on structured-grid model problems, the cleanest extension would be to check the same alternating-trained smoothers on non-uniform grids and on advection-diffusion-reaction equations, where the paper itself says future work lies.
  • The WANS comparison at large N shows a modest iteration reduction (336.6 vs 344.3 at N=4096); an editor's inference is that the practical advantage there may be the removed eigenvalue/power-method cost rather than dramatically faster convergence.
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

5 major / 6 minor

Summary. The manuscript proposes a family of learned long-step Richardson iterative solvers, termed Richardson(m)-NS, together with momentum-accelerated variants and SSOR preconditioning, and integrates them as smoothers into the multilevel neural solvers FNS and WANS. The central claims are that these solvers achieve faster convergence and lower computational complexity than Chebyshev iteration and Chebyshev semi-iteration, and that the multilevel versions Richardson(m)-FNS and NAG-Richardson(m)-WANS overcome the grid- and parameter-dependence of single-level solvers, with the WANS version having a parameter-independent convergence rate. These claims are supported by tables of iteration counts from numerical experiments on anisotropic diffusion and high-wavenumber Helmholtz problems.

Significance. The algorithmic contribution is interesting and the experiments are broad in their coverage of PDE parameters. The Richardson(m)-NS framework with momentum and preconditioning is clearly described, and the alternating optimization training for the multilevel FNS variant is a practically useful technique. If the performance were reproducible and the claims were restricted to the settings actually tested, the paper would be a useful addition to the neural-solver literature. However, the headline claims of parameter independence and lower computational complexity are not supported by the data presented, and the multilevel scaling results in the paper's own tables contradict the stated robustness. The current form therefore overstates the significance of the experiments.

major comments (5)
  1. [Abstract and §3.2, Table 9] The abstract and §3.2 state that NAG-Richardson(m)-WANS has a 'parameter-independent convergence rate,' but Table 9 shows FGMRES iteration counts growing from 8.8 at N=128 to 336.6 at N=4096, i.e., roughly a 38-fold increase over a 32-fold grid-size increase and an approximately linear growth at the largest sizes. This is not parameter-independent, and the improvement over the original WANS at N=4096 is marginal (336.6 vs. 344.3 iterations). The claim should be removed or replaced with a precise statement about the range and behavior actually observed.
  2. [Introduction, §3.1, Table 8] The Introduction asserts that Richardson(m)-FNS is 'mildly dependent on grid size,' but Table 8 shows iteration counts rising from 12 at N=256 to 83 at N=512, a six-fold increase over a single grid doubling. The single-level solver in Table 4 grows from 27 iterations at N=64 to 977 at N=512. The data therefore do not support the assertion that the multilevel versions 'effectively overcome the drawback of single-level methods' with respect to grid-size dependence. The authors should either report experiments at larger N that demonstrate a bounded or slowly growing iteration count, or substantially weaken this claim.
  3. [Abstract and §2.3] The claim of 'lower computational complexity' relative to Chebyshev iteration and Chebyshev semi-iteration is not established by the evidence. Only iteration counts are reported; no runtimes, flop counts, or matrix-vector product counts are given. The Chebyshev iteration in Table 2 is based on Eq. (6), which requires knowledge of λmax and λmin, but the computational cost of obtaining these (for example, via the power method or a dense eigensolver) is not accounted for in the comparison. Per-iteration costs also differ among the proposed variants because of momentum terms, the SSOR preconditioner, and the Meta-omega network evaluation. The paper should either provide a complexity analysis that includes these costs or restrict the claims to iteration counts.
  4. [§2.2, Eq. (15), §2.3, §3.1] The training loss L2 (Eq. (15)) is the same relative-residual metric used as the stopping criterion in all reported tables. This means the reported iteration counts measure the objective for which Meta-omega was trained, and the test parameters in the single-level experiments are within the training distribution (ε ∈ [10^-6,1], θ ∈ [0,π]) at the training grid size N=64. The degradation observed at N=512 in Tables 4 and 8 is an out-of-distribution extrapolation, and the Meta-omega network has no input that encodes grid size or spectral information beyond (lg ε, θ). To support claims of robustness or parameter independence, the authors should test on parameters outside the training distribution and report statistics over many random samples rather than single deterministic points.
  5. [§2.3, Table 2] The comparison to 'Chebyshev iteration (m=3 in (6))' is not apples-to-apples because Eq. (6) requires the exact extreme eigenvalues λmax and λmin, and the text itself notes that computing λmin is costly. The paper does not state how these reference results were obtained. If the eigenvalues were computed exactly, the reference method is an oracle that does not correspond to an implementable solver with the same per-iteration cost; otherwise, the reader cannot judge whether the reported iteration savings translate into real computational savings. The authors should describe the reference computation and account for its cost.
minor comments (6)
  1. [Abstract] The phrase 'This paper first introduce' should be 'This paper first introduces'; there are several similar grammar errors throughout.
  2. [Introduction] The sentence 'we obtain a Richardson(m)-FNS that is independent of PDE parameters and mildly dependent on grid size' is imprecise; Table 8 does not support 'mildly dependent.'
  3. [§3.2] There is a typo in 'x− and y− directionss' and the spacing in 'W ANS' is inconsistent across the text and tables.
  4. [Conclusions] The phrase 'preprocessing techniques' in the Conclusions should read 'preconditioning techniques.'
  5. [§2.1] The description of m as 'continuous m (with m ≥ 1)' is odd, since m is an integer number of inner iterations.
  6. [Introduction and §2.3] The claim that NAGex-Richardson(3)-NS with SSOR achieves 'nearly a 20 times speedup' is not consistent with Table 2, where the iteration ratio relative to Richardson(3)-NS is about 77 at ε=10^-6 and about 28 at ε=1; please re-check the calculation and state the baseline clearly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the trained weights are evaluated against external baselines, and the residual-based loss is a standard training objective rather than a disguised prediction.

full rationale

Walking the paper's derivation chain, I find no step that reduces a claimed prediction to its own inputs by construction. The Meta-omega network is trained by minimizing the relative residual loss L2 (Eq. 15) over a distribution of PDE parameters, and the same relative residual is later used as the stopping criterion; this is ordinary supervised training and evaluation, not a fitted parameter being renamed as a prediction. The central comparisons to Chebyshev iteration, Chebyshev semi-iteration, and the original FNS/WANS are external baselines computed in the same tables, so the reported gains do not follow from the training objective alone. The multilevel FNS and WANS architectures are imported from the authors' prior work, but they are used as modifiable components, the paper replaces their smoothers, and it reports direct numerical comparisons rather than invoking an unverified uniqueness theorem; this is self-citation but not load-bearing circularity. The abstract's 'parameter-independent convergence rate' is not well supported by Table 9 (iteration counts rise from 8.8 at N=128 to 336.6 at N=4096), and Table 4 shows strong grid-size dependence for the single-level solver, but an unsupported empirical claim is a correctness and robustness concern, not a circularity. The derivation of the iterative schemes from optimization methods is explicit and independent of the learned parameters. Therefore the paper is not circular in any of the defined senses.

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

The paper introduces no new physical entities. The learned step sizes and momentum coefficients are free parameters, and the central assumption is the transferability of the meta-network from its training distribution to larger grids and different PDE regimes.

free parameters (5)
  • step sizes omega_i for Richardson(m) = not reported (learned)
    Outputs of the Meta-omega FNN; trained to minimize the relative residual loss on anisotropic diffusion systems. Chebyshev theory (Eq. 6) would fix them from eigenvalues, but here they are fit to data.
  • momentum coefficients alpha_i and NAG-extension tilde_alpha_i = not reported (learned)
    In MOM-Richardson(m), NAG-Richardson(m), and NAGex variants, momentum and extrapolation coefficients are also meta-network outputs, fit on the same training set.
  • initial smoother weight omega^(0)=0.5 in Richardson(m)-FNS = 0.5
    Hand-chosen initial value for the frozen Meta-omega during the first phase of alternating optimization (Remark 3).
  • smoothing iteration count M=10 = 10
    Fixed number of smoothing passes per outer iteration in Richardson(m)-FNS; set by hand and affects all reported iteration counts.
  • epoch schedule Ninterval in alternating optimization = not specified
    Chosen based on when training loss stops decreasing (Remark 3); no concrete value or rule is given.
assumptions (4)
  • domain assumption The coefficient matrix A for the anisotropic diffusion problem is symmetric positive definite (SPD).
    The Chebyshev optimal weight formula (Eq. 6) and the equivalence to gradient descent (Eq. 11) require A SPD; for Helmholtz the matrix is indefinite, which is why the authors switch to WANS.
  • domain assumption The training data distribution (epsilon uniform in [10^-6,1], theta uniform in [0,pi], N=64) is representative of the test cases, including larger grids and all wavenumbers tested.
    No out-of-distribution test is reported; the solver's weights are fit on this distribution, and the paper's central empirical claim depends on this transfer.
  • ad hoc to paper Evaluating the Meta-omega FNN is computationally negligible compared with matrix-vector products, so iteration count alone determines computational complexity.
    The abstract claims lower computational complexity, but only iterations are measured; wall-clock time and training cost are not reported.
  • domain assumption For the Helmholtz experiments, the discretization (Eq. 21) with at least 10 points per wavelength yields linear systems whose AMG smoothing can be learned by the same architecture.
    The WANS data is taken from [12] and not described in this paper; the claim of a parameter-independent convergence rate relies on that setup.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Momentum-Accelerated Richardson(m) and Their Multilevel Neural Solvers." pith.science (2026). https://pith.science/paper/S4UBJEZY

@misc{pith2026241208076,
  author       = {Pith},
  title        = {Pith review of: Momentum-Accelerated Richardson(m) and Their Multilevel Neural Solvers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S4UBJEZY}},
  note         = {Machine review of arXiv:2412.08076}
}
read the original abstract

Recently, designing neural solvers for large-scale linear systems of equations has emerged as a promising approach in scientific and engineering computing. This paper first introduce the Richardson(m) neural solver by employing a meta network to predict the weights of the long-step Richardson iterative method. Next, by incorporating momentum and preconditioning techniques, we further enhance convergence. Numerical experiments on anisotropic second-order elliptic equations demonstrate that these new solvers achieve faster convergence and lower computational complexity compared to both the Chebyshev iterative method with optimal weights and the Chebyshev semi-iteration method. To address the strong dependence of the aforementioned single-level neural solvers on PDE parameters and grid size, we integrate them with two multilevel neural solvers developed in recent years. Using alternating optimization techniques, we construct Richardson(m)-FNS for anisotropic equations and NAG-Richardson(m)-WANS for the Helmholtz equation. Numerical experiments show that these two multilevel neural solvers effectively overcome the drawback of single-level methods, providing better robustness and computational efficiency.

Figures

Figures reproduced from arXiv: 2412.08076 by the authors.

Figure 1
Figure 1. During the validation or testing process, we use the relative residual ∥f − Auk∥/∥f∥ to measure the approximation accuracy of the numerical solution to the true solution. For solvers with converging iterative processes, the iteration will stop if the relative residual is less than 10−6 . FC+Tanh FC+Tanh FC [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Computational flow of Richardson(m)-FNS. [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Network architecture of Meta-T and Meta-Λ. ˜ 11 [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Computational flow of NAG-Richardson(m)-WANS. [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

45 extracted references · 31 canonical work pages

  1. [1]

    Quarteroni, A

    A. Quarteroni, A. Valli, Numerical approximation of partial differential equations, Vol. 23, Springer Science & Business Media, 2008

    2008

  2. [2]

    Saad, Iterative methods for sparse linear systems, SIAM, 2003

    Y. Saad, Iterative methods for sparse linear systems, SIAM, 2003

    2003

  3. [3]

    Nesterov, Introductory lectures on convex optimization: A basic course, Vol

    Y. Nesterov, Introductory lectures on convex optimization: A basic course, Vol. 87, Springer Science & Business Media, 2003

    work page 2003

  4. [4]

    Nesterov, A method for unconstrained convex minimization problem with the rate of convergence, Doklady AN SSSR 269 (1983) 543–547

    Y. Nesterov, A method for unconstrained convex minimization problem with the rate of convergence, Doklady AN SSSR 269 (1983) 543–547

    work page 1983

  5. [5]

    Trottenberg, C

    U. Trottenberg, C. W. Oosterlee, A. Schuller, Multigrid, Elsevier, 2000

    work page 2000

  6. [6]

    W. E, C. Ma, L. Wu, Barron spaces and the compositional function spaces for neural network models, arXiv preprint arXiv:1906.08039 (2019)

    arXiv 2019

  7. [7]

    L. Lu, R. Pestourie, W. Yao, Z. Wang, F. Verdugo, S. G. Johnson, Physics-informed neural networks with hard constraints for inverse design, SIAM Journal on Scientific Computing 43 (6) (2021) B1105–B1132. 15

    work page 2021

  8. [8]

    Raissi, P

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

    work page 2019

Show all 45 references
  1. [9]

    Yu, et al., The deep ritz method: a deep learning-based numerical algorithm for solving variational problems, Communications in Mathematics and Statistics 6 (1) (2018) 1–12

    B. Yu, et al., The deep ritz method: a deep learning-based numerical algorithm for solving variational problems, Communications in Mathematics and Statistics 6 (1) (2018) 1–12

    2018

  2. [10]

    L. Lu, P. Jin, G. Pang, Z. Zhang, G. E. Karniadakis, Learning nonlinear operators via deeponet based on the universal approximation theorem of operators, Nature machine intelligence 3 (3) (2021) 218–229

    2021

  3. [11]

    Z. Li, N. B. Kovachki, K. Azizzadenesheli, B. liu, K. Bhattacharya, A. Stuart, A. Anandkumar, Fourier neural operator for parametric partial differential equations, International Conference on Learning Representations (2021)

    2021

  4. [12]

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

    2024 arXiv

  5. [13]

    Kaneda, O

    A. Kaneda, O. Akar, J. Chen, V. Kala, D. Hyde, J. Teran, A deep gradient correction method for iteratively solving linear systems, arXiv preprint arXiv:2205.10763 (2022)

    2022 arXiv

  6. [14]

    Trifonov, A

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

    2024 arXiv

  7. [15]

    Katrutsa, T

    A. Katrutsa, T. Daulbaev, I. Oseledets, Black-box learning of multigrid parameters, Journal of Computational and Applied Mathematics 368 (2020) 112524

    2020

  8. [16]

    Huang, R

    R. Huang, R. Li, Y. Xi, Learning optimal multigrid smoothers via neural networks, SIAM Journal on Scientific Computing 0 (0) (2022) S199–S225

    2022

  9. [17]

    Y. Chen, B. Dong, J. Xu, Meta-mgnet: Meta multigrid networks for solving parameterized partial differential equations, Journal of Computational Physics 455 (2022) 110996

    2022

  10. [18]

    Greenfeld, M

    D. Greenfeld, M. Galun, R. Basri, I. Yavneh, R. Kimmel, Learning to optimize multigrid pde solvers, in: International Conference on Machine Learning, Vol. 97, 2019, pp. 2415–2423

    2019

  11. [19]

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

    2020

  12. [20]

    Kopaniˇ c´ akov´ a, G

    A. Kopaniˇ c´ akov´ a, G. E. Karniadakis, Deeponet based preconditioning strategies for solving parametric linear systems of equations, arXiv preprint arXiv:2401.02016 (2024)

    2024 arXiv

  13. [21]

    Y. Liu, S. Fu, Y. Zhou, C. Ye, E. T. Chung, Learning a generalized multiscale prolongation operator, arXiv:2410.06832 (Oct. 2024)

    2024 arXiv

  14. [22]

    Taghibakhshi, S

    A. Taghibakhshi, S. MacLachlan, L. Olson, M. West, Optimization-based algebraic multigrid coarsening using reinforcement learning, Advances in Neural Information Processing Systems 34 (2021) 12129–12140. 16

    2021

  15. [23]

    Caldana, P

    M. Caldana, P. F. Antonietti, et al., A deep learning algorithm to accelerate algebraic multi- grid methods in finite element solvers of 3d elliptic pdes, Computers & Mathematics with Applications 167 (2024) 217–231

    2024

  16. [24]

    H. Zou, X. Xu, C.-S. Zhang, Z. Mo, Autoamg ( θ): An auto-tuned amg method based on deep learning for strong threshold, Communications in Computational Physics 36 (1) (2024) 200–220

    2024

  17. [25]

    Klawonn, M

    A. Klawonn, M. Lanser, J. Weber, Learning adaptive coarse basis functions of feti-dp, Journal of Computational Physics 496 (2024) 112587

    2024

  18. [26]

    Taghibakhshi, N

    A. Taghibakhshi, N. Nytko, T. Zaman, S. MacLachlan, L. Olson, M. West, Learning inter- face conditions in domain decomposition solvers, Advances in Neural Information Processing Systems 35 (2022) 7222–7235

    2022

  19. [27]

    Knoke, S

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

    2023 arXiv

  20. [28]

    Taghibakhshi, N

    A. Taghibakhshi, N. Nytko, T. U. Zaman, S. MacLachlan, L. Olson, M. West, Mg-gnn: Multigrid graph neural networks for learning multilevel domain decomposition methods, Pro- ceedings of the 40th International Conference on Machine Learning 202 (2023) 33381–33395

    2023

  21. [29]

    Zhang, A

    E. Zhang, A. Kahana, A. Kopaniˇ c´ akov´ a, E. Turkel, R. Ranade, J. Pathak, G. E. Karniadakis, Blending neural operators and relaxation methods in pde numerical solvers, Nature Machine Intelligence (2024) 2522–5839

    2024

  22. [30]

    C. Cui, K. Jiang, Y. Liu, S. Shu, Fourier neural solver for large sparse linear algebraic systems, Mathematics 10 (21) (2022) 4014

    2022

  23. [31]

    C. Cui, K. Jiang, Y. Liu, S. Shu, Convergence framework of deep learning-based hybrid iterative methods and the application to designing a fourier neural solver for parametric pdes, arXiv preprint arXiv:2408.08540 (2024)

    2024 arXiv

  24. [32]

    J. Hu, P. Jin, A hybrid iterative method based on mionet for pdes: Theory and numerical examples, arXiv preprint arXiv:2402.07156 (2024)

    2024 arXiv

  25. [33]

    Y. Xie, M. Lv, C. Zhang, Mgcnn: a learnable multigrid solver for linear pdes on structured grids, arXiv preprint arXiv:2312.11093 (2023)

    2023 arXiv

  26. [34]

    Rudikov, V

    A. Rudikov, V. Fanaskov, E. Muravleva, Y. M. Laevsky, I. Oseledets, Neural operators meet conjugate gradients: The fcg-no method for efficient pde solving, arXiv preprint arXiv:2402.05598 (2024)

    2024 arXiv

  27. [35]

    Rahaman, A

    N. Rahaman, A. Baratin, D. Arpit, F. Draxler, M. Lin, F. Hamprecht, Y. Bengio, A. Courville, On the spectral bias of neural networks, in: International Conference on Machine Learning, PMLR, 2019, pp. 5301–5310

    2019

  28. [36]

    Q. Hong, Q. Tan, J. W. Siegel, J. Xu, On the activation function dependence of the spectral bias of neural networks, arXiv preprint arXiv:2208.04924 (2022). 17

    2022 arXiv

  29. [37]

    Z.-Q. J. Xu, Y. Zhang, T. Luo, Y. Xiao, Z. Ma, Frequency principle: Fourier analysis sheds light on deep neural networks, Communications in Computational Physics 28 (5) (2020) 1746– 1767

    2020

  30. [38]

    Grimmer, Provably faster gradient descent via long steps, SIAM Journal on Optimization 34 (3) (2024) 2588–2608

    B. Grimmer, Provably faster gradient descent via long steps, SIAM Journal on Optimization 34 (3) (2024) 2588–2608

    2024

  31. [39]

    G. H. Golub, R. S. Varga, Chebyshev semi-iterative methods, successive overrelaxation itera- tive methods, and second order richardson iterative methods, Numerische Mathematik 3 (1) (1961) 157–168

    1961

  32. [40]

    Axelsson, P

    O. Axelsson, P. S. Vassilevski, Algebraic multilevel preconditioning methods, ii, SIAM Journal on Numerical Analysis 27 (6) (1990) 1569–1590

    1990

  33. [41]

    J. C. Mason, D. C. Handscomb, Chebyshev polynomials, CRC press, 2002

    2002

  34. [42]

    Adams, M

    M. Adams, M. Brezina, J. Hu, R. Tuminaro, Parallel multigrid smoothing: polynomial versus gauss–seidel, Journal of Computational Physics 188 (2) (2003) 593–610

    2003

  35. [43]

    Qian, On the momentum term in gradient descent learning algorithms, Neural networks 12 (1) (1999) 145–151

    N. Qian, On the momentum term in gradient descent learning algorithms, Neural networks 12 (1) (1999) 145–151

    1999

  36. [44]

    Botev, G

    A. Botev, G. Lever, D. Barber, Nesterov’s accelerated gradient and momentum as approxima- tions to regularised update descent, 2017 International Joint Conference on Neural Networks (IJCNN) (2016) 1899–1903

    2016

  37. [45]

    Treister, E

    E. Treister, E. Haber, A multigrid solver to the helmholtz equation with a point source based on travel time and amplitude, Numerical Linear Algebra with Applications 26 (1) (2019) e2206. 18

    2019

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.