Pith. sign in

REVIEW 2 major objections 4 minor 1 cited by

Matrix-free Neural Preconditioner for the Dirac Operator in Lattice Gauge Theory

T0 review · 2 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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

Pith's one-line read A matrix-free neural network that rewrites the gauge field roughly halves Dirac solver iterations in the Schwinger model and transfers to new lattice sizes.

desk verdict A well-executed proof-of-concept for a matrix-free learned preconditioner, but the paper's own Section 4.3 shows a trivial analytic map does almost as well, which weakens the operator-learning claim. read the letter →

arxiv 2509.10378 v1 pith:GKJ4LBXK submitted 2025-09-12 hep-lat cs.LG

Yixuan Sun , Srinivas Eswar , Yin Lin , William Detmold , Phiala Shanahan , Xiaoye Li , Yang Liu , Prasanna Balaprakash This is my paper
classification hep-latcs.LG MSC 65F0865F1068T0781T25 PACS 11.15.Ha12.38.Gc
keywords preconditionerneuraloperatorlatticegaugetheoryWilson-Diracconjugategradientmatrix-freezero-shottransferSchwingermodel
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 proposes a way to precondition the large, ill-conditioned linear systems that dominate lattice gauge theory calculations, without ever forming or storing a matrix. The central idea is to train an operator-learning network to map a gauge field configuration U to a new configuration Ũ such that the Wilson-Dirac normal operator built from Ũ, M^{-1}=D†D[Ũ], acts as an approximate inverse of the original operator A=D†D[U]. Because M^{-1} has the same sparse stencil as A, applying it costs one matrix-vector product, and the whole pipeline is matrix-free from training through solver use. In the 2D Schwinger model the preconditioner reduces the condition number by roughly a factor of four and cuts CG iterations approximately in half, and a network trained on one lattice size works on other sizes and couplings without retraining. The learned mapping turns out to be nearly Ũ ≈ -½U, a simple analytic rule that itself already provides much of the acceleration.

What carries the argument

The load-bearing object is the map Gθ taking a U(1) gauge field U to a transformed field Ũ, together with the fixed rule that the preconditioner is M^{-1}=D†D[Ũ]. This ties the preconditioner to the same sparse stencil as A, so matrix-vector products are cheap and no matrix is ever assembled; two operator-learning architectures (a local convolutional network and a global spectral-mode network) are trained to produce Ũ. Training uses an unsupervised random-projection loss, Eq. (8): the squared norm of M^{-1}A v - v averaged over K random vectors v, which proxies for the Frobenius norm of M^{-1}A - I without computing spectra. A final piece is the identity A=D†D[U]=|1-2κH|², with H linear in U

What would settle it

Take the analytic rule Ũ = -½U, build M^{-1}=D†D[Ũ], and measure the condition number of M^{-1/2} A M^{-1/2} (or the CG iteration count) on SU(3) Wilson-Dirac operators at physical parameters in 4D. If the iteration reduction does not approach the roughly twofold gain seen in the Schwinger model—or if the spectrum of M^{-1}A develops clusters far from 1—then the learned mapping and the simple analytic form are artifacts of the Abelian 2D model, and the zero-shot transfer claim is restricted to that setting.

Watch

Extended reading notes

Core claim

The central claim is that an effective preconditioner for Wilson-Dirac normal equations can be learned as a function of the gauge field alone. The construction fixes M^{-1}=D†D[Ũ(x)] with Ũ(x)=Gθ(U(x)), so the preconditioner is the same discretized normal operator evaluated on a transformed gauge field and therefore shares the original matrix's sparse structure. Training is unsupervised and matrix-free: the loss is the average squared norm of (M^{-1}A - I) applied to K random vectors from an isotropic standard normal distribution. With this loss, a network trained on 1600 configurations of the two-flavor Schwinger model reduces the condition number of A by roughly a factor of four across L=8

Load-bearing premise

The load-bearing premise is that forcing the preconditioner to have exactly the same sparse structure as the original Wilson-Dirac normal operator—namely, evaluating D†D on a transformed gauge field—still leaves enough flexibility for M^{-1} to approximate A^{-1} well enough to accelerate conjugate gradient, and that the training loss, which checks the error only on random vectors, is a faithful enough proxy for that approximation.

Editorial extensions

If this is right

  • Preconditioners for lattice fermion solves can be generated on the fly from gauge fields with no matrix assembly or setup step, making the approach scalable to the very large operators (dimension greater than 10^8) encountered in lattice QCD.
  • A single training run on a small lattice can be amortized across many action parameters and lattice geometries: zero-shot volume transfer means no retraining for unseen configurations within the explored parameter ranges.
  • Since per CG iteration the neural preconditioner costs one matrix-vector product and no triangular solve, it avoids the per-iteration overhead of incomplete-factorization or even-odd preconditioners, at the cost of somewhat more iterations than the incomplete-factorization approach.
  • Using (M^{-1})^p with p=2, 3, or 4 as the preconditioner reduces iterations further (e.g., the FCN-based model at L=16 drops from 99 to 79–81 iterations), providing a tunable iteration-vs-per-step-cost knob.
  • The explicit analytic choice Ũ = -½U already gives most of the acceleration, implying that a useful preconditioner may be constructible without training at all in the Schwinger-model setting.

Reading between the lines

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

  • If the learned mapping generalizes to non-Abelian groups, the simple form Ũ ≈ -½U is unlikely to survive because link variables are unitary matrices rather than phases; a non-Abelian analog would more plausibly be a group-specific redefinition, so the preconditioner structure should be tailored to the gauge group.
  • The random-projection loss, which the paper acknowledges lacks theoretical grounding, could be replaced by a spectral proxy such as stochastic estimates of the condition number; the paper's own sensitivity study suggests K=128 is a pragmatic choice, and a principled objective might close the remaining gap to classical incomplete-factorization preconditioners.
  • The zero-shot volume transfer indicates the mapping depends mostly on local features of the gauge field, consistent with the local-network's superior transfer at L=8. A testable extension is that the preconditioner will continue to transfer to even larger lattices (e.g., L=128) with bounded loss of efficiency; if instead iteration counts grow as in the unpreconditioned case, the local-feature assu
  • One could apply the same construction to other fermion discretizations (staggered, domain-wall) by using the corresponding normal operator as the fixed stencil; this would test whether the learned mapping again reduces to a simple field redefinition or requires nontrivial correlations.
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

2 major / 4 minor

Summary. The paper proposes a matrix-free neural preconditioner for Wilson-Dirac normal equations in lattice gauge theory. Given a gauge configuration U, a neural network Gθ (FNO or FCN) outputs a transformed configuration Ũ; the preconditioner M^{-1}=D†D[Ũ] is then applied as a linear operator inside preconditioned CG. Training uses a random-projection loss that minimizes ||M^{-1}D†D v - v||² without ever forming matrices explicitly. Experiments in the 2D U(1) Schwinger model with two Wilson fermions show that, for lattices L=8–64 at κ=0.276, β=2.0, the preconditioner reduces the condition number and roughly halves CG iterations; a model trained at L=16 transfers zero-shot to other volumes and to modified κ and β. Section 4.3 reports that the learned transformation is approximately Ũ = -1/2 e^{iπ} U, and Table 4 shows that the manually constructed analytic preconditioner D†D[-1/2 U] gives nearly identical CG iteration counts to the trained networks.

Significance. If the claimed gains hold, the matrix-free construction is a useful step toward ML preconditioners for large-scale lattice calculations, since it avoids explicit matrix storage and setup/decomposition costs and offers zero-shot volume transfer. The paper is transparent about its limitations: the loss is an acknowledged proxy, the simple learned map is identified, and higher powers of M^{-1} are proposed but not fully analyzed. However, the central added value of the learned operator over the analytic transformation is not established. The paper's own Table 4 shows the neural network outperforms the analytic pointwise mapping by only 3–11 CG iterations out of 78–548, which is a small fraction of the total solve. This weakens the 'operator learning' claim: the evidence suggests the network has essentially rediscovered a simple scaling/phase relation. The framework is plausible and the empirical methodology is sound, but the current manuscript does not yet demonstrate that the neural component is necessary or advantageous relative to a parameter-free analytic preconditioner.

major comments (2)
  1. [Section 4.3, Table 4] This section reports that the learned maps satisfy |Ũ|≈1/2 and a phase difference of π, i.e., Ũ≈-1/2 U. Table 4 then shows that the manually constructed preconditioner D†D[-0.5U] gives CG iteration counts of 43±2, 104±7, 278±20 for L=8,16,32, while the trained FNO gives 40±2, 99±7, 267±21. The differences are only 3, 5, and 11 iterations, out of unpreconditioned counts of 78, 201, and 548. This is load-bearing: the paper's main contribution is that an operator-learning framework learns a useful preconditioner, but the evidence indicates the network has recovered a simple pointwise scaling/rotation that can be implemented without any training. The authors should (i) quantify how much Gθ deviates from -0.5U over the whole test set rather than a single L=64 instance (Fig. 7), (ii) test regimes where the small-κ expansion argument in §4.3 is expected to fail (e.g., κ further from criticality
  2. [Section 4.1 and Remark after Table 2] The abstract and Section 4.1 claim computational efficiency and reduced complexity, and the Remark states that the NN preconditioner avoids IChol's setup cost and triangular solve. However, no wall-clock timings are reported anywhere. Figure 4 shows only iteration counts, even though the text in §4.1 says 'we compare the average number of iterations and the time required to reach convergence.' Since CG per-iteration cost differs between one sparse matrix-vector product with M^{-1} and an IChol triangular solve, iteration counts alone do not establish a computational advantage. The paper should report setup time, per-iteration cost, and total solve time on the same hardware (A100) for unpreconditioned, IChol, even-odd, and NN-preconditioned solves, at least for L=32 and L=64. Without these data, the 'computationally efficient' claim is unsubstantiated.
minor comments (4)
  1. [Title page / affiliations] The affiliation 'Lawrance Berkeley National Laboratory' should be 'Lawrence Berkeley National Laboratory'.
  2. [Eq. (2)] The fermion action term is rendered as '1X f=0' due to a formatting issue; it should be a sum over flavor index f=0,1.
  3. [Eq. (5) and Fig. 6] The notation 'U(1)^{X×T×d} ⊂ C^{X×T×d}' is confusing: U(1) is the target space of each component, not a power of U(1). Clarify the spaces. In Fig. 6, the scatter plots would be more informative with a quantitative similarity metric such as a correlation coefficient or normalized distance.
  4. [Code availability] The text says the code is available at the given URL, but the footnotes state the repository 'will be publicly accessible upon acceptance.' These statements should be made consistent.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: training objective is a direct preconditioning objective evaluated on held-out/zero-shot data; the discovered simple mapping is a post-hoc control.

full rationale

The paper's derivation chain is not circular. The central object is a neural network Gθ mapping gauge configurations U to preconditioner-generating configurations U~, trained by minimizing the random-projection loss L(θ) in Eq. (8), which directly targets M^{-1}A ≈ I. All headline quantities (condition numbers, CG iterations) are measured on a held-out test set of 200 configurations, and the volume-transfer results apply the L=16 model to unseen sizes and parameters; these are not quantities fitted during training. The self-citation to Ref. [6] (Calì et al., which shares authors Lin and Shanahan) is contextual and methodological: the paper states 'Our work closely follows [6]' but the present contribution (matrix-free operator-learning preconditioner) is independently evaluated against IChol and even-odd baselines. No uniqueness theorem or ansatz is imported from the authors' prior work to force the conclusion. The post-hoc observation in Sec. 4.3 that the learned map approximates U~ = -0.5U, and the control in Table 4 showing the analytic preconditioner nearly matches the NN, is a critique of novelty/utility but not circularity: the simple map was not used to construct labels or training data, and the paper explicitly tests it as a check. The paper also openly notes the loss is a proxy and not theoretically grounded. Thus there is no step where a prediction reduces to an input by construction.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the specific lattice discretization, the structural constraint that the preconditioner is itself a Dirac normal operator on a modified gauge field, and the empirical assumption that the random-projection loss trains a useful preconditioner. No new physical entities are introduced. The fitted parameters are the network weights and the hyperparameters K and p.

free parameters (3)
  • Network weights (FNO/FCN) = learned during training
    All model parameters are fitted by minimizing the random-projection loss Eq. (8); the central claim depends on the specific learned mapping.
  • Number of random vectors K = 128
    Hyperparameter controlling the loss estimate; Appendix B shows performance plateaus above K=64.
  • Power p of preconditioner = 1,2,3,4
    Higher powers (M^{-1})^p densify the preconditioner; p=4 is fully dense for L=16 and reduces iterations further, at extra per-step cost.
assumptions (4)
  • domain assumption Wilson discretization of the Dirac operator and the 2D Schwinger action define the linear system.
    The paper's entire setup is this lattice action; if this discretization is not representative of target applications, the method's value is limited.
  • standard math D†D is Hermitian positive definite and CG is an appropriate solver.
    Standard result used implicitly.
  • domain assumption The random-projection loss is a sufficient proxy for the true objective of minimizing the condition number of the preconditioned system.
    The paper acknowledges this is not theoretically grounded (Section 3.3) and uses an empirical proxy.
  • domain assumption A neural operator trained on one lattice size learns a mapping that generalizes to other sizes and parameters.
    This is the zero-shot claim; tested empirically on L=8,16,32,64 with mixed success.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Matrix-free Neural Preconditioner for the Dirac Operator in Lattice Gauge Theory." pith.science (2026). https://pith.science/paper/GKJ4LBXK

@misc{pith2026250910378,
  author       = {Pith},
  title        = {Pith review of: Matrix-free Neural Preconditioner for the Dirac Operator in Lattice Gauge Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GKJ4LBXK}},
  note         = {Machine review of arXiv:2509.10378}
}
read the original abstract

Linear systems arise in generating samples and in calculating observables in lattice quantum chromodynamics~(QCD). Solving the Hermitian positive definite systems, which are sparse but ill-conditioned, involves using iterative methods, such as Conjugate Gradient (CG), which are time-consuming and computationally expensive. Preconditioners can effectively accelerate this process, with the state-of-the-art being multigrid preconditioners. However, constructing useful preconditioners can be challenging, adding additional computational overhead, especially in large linear systems. We propose a framework, leveraging operator learning techniques, to construct linear maps as effective preconditioners. The method in this work does not rely on explicit matrices from either the original linear systems or the produced preconditioners, allowing efficient model training and application in the CG solver. In the context of the Schwinger model U(1) gauge theory in 1+1 spacetime dimensions with two degenerate-mass fermions), this preconditioning scheme effectively decreases the condition number of the linear systems and approximately halves the number of iterations required for convergence in relevant parameter ranges. We further demonstrate the framework learns a general mapping dependent on the lattice structure which leads to zero-shot learning ability for the Dirac operators constructed from gauge field configurations of different sizes.

Figures

Figures reproduced from arXiv: 2509.10378 by the authors.

Figure 1
Figure 1. Matrix-free neural preconditioner for the discretized Dirac operator calculated from a given gauge configura [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. The sparsity pattern of the powers of M−1 for L = 16. Higher power shows denser structure. It becomes fully dense when p = 4. 4 Numerical Experiments We train models using lattice gauge configurations with various sizes and couplings ( [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Comparison of the average condition numbers among the unpreconditioned systems and neural network [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: (a) comparison of the average number of iterations required for a given tolerance in the CG solver for the [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: Zero-shot performance of NL16 models on gauges configurations of (a) L = 8 and (b) L = 32. The figures show the CG solve residual norm vs. the number of iterations, where the neural preconditioner effective reduces the number iterations required for convergence. In par…
Figure 6
Figure 6. Figure 6: Scatter plots of model weights learned from different lattice gauge fields. (a) FNO-based; (b) FCN-based. [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: Visualization of an L = 64 instance comparing the FNO-predicted gauge field U˜ µ(x) to the corresponding input configuration Uµ(x) across all lattice sites. (a) shows the magnitude of U˜ µ(x) (note that |Uµ(x)| = 1 for the input configuration, as Uµ(x) ∈ U(1)); (b) dis…
Figure 8
Figure 8. Figure 8: Validation loss (offset by 6.869) comparison among models using various number of random vectors for [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

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

  1. A novel gauge-equivariant neural-network architecture for preconditioners in lattice QCD

    hep-lat 2026-02 unverdicted novelty 8.0 of 10

    A novel gauge-equivariant neural-network preconditioner for the Dirac equation in lattice QCD mitigates critical slowing down and transfers to unseen configurations without retraining.

Reference graph

Works this paper leans on

63 extracted references · 11 linked inside Pith · cited by 1 Pith paper

  1. [1]

    Hadrons and nuclei.The European Physical Journal A, 55:1–27, 2019

    USQCD collaboration, William Detmold, Robert G Edwards, Jozef J Dudek, Michael Engelhardt, Huey-Wen Lin, Stefan Meinel, Kostas Orginos, and Phiala Shanahan. Hadrons and nuclei.The European Physical Journal A, 55:1–27, 2019

    2019

  2. [2]

    Review of particle physics.Progress of Theoretical and Experimental Physics, 2020(8):083C01, 2020

    Particle Data Group, PAea Zyla, RM Barnett, J Beringer, O Dahl, DA Dwyer, DE Groom, C-J Lin, KS Lugovsky, E Pianori, et al. Review of particle physics.Progress of Theoretical and Experimental Physics, 2020(8):083C01, 2020

    2020

  3. [3]

    Hot-dense lattice qcd.The European Physical Journal A, 55:1–11, 2019

    USQCD Collaboration, Alexei Bazavov, Frithjof Karsch, Swagato Mukherjee, and Peter Petreczky. Hot-dense lattice qcd.The European Physical Journal A, 55:1–11, 2019

    2019

  4. [4]

    Status and future perspectives for lattice gauge theory calculations to the exascale and beyond.The European Physical Journal A, 55:1–26, 2019

    Bálint Joó, Chulwoo Jung, Norman H Christ, William Detmold, Robert G Edwards, Martin Savage, and Phiala Shanahan. Status and future perspectives for lattice gauge theory calculations to the exascale and beyond.The European Physical Journal A, 55:1–26, 2019

    2019

  5. [5]

    SIAM, 2003

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

    2003

  6. [6]

    Hackett, Yin Lin, Phiala E

    Salvatore Calì, Daniel C. Hackett, Yin Lin, Phiala E. Shanahan, and Brian Xiao. Neural-network preconditioners for solving the Dirac equation in lattice gauge theory.Physical Review D, 107(3):034508, February 2023. 12

    2023

  7. [7]

    Brannick and K

    J. Brannick and K. Kahl. Bootstrap Algebraic Multigrid for the 2D Wilson Dirac system.SIAM Journal on Scientific Computing, 36(3):B321–B347, January 2014. Publisher: Society for Industrial and Applied Mathematics

    2014

  8. [8]

    Adaptive multigrid algorithm for lattice qcd.Physical review letters, 100(4):041601, 2008

    James Brannick, Richard C Brower, MA Clark, James C Osborn, and Claudio Rebbi. Adaptive multigrid algorithm for lattice qcd.Physical review letters, 100(4):041601, 2008

    2008

Show all 63 references
  1. [9]

    Adaptive multigrid algorithm for the lattice wilson-dirac operator.Physical review letters, 105(20):201602, 2010

    Ronald Babich, James Brannick, Richard C Brower, MA Clark, Thomas A Manteuffel, SF McCormick, JC Osborn, and C Rebbi. Adaptive multigrid algorithm for the lattice wilson-dirac operator.Physical review letters, 105(20):201602, 2010

    2010

  2. [10]

    Brower, M

    Richard C. Brower, M. A. Clark, Alexei Strelchenko, and Evan Weinberg. Multigrid algorithm for staggered lattice fermions.Phys. Rev. D, 97(11):114513, 2018

    2018

  3. [11]

    Brower, M

    Richard C. Brower, M. A. Clark, Dean Howarth, and Evan S. Weinberg. Multigrid for chiral lattice fermions: Domain wall.Phys. Rev. D, 102(9):094517, 2020

    2020

  4. [12]

    Babich, M

    R. Babich, M. A. Clark, B. Joo, G. Shi, R. C. Brower, and S. Gottlieb. Scaling lattice QCD beyond 100 GPUs. In International Conference for High Performance Computing, Networking, Storage and Analysis, 9 2011

    2011

  5. [13]

    M. A. Clark, Bálint Joó, Alexei Strelchenko, Michael Cheng, Arjun Gambhir, and Richard. C. Brower. Acceler- ating lattice QCD multigrid on GPUs using fine-grained parallelization. InInternational Conference for High Performance Computing, Networking, Storage and Analysis, 12 2016

    2016

  6. [14]

    Adaptive Aggregation-Based Domain Decomposition Multigrid for the Lattice Wilson–Dirac Operator.SIAM J

    Andreas Frommer, Karsten Kahl, Stefan Krieg, Björn Leder, and Matthias Rottmann. Adaptive Aggregation-Based Domain Decomposition Multigrid for the Lattice Wilson–Dirac Operator.SIAM J. Sci. Comput., 36(4):A1581– A1608, 2014

    2014

  7. [15]

    Comparison of Domain Wall Fermion Multigrid Methods

    Peter Boyle and Azusa Yamaguchi. Comparison of Domain Wall Fermion Multigrid Methods. 3 2021

    2021

  8. [16]

    Adaptive Aggregation-based Domain Decomposition Multigrid for Twisted Mass Fermions.Phys

    Constantia Alexandrou, Simone Bacchio, Jacob Finkenrath, Andreas Frommer, Karsten Kahl, and Matthias Rottmann. Adaptive Aggregation-based Domain Decomposition Multigrid for Twisted Mass Fermions.Phys. Rev. D, 94(11):114509, 2016

    2016

  9. [17]

    Multigrid Preconditioning for the Overlap Operator in Lattice QCD.Numer

    James Brannick, Andreas Frommer, Karsten Kahl, Björn Leder, Matthias Rottmann, and Artur Strebel. Multigrid Preconditioning for the Overlap Operator in Lattice QCD.Numer. Math., 132(3):463–490, 2016

    2016

  10. [18]

    Brower, Robert G

    Richard C. Brower, Robert G. Edwards, Claudio Rebbi, and Ettore Vicari. Projective multigrid for Wilson fermions.Nucl. Phys. B, 366:689–705, 1991

    1991

  11. [19]

    Arjan Hulsebos, Jan Smit, and Jeroen C. Vink. Multigrid inversion of the staggered fermion matrix.Nucl. Phys. B Proc. Suppl., 20:94–97, 1991

    1991

  12. [20]

    Accelerating multigrid with streaming chiral SVD for Wilson fermions in lattice QCD

    Travis Whyte, Andreas Stathopoulos, and Eloy Romero. Accelerating multigrid with streaming chiral SVD for Wilson fermions in lattice QCD. 5 2025

    2025

  13. [21]

    Multigrid low-mode averaging.Phys

    Roman Gruber, Tim Harris, and Marina Krstic Marinkovic. Multigrid low-mode averaging.Phys. Rev. D, 111(7):074508, 2025

    2025

  14. [22]

    Peter A. Boyle. Multiple right hand side multigrid for domain wall fermions with a multigrid preconditioned block conjugate gradient algorithm. 9 2024

    2024

  15. [23]

    Peter A. Boyle. Advances in algorithms for solvers and gauge generation. 1 2024

    2024

  16. [24]

    Deflated Multigrid Multilevel Monte Carlo.PoS, LAT- TICE2022:030, 2023

    Andreas Frommer and Gustavo Ramirez-Hidalgo. Deflated Multigrid Multilevel Monte Carlo.PoS, LAT- TICE2022:030, 2023

    2023

  17. [25]

    MRHS multigrid solver for Wilson-clover fermions.PoS, LATTICE2022:285, 2023

    Daniel Richtmann, Nils Meyer, and Tilo Wettig. MRHS multigrid solver for Wilson-clover fermions.PoS, LATTICE2022:285, 2023

    2023

  18. [26]

    Boyle, and Tilo Wettig

    Daniel Richtmann, Peter A. Boyle, and Tilo Wettig. Multigrid for Wilson Clover Fermions in Grid.PoS, LATTICE2018:032, 2019

    2019

  19. [27]

    Coarsest-level improvements in multigrid for lattice QCD on large-scale computers.Comput

    Jesus Espinoza-Valverde, Andreas Frommer, Gustavo Ramirez-Hidalgo, and Matthias Rottmann. Coarsest-level improvements in multigrid for lattice QCD on large-scale computers.Comput. Phys. Commun., 292:108869, 2023

    2023

  20. [28]

    SIAM, 1994

    Richard Barrett, Michael Berry, Tony F Chan, James Demmel, June Donato, Jack Dongarra, Victor Eijkhout, Roldan Pozo, Charles Romine, and Henk Van der V orst.Templates for the solution of linear systems: building blocks for iterative methods. SIAM, 1994

    1994

  21. [29]

    Neural incomplete factorization: learning preconditioners for the conjugate gradient method, October 2024

    Paul Häusner, Ozan Öktem, and Jens Sjölund. Neural incomplete factorization: learning preconditioners for the conjugate gradient method, October 2024. arXiv:2305.16368 [math] version: 3. 13

    2024 arXiv

  22. [30]

    Hicken, and Shaowu Pan

    Soha Yusuf, Jason E. Hicken, and Shaowu Pan. Constructing ILU Preconditioners for Advection-Dominated Problems Using Graph Neural Networks. InAIAA AVIATION FORUM AND ASCEND 2024, AIAA Aviation Forum and ASCEND co-located Conference Proceedings. American Institute of Aeronautic...

    2024

  23. [31]

    NeuralPCG: Learning Preconditioner for Solving Partial Differential Equations with Graph Neural Network

    Yichen Li, Tao Du, Peter Yichen Chen, and Wojciech Matusik. NeuralPCG: Learning Preconditioner for Solving Partial Differential Equations with Graph Neural Network. September 2022

    2022

  24. [32]

    Multigrid-augmented deep learning preconditioners for the Helmholtz equation, March 2022

    Yael Azulay and Eran Treister. Multigrid-augmented deep learning preconditioners for the Helmholtz equation, March 2022. arXiv:2203.11025 [cs, math]

    2022 arXiv

  25. [33]

    Deep Learning of Preconditioners for Conjugate Gradient Solvers in Urban Water Related Problems, June 2019

    Johannes Sappl, Laurent Seiler, Matthias Harders, and Wolfgang Rauch. Deep Learning of Preconditioners for Conjugate Gradient Solvers in Urban Water Related Problems, June 2019. arXiv:1906.06925 [cs, math, stat]

    2019 arXiv

  26. [34]

    Graph Neural Networks for Selection of Preconditioners and Krylov Solvers

    Ziyuan Tang, Hong Zhang, and Jie Chen. Graph Neural Networks for Selection of Preconditioners and Krylov Solvers

  27. [35]

    Preconditioning.Acta Numerica, 24:329–376, 2015

    Andrew J Wathen. Preconditioning.Acta Numerica, 24:329–376, 2015

    2015

  28. [36]

    SIAM, 1997

    Anne Greenbaum.Iterative methods for solving linear systems. SIAM, 1997

    1997

  29. [37]

    Number 13

    Henk A Van der V orst.Iterative Krylov methods for large linear systems. Number 13. Cambridge University Press, 2003

    2003

  30. [38]

    SIAM, 2014

    Maxim A Olshanskii and Eugene E Tyrtyshnikov.Iterative methods for linear systems: theory and applications. SIAM, 2014

    2014

  31. [39]

    Preconditioning discretizations of systems of partial differential equations

    Kent-Andre Mardal and Ragnar Winther. Preconditioning discretizations of systems of partial differential equations. Numerical Linear Algebra with Applications, 18(1):1–40, 2011

    2011

  32. [40]

    A note on preconditioners and scalar products in krylov subspace methods for self-adjoint problems in hilbert space.Electron

    Andreas Günnel, Roland Herzog, and Ekkehard Sachs. A note on preconditioners and scalar products in krylov subspace methods for self-adjoint problems in hilbert space.Electron. Trans. Numer. Anal, 41:13–20, 2014

    2014

  33. [41]

    SIAM, 2014

    Josef Málek and Zdenˇek Strakoš.Preconditioning and the conjugate gradient method in the context of solving PDEs. SIAM, 2014

    2014

  34. [42]

    Preconditioning techniques for large linear systems: a survey.Journal of computational Physics, 182(2):418–477, 2002

    Michele Benzi. Preconditioning techniques for large linear systems: a survey.Journal of computational Physics, 182(2):418–477, 2002

    2002

  35. [43]

    Graph Networks as a Universal Machine Learning Framework for Molecules and Crystals.Chemistry of Materials, 31(9):3564–3572, May 2019

    Chi Chen, Weike Ye, Yunxing Zuo, Chen Zheng, and Shyue Ping Ong. Graph Networks as a Universal Machine Learning Framework for Molecules and Crystals.Chemistry of Materials, 31(9):3564–3572, May 2019

    2019

  36. [44]

    Düben, Tim N

    Jan Ackmann, Peter D. Düben, Tim N. Palmer, and Piotr K. Smolarkiewicz. Machine-Learned Preconditioners for Linear Solvers in Geophysical Fluid Flows, October 2020. arXiv:2010.02866 [physics] version: 1

    2020

  37. [45]

    Graph neural preconditioners for iterative solutions of sparse linear systems, 2025

    Jie Chen. Graph neural preconditioners for iterative solutions of sparse linear systems, 2025

    2025

  38. [46]

    Gauge-equivariant neural networks as preconditioners in lattice QCD, February

    Christoph Lehner and Tilo Wettig. Gauge-equivariant neural networks as preconditioners in lattice QCD, February

  39. [47]

    G. Cybenko. Approximation by superpositions of a sigmoidal function.Mathematics of Control, Signals and Systems, 2(4):303–314, December 1989

    1989

  40. [48]

    Multilayer feedforward networks are universal approxi- mators.Neural Networks, 2(5):359–366, January 1989

    Kurt Hornik, Maxwell Stinchcombe, and Halbert White. Multilayer feedforward networks are universal approxi- mators.Neural Networks, 2(5):359–366, January 1989

    1989

  41. [49]

    Fourier Neural Operator for Parametric Partial Differential Equations, May 2021

    Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Fourier Neural Operator for Parametric Partial Differential Equations, May 2021. arXiv:2010.08895 [cs, math]

    2021 arXiv

  42. [50]

    Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators.Nature Machine Intelligence, 3(3):218–229, March 2021

    Lu Lu, Pengzhan Jin, Guofei Pang, Zhongqiang Zhang, and George Em Karniadakis. Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators.Nature Machine Intelligence, 3(3):218–229, March 2021

    2021

  43. [51]

    Neural Operator: Learning Maps Between Function Spaces, April 2023

    Nikola Kovachki, Zongyi Li, Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Neural Operator: Learning Maps Between Function Spaces, April 2023. arXiv:2108.08481 [cs, math]

    2023 arXiv

  44. [52]

    Patel, Nathaniel A

    Ravi G. Patel, Nathaniel A. Trask, Mitchell A. Wood, and Eric C. Cyr. A physics-informed operator regres- sion framework for extracting data-driven continuum models.Computer Methods in Applied Mechanics and Engineering, 373:113500, January 2021. arXiv: 2009.11992

    2021 arXiv

  45. [53]

    A physics-informed variational Deep- ONet for predicting the crack path in brittle materials.arXiv:2108.06905 [cs, math], September 2021

    Somdatta Goswami, Minglang Yin, Yue Yu, and George Karniadakis. A physics-informed variational Deep- ONet for predicting the crack path in brittle materials.arXiv:2108.06905 [cs, math], September 2021. arXiv: 2108.06905. 14

    2021 arXiv

  46. [54]

    DeepGraphONet: A Deep Graph Operator Network to Learn and Zero-Shot Transfer the Dynamic Response of Networked Systems.IEEE Systems Journal, 17(3):4360– 4370, September 2023

    Yixuan Sun, Christian Moya, Guang Lin, and Meng Yue. DeepGraphONet: A Deep Graph Operator Network to Learn and Zero-Shot Transfer the Dynamic Response of Networked Systems.IEEE Systems Journal, 17(3):4360– 4370, September 2023. Conference Name: IEEE Systems Journal

    2023

  47. [55]

    A Mathematical Guide to Operator Learning, December 2023

    Nicolas Boullé and Alex Townsend. A Mathematical Guide to Operator Learning, December 2023. arXiv:2312.14688 [cs, math]

    2023 arXiv

  48. [56]

    Fully Convolutional Networks for Semantic Segmentation

    Jonathan Long, Evan Shelhamer, and Trevor Darrell. Fully Convolutional Networks for Semantic Segmentation. arXiv:1411.4038 [cs], March 2015. arXiv: 1411.4038

    2015 arXiv

  49. [57]

    Confinement of quarks.Physical review D, 10(8):2445, 1974

    Kenneth G Wilson. Confinement of quarks.Physical review D, 10(8):2445, 1974

    1974

  50. [58]

    CASK: A Gauge Covariant Transformer for Lattice Gauge Theory

    Yuki Nagai, Hiroshi Ohno, and Akio Tomiya. CASK: A Gauge Covariant Transformer for Lattice Gauge Theory. PoS, LATTICE2024:030, 2025

    2025

  51. [59]

    Differentiating through linear solvers.arXiv preprint arXiv:2404.17039, 2024

    Paul Hovland and Jan Hückelheim. Differentiating through linear solvers.arXiv preprint arXiv:2404.17039, 2024

    2024 arXiv

  52. [60]

    A taxonomy of automatic differentiation pitfalls.Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery, 14(6):e1555, 2024

    Jan Hückelheim, Harshitha Menon, William Moses, Bruce Christianson, Paul Hovland, and Laurent Hascoët. A taxonomy of automatic differentiation pitfalls.Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery, 14(6):e1555, 2024

    2024

  53. [61]

    Scaling test of fermion actions in the schwinger model

    N Christian, K Jansen, K Nagai, and B Pollakowski. Scaling test of fermion actions in the schwinger model. Nuclear Physics B, 739(1-2):60–84, 2006

    2006

  54. [62]

    Cambridge University Press, 2003

    Jan Smit.Introduction to Quantum Fields on a Lattice, volume 15. Cambridge University Press, 2003. A Model architecture and training details We use the same set of hyperparameters to train all models and implement early stopping with patience of 50 epochs. The specifications o...

    2003

  55. [2023]

    arXiv:2302.05419 [hep-lat]

    arXiv

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.