Learning-Guided Integration Contours Construction for Fast Large-Scale Generalized Eigensolvers

Hong Wang; Lei Liu; Yeqiu Chen; Ziyan Liu

REVIEW 4 major objections 6 minor 46 references

Learning-Guided Integration Contours Construction for Fast Large-Scale Generalized Eigensolvers

T0 review · 4 major / 6 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 DeepContour claims that a learned spectral predictor plus kernel-density-based contour construction can replace scouting in contour integral eigensolvers, cutting solve time up to 5.63x without missing target eigenvalues.

desk verdict A genuinely new hybrid—learned eigenvalue prediction plus KDE-guided contours for CI eigensolvers—with extensive experiments, but Algorithm 1 omits the promised safety margin and the main tables lack missed-eigenvalue counts. read the letter →

arxiv 2511.01927 v2 pith:37QHOPZR submitted 2025-11-02 cs.LG cs.AIcs.NAmath.NA

Yeqiu Chen , Ziyan Liu , Hong Wang , Lei Liu This is my paper

classification cs.LGcs.AIcs.NAmath.NA MSC 65F1565N2568T07

keywords generalizedeigenvalueproblemcontourintegralmethodspectralpredictionneuraloperatorkerneldensityestimationselectionCIRRFEAST

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

Contour integral eigensolvers are a fast, parallelizable way to find many interior eigenvalues of a large generalized eigenvalue problem, but their performance hinges on choosing integration contours that enclose the target eigenvalues without being wastefully large. The standard method, running a cheap iterative solver to 'scout' the spectrum, is slow and inaccurate enough that the contours end up oversized or misplaced. This paper claims that a neural operator (ENO) can learn the mapping from a physical system's parameter field to its eigenvalue spectrum, and that kernel density estimation (KDE) can then split that predicted spectrum into tight, well-placed contours. The pipeline replaces the scouting step entirely, and the paper reports end-to-end speedups up to 5.63x on five PDE-derived datasets at matrix size 50,000, with zero missed eigenvalues on its 100-instance ablation. The gain comes from smaller contour area (3.8–6.5x smaller), which directly reduces the number of shifted linear systems the CI solver must solve.

What carries the argument

The central mechanism is the learned operator G: a(x) ↦ Λ implemented by the Eigen-Neural-Operator (ENO), and the interval-sparsity kernel G_k(t) = Σ_j exp(−w(t−λ̂_j)²/(λ_end−λ_start)²) used by KDE to choose split points. The spectral projector P_k = (1/2πi)∮_{Γ_k}(zB−A)^{-1}B dz does the mathematical work of isolating invariant subspaces; the contours determine how expensive that projector is. By placing multiple tight circular contours at spectral gaps, DeepContour reduces quadrature points and the size of the projected dense problem, which is the direct source of the solver speedup.

What would settle it

Run DeepContour on a GEP from the same five PDEs but with parameter fields drawn from a distribution with a correlation length or amplitude far outside the training GRF range, or on a different PDE discretized at a mesh resolution the ENO never saw; count missed target eigenvalues and measure end-to-end time. If the zero-missed-eigenvalue record breaks or the speedup falls below 1, the central claim fails for out-of-distribution inputs.

Watch

Extended reading notes

Core claim

The paper's central claim is that the bottleneck of contour selection in CI eigensolvers can be automated and accelerated by a two-stage 'predict-then-guide' framework. The ENO, an FNO-based neural operator with an MLP head, maps the parameter function a(x) to the M smallest eigenvalues in a few milliseconds. A KDE-based recursive splitting algorithm builds an interval-sparsity function whose minima mark the sparsest points in the predicted spectrum, partitioning it into circular contours that are then fed to CIRR or FEAST. The key evidence is quantitative: on N=50,000 problems the ENO predicts the spectrum with normalized MSE one to two orders of magnitude lower than the Ritz values from an

Load-bearing premise

The load-bearing premise is that the neural predictor's training distribution matches the test problems—same PDE family, discretization, and parameter-field statistics—so that the predicted spectrum, and therefore the contour placement, is trustworthy; a problem outside that distribution can break both the speedup and the completeness guarantee.

Editorial extensions

If this is right

The speedup holds across both major CI solver families (CIRR and FEAST), across five scientific domains, and across accuracy tolerances from 1e-2 to 1e-12; the only variation is magnitude, not sign.
The speedup grows with the matrix dimension, meaning the method is most attractive for very large problems where scouting is most expensive.
The ENO's spectral prediction cost is ~8 ms versus 7–12 s for scouting, so the overhead of the learned contour design is effectively negligible.
The ablation shows that neither component alone is sufficient: replacing ENO with a scout breaks reliability (41.8 missed eigenvalues), while replacing KDE with naive interval expansion costs 1.5x in solve time.

Reading between the lines

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

Because the speedup is tied to contour tightness, any improvement in spectral estimation—uncertainty-quantified predictions, active learning, or higher-resolution KDE—should compound the benefit; the framework is a template for learned priors for classical solvers rather than a single tuned model.
The same predict-then-guide pipeline should transfer to nonlinear eigenvalue problems, rational-approximation pole selection, and preconditioner design, where a cheap learned prior could replace an expensive scan; the paper mentions nonlinear eigenvalue problems as future work.
The method assumes a parametric physical description; for a truly black-box matrix pencil, the ENO has no input, so the framework would need to be paired with an auxiliary method that recovers a parameterization from the matrices.

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.

Desk Editor's Note

read the letter

Best read as an engineering paper with a real result: for a parametric family of GEPs, you can train an FNO once to predict the low end of the spectrum, use KDE on that prediction to carve the spectrum into tight intervals, and feed those intervals to a CI solver as circular contours. The authors show that on five PDE-derived datasets at N=50000 with M=500, this beats five scouting baselines by 2-5.6x, and the speedup grows with matrix size. That is plausible—their contours are much smaller than the scout's safety-margin-expanded box, so the CI solver uses fewer quadrature points and smaller projected problems. The experimental work is extensive: two CI solvers, five baselines, five tolerances, ablations on every component, and the training times and hyperparameter sensitivity are disclosed. The combination is genuinely new as far as their citations show, and they do not oversell the neural operator as a solver—ENO is a guide, the classical solver does the final work. Per-instance overhead is small (~1.5s vs 7-12s for scouting) once the model is trained.

The soft spots are real. The big one: Algorithm 1, as written, constructs each contour as the circle with the interval's endpoints as diameter—no safety margin. Section 4.2 says the initial range is padded with a small safety margin, but the pseudocode does not implement it. That means extreme predicted eigenvalues lie on the contour boundary, and CI methods assume eigenvalues strictly inside. The resolvent (zB-A)^{-1} is singular at an eigenvalue on the path. The "strict numerical rigor" claim is unsupported while this discrepancy stands. It is likely fixable—add a margin parameter and state it—but as written it is an internal inconsistency.

Second, missed-eigenvalue counts appear only in one ablation (Table 2) and the KDE weight sweep (Table 14), not in the main speedup tables. For a numerical method, speedup without accuracy counts is incomplete; the authors should report missed counts (and residuals) for every dataset and tolerance.

Third, training costs 7-14 hours per dataset and is not amortized into the speedup. That is a poor trade for a one-off problem and fine for repeated in-family solves; the paper should say so explicitly. Appendix G is honest about out-of-distribution generalization being untested and about requiring the parameter function a(x) rather than arbitrary matrix pairs.

In sum, a solid empirical contribution with a real algorithmic gap to fix. I would send it to peer review—a referee can help close the margin issue and the accuracy-reporting gap. Not a desk reject.

Referee Report

4 major / 6 minor

Summary. The paper proposes DeepContour, a hybrid pipeline for contour-integral (CI) eigensolvers. A Fourier-Neural-Operator-based Eigen-Neural-Operator (ENO) predicts the M smallest eigenvalues of a parametrized generalized eigenvalue problem from a coefficient field a(x); a KDE-based procedure then partitions the predicted spectrum into intervals and constructs circular contours; a standard CI solver (CIRR or FEAST) computes the eigenpairs. The paper claims up to 5.63x end-to-end speedups over five scouting-based localization baselines on five PDE-derived datasets at N=50000, while maintaining full numerical accuracy. Extensive experiments, ablations, and runtime breakdowns are reported, and training times for the ENO are disclosed.

Significance. If the central speedup claim survives scrutiny, the work is a useful contribution to the growing area of learned numerical linear algebra: it replaces an expensive heuristic scouting phase with a trained surrogate and an automated contouring rule, and it evaluates the idea on several nontrivial Hermitian GEPs with a standard CI solver. The paper is also honest about the supervised nature of the ENO and reports training times and limitation statements. The main potential significance is not a new eigensolver but a practical contour-placement layer that can be dropped in front of CIRR/FEAST. The strength of the empirical evaluation is considerable: five datasets, five tolerances, five baselines, CIRR and FEAST, and multiple ablations. However, the claims of 'strict numerical rigor' and 'end-to-end' speedup are currently not fully supported by the reported evidence.

major comments (4)

[Algorithm 1 and Section 4.2] Algorithm 1 (lines 3 and 10) builds the initial interval as [min(Λhat), max(Λhat)] and constructs a circular contour centered at (λ_start+λ_end)/2 with radius (λ_end−λ_start)/2. For the initial interval, and for any interval that retains a predicted extremal eigenvalue as an endpoint, the contour passes exactly through that predicted eigenvalue. Section 4.2 states that the initial range should include 'a small safety margin,' but Algorithm 1 contains no such margin. This is an internal inconsistency. Since CI methods capture only eigenvalues strictly inside Γ, and the resolvent is singular on Γ, a contour through an eigenvalue can produce missed eigenvalues or corrupted quadrature. The paper's 'strict numerical rigor' and the 0-missed-eigenvalue claim rest on this point. Please add an explicit margin to Algorithm 1, or justify why the current construction is safe for all intervals, and r
[Section 5.1, Tables 1, 5, 11, 12] The speedup tables report no accuracy metric. The only quantitative evidence that no eigenvalues are missed is the ablation in Table 2 (Kirchhoff-Love, tol=1e-7) and the related appendix ablations (Tables 10 and 14). The main speedup tables, including the FEAST results and all other tolerances/datasets, do not state how many of the M target eigenvalues were actually recovered by the CI solve. A faster contour can trivially be a smaller or more aggressive contour that omits eigenvalues. Please report missed-eigenvalue counts (or a certified containment check) for every configuration in the main speedup tables, or clearly restrict the claim 'maintaining strict numerical rigor' to the configurations with verified counts.
[Section 5.1 (Metrics) and Table 4] The 'End-to-End Time Speedup' metric compares Time_Baseline(Scouting + CI Solve) with Time_Ours(Hybrid Contour Design + CI Solve), where the hybrid phase includes only ENO inference and KDE construction. Offline ENO training time (Table 4: 7.6–14.21 hours per dataset) and the offline generation of ground-truth eigenvalues are excluded. For a one-off solve this is not end-to-end; for an amortized workload the speedup is only realized after enough solves to pay for training. The abstract's 'up to a 5.63x speedup' is therefore stated without the amortization regime. Please either include training cost in the reported trade-off, state the break-even number of solves, or explicitly define the claim as an amortized/inference-time speedup. The disclosed training times are appreciated, but the metric name should match the quantity.
[Appendix G and Section 4.1] The method requires a known parameter function a(x) and a trained ENO for that parametric family; for an arbitrary GEP given only as matrices A, B, there is no input for the ENO. Appendix G correctly notes that generalization to systems outside the training distribution remains unexplored. This is acceptable as a scope limitation if stated prominently, but the introduction and abstract describe DeepContour as a general framework for 'large-scale GEPs.' Please add a sentence in the introduction or method section making the parametric-family assumption explicit, so that the scope is clear before the experiments.

minor comments (6)

[Table 1] Kirchhoff-Love Plate, tol=1e-12, KrylovSchur column reads '2.39 / 187'; the solver speedup should almost certainly be 1.87. Please check all numeric entries for typos.
[Section 5.1 and Appendix E.1] The manuscript says 'For details of experimental settings and hyperparameters, please refer to Appendix E,' but the relevant hyperparameters for KDE and FNO are given in Appendix E.1; consider moving the KDE hyperparameters to the main text or to Table 1 so the reader can see them without reading the appendix.
[Algorithm 1, line 21] The merge step says 'Merge any contour in C_final containing fewer than N_min eigenvalues with its nearest neighbor.' The notion of 'nearest neighbor' for intervals and the resulting contour radius after merging are not defined. Please specify the rule precisely, since it affects the final contours.
[Section 6] The conclusion states the method 'successfully transforms the manual, heuristic-based contour construction process into a fully automated, data-driven strategy.' This overstates the automation: training data still requires expensive ground-truth GEP solves. Rephrase to acknowledge the offline cost.
[Figure 3] The caption labels (c1) and (c2) are not explained in the caption text. Consider adding a sentence describing what each panel shows.
[Throughout] There are minor typos, e.g., 'comparsion' in Section 5.2 and 'Eigenvalue Prediction' in Figure 3. A careful proofread is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the ENO is an explicitly trained supervised surrogate, final eigenvalues come from the CI solver, and the speedup claims are empirical against external baselines.

full rationale

The paper's derivation chain is not circular. The ENO is trained by minimizing MSE against ground-truth eigenvalues (Eq. 5), which is a disclosed supervised regression, not a first-principles derivation. The paper does not present the ENO output as the final scientific result; it explicitly frames ENO as an 'intelligent guide' and Algorithm 2 shows that the final eigenpairs are computed by the CI solver from the matrices A and B, independent of the ENO except for contour placement. The central speedup claims are measured end-to-end against scouting-based baselines implemented with SLEPc/PETSc, and the baselines' safety margins are calibrated to ground-truth coverage, which is favorable to the baselines rather than to DeepContour. The KDE contour construction is driven by predicted eigenvalues, but this is the intended predictive mechanism, not a case where a fitted parameter is renamed as a prediction of the same quantity on the same instances; the reported speedups and accuracy are on held-out test instances. There are no load-bearing self-citations, no imported uniqueness theorems, and no ansatz smuggled via citation. The internal inconsistency in Algorithm 1 regarding the promised safety margin is a correctness/reproducibility concern, not a circularity. The acknowledged limitation in Appendix G about generalization outside the training distribution is a standard machine-learning caveat, not a circular step. Thus no circular step reaches the threshold required by the analysis rules.

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

The central claim is an empirical speedup, so the free parameters are the learned network weights and the hand-set KDE/architecture hyperparameters. The method rests on the domain assumption that the target GEPs are parametric PDE discretizations and that test instances share the training distribution. No new physical entities are introduced.

free parameters (4)

ENO network weights (FNO backbone + MLP head) = trained on 1000 samples per dataset
The spectral predictor is a supervised regression fit to ground-truth eigenvalues; the entire predictive power of the method comes from these fitted weights.
KDE hyperparameters (Nmin=10, Nmax=50, w=10) = 10, 50, 10
Chosen by hand with 'minimal tuning'; ablation shows w in [5,20] is stable, but Nmin/Nmax are not ablated.
FNO architecture (4 layers, width 64, 20 modes) = 4/64/20
Selected via hyperparameter ablation (Table 9); affects prediction accuracy and hence contour quality.
Target fraction M/N = 1% = M = 0.01N
Experimental design choice; the method's behavior for other fractions is untested.

assumptions (5)

standard math Contour integral spectral projector (Eq. 2) correctly isolates invariant subspaces; CIRR/FEAST converge to stated tolerances
Standard numerical analysis background, not proved in paper.
domain assumption GEPs come from FEM-discretized parametric PDEs; an input function a(x) determines the spectrum
The whole learning pipeline requires this parametric structure; arbitrary matrix pencils have no ENO input.
ad hoc to paper Test instances are drawn from the same distribution (PDE family, GRF parameters, mesh) as training
No out-of-distribution evaluation; acknowledged as a limitation in Appendix G.
domain assumption The M smallest eigenvalues of a Hermitian pencil are real and sufficient to define good contours
The paper restricts to Hermitian GEPs (Appendix E.3); complex spectra are deferred.
standard math KDE sparsity function minima identify useful spectral gaps
Heuristic from spectral density estimation; no guarantee for finite predicted sets.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Learning-Guided Integration Contours Construction for Fast Large-Scale Generalized Eigensolvers." pith.science (2026). https://pith.science/paper/37QHOPZR

@misc{pith2026251101927,
  author       = {Pith},
  title        = {Pith review of: Learning-Guided Integration Contours Construction for Fast Large-Scale Generalized Eigensolvers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/37QHOPZR}},
  note         = {Machine review of arXiv:2511.01927}
}

read the original abstract

Solving large-scale Generalized Eigenvalue Problems (GEPs) is a fundamental yet computationally prohibitive task in science and engineering. As a promising direction, contour integral (CI) methods offer an efficient and parallelizable framework. However, their performance is critically dependent on the selection of integration contours -- improper selection without reliable prior knowledge of eigenvalue distribution can incur significant computational overhead and compromise numerical accuracy. To address this challenge, we propose Deepcontour, a novel hybrid framework that integrates a deep learning-based spectral predictor with Kernel Density Estimation (KDE) for principled contour design. Specifically, Deepcontour utilizes its specialized Eigen-Neural-Operator (ENO) to provide rapid spectral distribution priors, driving a KDE module to automatically construct the optimized integration contours, which guide the CI solver to efficiently find the desired eigenvalues. Deepcontour achieves up to a 5.63x speedup across diverse scientific datasets while maintaining strict numerical rigor. By merging the predictive power of deep learning with the numerical rigor of classical solvers, this work establishes an efficient and robust paradigm for solving large-scale GEPs.

Figures

Figures reproduced from arXiv: 2511.01927 by the authors.

**Figure 1.** The variation in tolerance for DeepContour compared to scouting methods. Each line shows experimental results using a contour selection strategy and a CI-based solver. Notably, DeepContour substantially enhances the efficiency of solving Generalized Eigenvalue Problem, achieving a speed-up of up to 5.63 times. A comparison with traditional iterative eigensolvers is provided in Appendix F.1. Their significantly slo… view at source ↗

**Figure 2.** Experiments demonstrate the CIRR’s high sensi [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Overall architecture of DeepContour: (a) Construct contour Γ for CI solver to solve given matrices A and B. (b) Traditional Scout-based Method: An iterative solver (e.g., Arnoldi) is run for a fixed number of steps to obtain a rough spectral distribution, and a safety margin is then applied to define the integration contour. (c1) DeepContour Eigenvalue Prediction Module: Utilizing a specialized eigenvalue neural ope… view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Experiments on the Kirchhoff-Love Plate and EGFR Electronic problems with varying matrix sizes. The results indicate that as the matrix size increases, both time speedup and iteration speedup increase. (ENO + KDE): The complete model. (2) w/o ENO: Replace FNOwith a st…

**Figure 5.** Figure 5: KDE-based Contour Construction. Our automated contour construction follows the iterative process detailed in Algorithm 2. The core component is the interval sparsity function, Gk(t), defined in Eq. (6). The hyperparameters for this process, which have been noted to …

**Figure 5.** Figure 5: Training curves of our ENO model across five datasets. [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: plots the convergence curve of DeepContour, the scout-based CI methods, and these fully converged iterative solvers. The results reveal a three-tiered performance hierarchy. On our specific computational platform, the standalone iterative solvers are significantly sl…

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

46 extracted references

[1]

Finite element procedures, 1996

Klaus Jurgen Bathe. Finite element procedures, 1996

1996
[2]

Relaxation mode analysis for molecular dynamics simulations of pro- teins.Biophysical Reviews, 10:375–389, 2018

Ayori Mitsutake and Hideaki Takano. Relaxation mode analysis for molecular dynamics simulations of pro- teins.Biophysical Reviews, 10:375–389, 2018

2018
[3]

Courier Corporation, 2012

Attila Szabo and Neil S Ostlund.Modern quantum chemistry: introduction to advanced electronic struc- ture theory. Courier Corporation, 2012

2012
[4]

Waveland Press, 2010

Leonard Meirovitch.Fundamentals of Vibrations. Waveland Press, 2010

2010
[5]

SIAM, 2011

Yousef Saad.Numerical methods for large eigenvalue problems: revised edition. SIAM, 2011

2011
[6]

Springer, 2005

Daniel Kressner.Numerical Methods for General and Structured Eigenvalue Problems. Springer, 2005

2005
[7]

Moler and Gilbert W

Cleve B. Moler and Gilbert W. Stewart. An algorithm for generalized matrix eigenvalue problems.SIAM Journal on Numerical Analysis, 10(2):241–256, 1973

1973
[8]

Lehoucq, Danny C

Richard B. Lehoucq, Danny C. Sorensen, and Chao Yang.ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. SIAM, 1998

1998

Show all 46 references

[9]

Cirr: a rayleigh- ritz type method with contour integral for generalized eigenvalue problems.Hokkaido mathematical journal, 36(4):745–757, 2007

Tetsuya Sakurai and Hiroto Tadano. Cirr: a rayleigh- ritz type method with contour integral for generalized eigenvalue problems.Hokkaido mathematical journal, 36(4):745–757, 2007

2007
[10]

Density-matrix-based algorithm for solving eigenvalue problems.Physical Review B, 79(11):115112, 2009

Eric Polizzi. Density-matrix-based algorithm for solving eigenvalue problems.Physical Review B, 79(11):115112, 2009

2009
[11]

A projection method for generalized eigenvalue problems using nu- merical integration.Journal of computational and ap- plied mathematics, 159(1):119–128, 2003

Tetsuya Sakurai and Hiroshi Sugiura. A projection method for generalized eigenvalue problems using nu- merical integration.Journal of computational and ap- plied mathematics, 159(1):119–128, 2003

2003
[12]

A method for profiling the distribution of eigenvalues using the as method.Taiwanese Journal of Mathematics, pages 839–853, 2010

Kenta Senzaki, Hiroto Tadano, Tetsuya Sakurai, and Zhaojun Bai. A method for profiling the distribution of eigenvalues using the as method.Taiwanese Journal of Mathematics, pages 839–853, 2010

2010
[13]

PhD thesis, Uni- versity of Massachusetts Amherst, 2013

Brendan Gavin.Enhancing the Performance and Ro- bustness of the FEAST Eigensolver. PhD thesis, Uni- versity of Massachusetts Amherst, 2013

2013
[14]

Fourier neural op- erator for parametric partial differential equations

Zongyi Li, Nikola Kovachki, Kamyar Azizzade- nesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Fourier neural op- erator for parametric partial differential equations. In International Conference on Learning Representations (ICLR), 2021

2021
[15]

Applications of the fourier neural op- erator in regional ocean modeling.Frontiers in Marine Science, 11:1383997, 2024

Byeong-Jun Choi, Hyeong-Seok Jin, and Bayasgalan Lkhagvasuren. Applications of the fourier neural op- erator in regional ocean modeling.Frontiers in Marine Science, 11:1383997, 2024

2024
[16]

Approximat- ing spectral densities of large matrices.SIAM Review, 58(1):34–65, 2016

Lin Lin, Yousef Saad, and Chao Yang. Approximat- ing spectral densities of large matrices.SIAM Review, 58(1):34–65, 2016

2016
[17]

A krylov–schur algorithm for large eigenproblems.SIAM Journal on Matrix Analysis and Applications, 23(3):601–614, 2002

Gilbert W Stewart. A krylov–schur algorithm for large eigenproblems.SIAM Journal on Matrix Analysis and Applications, 23(3):601–614, 2002

2002
[18]

A jacobi–davidson iteration method for linear eigenvalue problems.SIAM review, 42(2):267–293, 2000

Gerard LG Sleijpen and Henk A Van der V orst. A jacobi–davidson iteration method for linear eigenvalue problems.SIAM review, 42(2):267–293, 2000

2000
[19]

An integral method for solving nonlinear eigenvalue problems.Linear Algebra and its Applications, 436(10):3839–3863, 2012

Wolf-J ¨urgen Beyn. An integral method for solving nonlinear eigenvalue problems.Linear Algebra and its Applications, 436(10):3839–3863, 2012

2012
[20]

Zolotarev quadrature rules and load balancing for the feast eigensolver.SIAM Journal on Scientific Computing, 37(4):A2100–A2122, 2015

Stefan Guttel, Eric Polizzi, Ping Tak Peter Tang, and Gautier Viaud. Zolotarev quadrature rules and load balancing for the feast eigensolver.SIAM Journal on Scientific Computing, 37(4):A2100–A2122, 2015

2015
[21]

Maziar Raissi, Paris Perdikaris, and George E Kar- niadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equa- tions.Journal of Computational physics, 378:686–707, 2019

2019
[22]

Physics-informed neural networks for quantum eigen- value problems

Henry Jin, Marios Mattheakis, and Pavlos Protopapas. Physics-informed neural networks for quantum eigen- value problems. In2022 International Joint Confer- ence on Neural Networks (IJCNN), pages 1–8. IEEE, 2022

2022
[23]

Enhanced fifth order weno shock-capturing schemes with deep learning.Results in Applied Math- ematics, 12:100201, 2021

Tatiana Kossaczk ´a, Matthias Ehrhardt, and Michael G¨unther. Enhanced fifth order weno shock-capturing schemes with deep learning.Results in Applied Math- ematics, 12:100201, 2021

2021
[24]

Machine learning for vibrational spec- troscopy.Nature Communications, 12(1):4834, 2021

Michael Gastegger, Florian Fiedler, and Philipp Mar- quetand. Machine learning for vibrational spec- troscopy.Nature Communications, 12(1):4834, 2021

2021
[25]

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

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

2021
[26]

Neural operator: Learn- ing maps between function spaces with applications to pdes.Journal of Machine Learning Research, 24(89):1–97, 2023

Nikola Kovachki, Zongyi Li, Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Andrew Stu- art, and Anima Anandkumar. Neural operator: Learn- ing maps between function spaces with applications to pdes.Journal of Machine Learning Research, 24(89):1–97, 2023

2023
[27]

Tapas Tripura and Souvik Chakraborty. Wavelet neu- ral operator for solving parametric partial differential equations in computational mechanics problems.Com- puter Methods in Applied Mechanics and Engineering, 404:115783, 2023

2023
[28]

Choose a transformer: Fourier or galerkin.Advances in neural information processing systems, 34:24924–24940, 2021

Shuhao Cao. Choose a transformer: Fourier or galerkin.Advances in neural information processing systems, 34:24924–24940, 2021

2021
[29]

The nonlinear eigenvalue problem.Acta Numerica, 26:1–94, 2017

Stefan G ¨uttel and Franc ¸oise Tisseur. The nonlinear eigenvalue problem.Acta Numerica, 26:1–94, 2017

2017
[30]

On Sylvester’s law of inertia for nonlinear eigenvalue problems.Elec- tronic Transactions on Numerical Analysis, 40:82–93, 2013

Aleksandra Kostic and Heinrich V oss. On Sylvester’s law of inertia for nonlinear eigenvalue problems.Elec- tronic Transactions on Numerical Analysis, 40:82–93, 2013

2013
[31]

A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines.Communications in Statistics-Simulation and Computation, 19(2):433–450, 1990

Michael F Hutchinson. A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines.Communications in Statistics-Simulation and Computation, 19(2):433–450, 1990

1990
[32]

Deep Learning

Ian Goodfellow, Yoshua Bengio, and Aaron Courville. Deep Learning. MIT Press, 2016. http://www. deeplearningbook.org

2016
[33]

Optimization algorithms on matrix manifolds

P-A Absil, Robert Mahony, and Rodolphe Sepulchre. Optimization algorithms on matrix manifolds. Prince- ton University Press, 2008

2008
[34]

Krylov-schur methods in slepc.Univer- sitat Politecnica de Valencia, Tech

Vicente Hern ´andez, Jose E Rom´an, Andr´es Tom´as, and Vicente Vidal. Krylov-schur methods in slepc.Univer- sitat Politecnica de Valencia, Tech. Rep. STR-7, 2007

2007
[35]

Adams, Jed Brown, Peter Brune, Kris Buschelman, Lisandro Dalcin, Victor Eijkhout, William D

Satish Balay, Shrirang Abhyankar, Mark F. Adams, Jed Brown, Peter Brune, Kris Buschelman, Lisandro Dalcin, Victor Eijkhout, William D. Gropp, Dmitry Karpeyev, Dinesh Kaushik, Matthew G. Knepley, Fande Kong, Scott Kruger, Dave A. May, Lois Curf- man McInnes, Richard Tran Mills,...

2023
[36]

Roman, and Vicente Vi- dal

Vicente Hernandez, Jose E. Roman, and Vicente Vi- dal. SLEPc: A scalable and flexible toolkit for the solu- tion of eigenvalue problems.ACM Trans. Math. Softw., 31(3):351–362, 2005

2005
[37]

Scientific and technical information divi- sion.National Aeronautics and Space Administration, Washington, DC, 1964

Data Book. Scientific and technical information divi- sion.National Aeronautics and Space Administration, Washington, DC, 1964

1964
[38]

CRC press, 2006

Junuthula Narasimha Reddy.Theory and analysis of elastic plates and shells. CRC press, 2006

2006
[39]

New develop- ments in molecular orbital theory.Reviews of modern physics, 23(2):69, 1951

Clemens Carel Johannes Roothaan. New develop- ments in molecular orbital theory.Reviews of modern physics, 23(2):69, 1951

1951
[40]

John Wiley & Sons, 2015

Jian-Ming Jin.The finite element method in electro- magnetics. John Wiley & Sons, 2015

2015
[41]

The linear theory of piezoelectricity

HF Tiersten. The linear theory of piezoelectricity. In Linear Piezoelectric Plate Vibrations: Elements of the Linear Theory of Piezoelectricity and the Vibrations Piezoelectric Plates, pages 33–39. Springer, 1969

1969
[42]

Finite element method for piezoelectric vibration.International jour- nal for numerical methods in engineering, 2(2):151– 157, 1970

Henno Allik and Thomas JR Hughes. Finite element method for piezoelectric vibration.International jour- nal for numerical methods in engineering, 2(2):151– 157, 1970

1970
[43]

Academic press, 1982

Morton E Gurtin.An introduction to continuum me- chanics, volume 158. Academic press, 1982

1982
[44]

Conduction of heat in solids

Ofodike A Ezekoye. Conduction of heat in solids. In SFPE handbook of fire protection engineering, pages 25–52. Springer, 2016

2016
[45]

Klaus- Jurgen Bathe, 2006

Klaus-J ¨urgen Bathe.Finite element procedures. Klaus- Jurgen Bathe, 2006. A Details of Related Work A.1 Numerical Implementation of Contour Integral Methods This section provides further details on the numerical imple- mentation of the contour integral methods discussed in th...

2006
[46]

scout-then- solve

The training was conducted for 200 epochs on the GPU. We provide training time and training curves in Table 4 and Figure 5. KDE-based Contour Construction.Our automated con- tour construction follows the iterative process detailed in Al- gorithm 2. The core component is the in...

Pith tools

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

[1] [1]

Finite element procedures, 1996

Klaus Jurgen Bathe. Finite element procedures, 1996

1996

[2] [2]

Relaxation mode analysis for molecular dynamics simulations of pro- teins.Biophysical Reviews, 10:375–389, 2018

Ayori Mitsutake and Hideaki Takano. Relaxation mode analysis for molecular dynamics simulations of pro- teins.Biophysical Reviews, 10:375–389, 2018

2018

[3] [3]

Courier Corporation, 2012

Attila Szabo and Neil S Ostlund.Modern quantum chemistry: introduction to advanced electronic struc- ture theory. Courier Corporation, 2012

2012

[4] [4]

Waveland Press, 2010

Leonard Meirovitch.Fundamentals of Vibrations. Waveland Press, 2010

2010

[5] [5]

SIAM, 2011

Yousef Saad.Numerical methods for large eigenvalue problems: revised edition. SIAM, 2011

2011

[6] [6]

Springer, 2005

Daniel Kressner.Numerical Methods for General and Structured Eigenvalue Problems. Springer, 2005

2005

[7] [7]

Moler and Gilbert W

Cleve B. Moler and Gilbert W. Stewart. An algorithm for generalized matrix eigenvalue problems.SIAM Journal on Numerical Analysis, 10(2):241–256, 1973

1973

[8] [8]

Lehoucq, Danny C

Richard B. Lehoucq, Danny C. Sorensen, and Chao Yang.ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. SIAM, 1998

1998