pith. sign in

arxiv: 2511.02481 · v4 · submitted 2025-11-04 · 💻 cs.LG

NOWS: Neural Operator Warm Starts for Accelerating Iterative Solvers

Mohammad Sadegh Eshaghi , Cosmin Anitescu , Navid Valizadeh , Yizheng Wang , Xiaoying Zhuang , Timon Rabczuk This is my paper

Pith reviewed 2026-05-18 01:30 UTC · model grok-4.3

classification 💻 cs.LG
keywords neural operatorsiterative solversPDE simulationwarm startsKrylov methodsconjugate gradientGMREShybrid numerical methods
0
0 comments X

The pith

Neural operators generate initial guesses that cut iterative PDE solver time by up to 90 percent.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces Neural Operator Warm Starts (NOWS) as a way to combine learned solution operators with classical iterative solvers for partial differential equations. Neural operators supply high-quality starting points for methods such as conjugate gradient and GMRES, so fewer iterations are required to reach the solution. The approach leaves existing discretizations and solver code unchanged and works with finite elements, finite differences, and other standard schemes. A sympathetic reader would care because it promises faster high-fidelity simulations for repeated queries while retaining the stability and convergence guarantees that pure data-driven models often lose.

Core claim

Neural Operator Warm Starts (NOWS) harness learned solution operators to produce high-quality initial guesses for Krylov methods such as conjugate gradient and GMRES. This hybrid strategy accelerates classical iterative solvers for PDEs while preserving stability and convergence guarantees. Across benchmarks the learned initialization reduces iteration counts and end-to-end runtime, delivering a computational-time reduction of up to 90 percent, and integrates directly with finite-difference, finite-element, isogeometric, and finite-volume discretizations.

What carries the argument

Neural Operator Warm Starts (NOWS), the use of a trained neural operator to supply an initial guess to an otherwise unchanged Krylov iterative solver.

If this is right

  • Iteration counts for conjugate gradient and GMRES drop consistently across the tested benchmarks.
  • End-to-end runtime falls by up to 90 percent while the underlying numerical algorithm's stability and convergence guarantees remain intact.
  • The same learned operator can be paired with finite-difference, finite-element, isogeometric, and finite-volume discretizations without code changes.
  • The method targets many-query, real-time, and design tasks where repeated PDE solves are the bottleneck.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same warm-start idea could be applied to time-dependent or nonlinear PDEs by training the operator on solution snapshots rather than steady-state fields.
  • An online version might retrain or fine-tune the operator on recent solves to maintain performance when problem statistics drift.
  • Because the iterative solver still runs to convergence, the approach could serve as a safe drop-in replacement inside existing engineering workflows that already trust Krylov methods.

Load-bearing premise

A neural operator trained on one distribution of right-hand sides, boundary conditions, and geometries will still produce initial guesses close enough to the true solution on new problems that the iteration-count savings stay large and reliable.

What would settle it

Run the method on a collection of right-hand sides, boundary conditions, or geometries deliberately drawn from outside the training distribution and measure whether the iteration reduction drops below 10 percent or the solver fails to converge within a preset budget.

Figures

Figures reproduced from arXiv: 2511.02481 by Cosmin Anitescu, Mohammad Sadegh Eshaghi, Navid Valizadeh, Timon Rabczuk, Xiaoying Zhuang, Yizheng Wang.

Figure 1
Figure 1. Figure 1: Workflow of the Neural Operator Warm Start (NOWS) framework. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: NOWS accelerates iterative solvers in various resolutions. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Impact of physics-informed training and neural-operator warm starts (NOWS) on Darcy flow simulations. (a) [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: NOWS accelerates the iterative solution of PDEs on irregular domains. (a) [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: NOWS for dynamic problems (a) Ensemble of initial conditions in the test dataset used for the Burgers’ equation. (b) Comparing runtime distributions of CG and NOWS: scatter plot of runtime versus L2 error for each sample (top), violin plots comparing the runtime distributions of the CG and NOWS solvers (bottom). (c) Spatiotemporal evolution of the solution filed for a representative test sample in the Burg… view at source ↗
Figure 1
Figure 1. Figure 1: Workflow of the Neural Operator Warm Start (NOWS) framework. [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: NOWS accelerates iterative solvers in various resolutions. [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Impact of physics-informed training and neural-operator warm starts (NOWS) on Darcy flow simulations. (a) [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: NOWS accelerates the iterative solution of PDEs on irregular domains. (a) [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: NOWS for dynamic problems (a) Ensemble of initial conditions in the test dataset used for the Burgers’ equation. (b) Comparing runtime distributions of CG and NOWS: scatter plot of runtime versus L2 error for each sample (top), violin plots comparing the runtime distributions of the CG and NOWS solvers (bottom). (c) Spatiotemporal evolution of the solution filed for a representative test sample in the Burg… view at source ↗
read the original abstract

Partial differential equations (PDEs) underpin quantitative descriptions across the physical sciences and engineering, yet high-fidelity simulation remains a major computational bottleneck for many-query, real-time, and design tasks. Data-driven surrogates can be strikingly fast but are often unreliable when applied outside their training distribution. Here we introduce Neural Operator Warm Starts (NOWS), a hybrid strategy that harnesses learned solution operators to accelerate classical iterative solvers by producing high-quality initial guesses for Krylov methods such as conjugate gradient and GMRES. NOWS leaves existing discretizations and solver infrastructures intact, integrating seamlessly with finite-difference, finite-element, isogeometric analysis, finite volume method, etc. Across our benchmarks, the learned initialization consistently reduces iteration counts and end-to-end runtime, resulting in a reduction of the computational time of up to 90 %, while preserving the stability and convergence guarantees of the underlying numerical algorithms. By combining the rapid inference of neural operators with the rigor of traditional solvers, NOWS provides a practical and trustworthy approach to accelerate high-fidelity PDE simulations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces Neural Operator Warm Starts (NOWS), a hybrid method that trains a neural operator to generate initial guesses for Krylov iterative solvers (CG, GMRES) applied to discretized PDEs. The approach leaves existing discretizations and solver code unchanged and is claimed to reduce iteration counts and end-to-end runtime by up to 90 % across benchmarks while inheriting the stability and convergence guarantees of the underlying numerical algorithms.

Significance. If the reported speed-ups prove robust outside the training distribution, the work supplies a practical route to accelerate many-query and real-time PDE simulations without sacrificing the reliability that pure data-driven surrogates often lack. The explicit preservation of classical convergence theory and the compatibility with standard discretizations (finite elements, finite volumes, etc.) are concrete strengths.

major comments (2)
  1. [Experiments] Experiments section: the headline claim of consistent iteration-count and runtime reductions (up to 90 %) across benchmarks is load-bearing for the paper’s contribution, yet the manuscript supplies no quantitative characterization of the training distribution versus the diversity of test instances (new RHS, BCs, or geometries). Without such characterization or worst-case distance bounds on the learned initial guess, it is impossible to verify that the observed speed-ups will persist rather than revert to baseline behavior.
  2. [§3 and §4] §3 (Method) and §4 (Numerical results): while the method correctly inherits convergence guarantees from the Krylov solver, no analysis or empirical quantification is given for how close the neural-operator output lies to the true solution on out-of-distribution problems. This distance directly controls the iteration reduction and therefore the practical utility of the warm-start strategy.
minor comments (2)
  1. [Abstract] The abstract states performance claims without reference to any table or figure; a single sentence pointing to the relevant result table would improve readability.
  2. [§2] Notation for the neural operator and the underlying linear system could be introduced earlier and used consistently to avoid occasional ambiguity between the learned map and the discrete operator.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback emphasizing the need for clearer characterization of training versus test distributions and direct quantification of warm-start quality on out-of-distribution instances. These points help strengthen the presentation of robustness. We respond to each major comment below and have revised the manuscript to incorporate additional details and experiments.

read point-by-point responses

  1. Referee: Experiments section: the headline claim of consistent iteration-count and runtime reductions (up to 90 %) across benchmarks is load-bearing for the paper’s contribution, yet the manuscript supplies no quantitative characterization of the training distribution versus the diversity of test instances (new RHS, BCs, or geometries). Without such characterization or worst-case distance bounds on the learned initial guess, it is impossible to verify that the observed speed-ups will persist rather than revert to baseline behavior.

    Authors: We agree that explicit characterization of the training distribution relative to test diversity strengthens the claims. In the revised manuscript we have added a new subsection to §4 that specifies the training distribution parameters (e.g., ranges of forcing terms, boundary condition types, and geometry variations) and documents the test instances, which include previously unseen RHS, BCs, and geometries. We also report empirical relative L2 distances between neural-operator predictions and reference solutions on these test cases. While deriving rigorous worst-case distance bounds would require additional theoretical assumptions beyond the scope of the current work, the added empirical metrics confirm that iteration and runtime reductions remain substantial across the evaluated distribution shifts. revision: yes

  2. Referee: §3 (Method) and §4 (Numerical results): while the method correctly inherits convergence guarantees from the Krylov solver, no analysis or empirical quantification is given for how close the neural-operator output lies to the true solution on out-of-distribution problems. This distance directly controls the iteration reduction and therefore the practical utility of the warm-start strategy.

    Authors: The referee correctly notes that the practical speed-up depends on the quality of the initial guess. Although the Krylov convergence theory holds for any initial vector, we have now included direct empirical quantification in the revised §4. Specifically, we added tables and figures reporting the initial residual norms and relative solution errors of the neural-operator outputs on out-of-distribution problems. These results show that the learned warm starts consistently produce smaller initial residuals than zero or random initializations, directly explaining the observed 50–90 % iteration reductions even when the test instances differ from the training distribution. revision: yes

Circularity Check

0 steps flagged

No significant circularity; speedup follows from standard Krylov theory plus external neural-operator training

full rationale

The paper presents NOWS as a hybrid that uses a separately trained neural operator to supply initial guesses to unmodified Krylov solvers (CG, GMRES, etc.). Convergence guarantees and the iteration-reduction mechanism are inherited from classical numerical linear algebra, not derived inside the paper. No equations equate the claimed runtime reduction to a fitted parameter by construction, and no load-bearing premise rests on a self-citation chain whose validity is presupposed. The training distribution and generalization behavior are treated as empirical questions outside the derivation itself, consistent with the reader's assessment of score 2.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Review performed on abstract only; detailed ledger cannot be completed without the full manuscript. Standard numerical linear algebra assumptions are invoked implicitly.

axioms (1)
  • domain assumption Krylov subspace methods converge for any initial guess, with iteration count depending on the quality of that guess.
    Implicit in the claim that better initial guesses reduce iteration counts while preserving guarantees.
invented entities (1)
  • Neural Operator Warm Starts (NOWS) no independent evidence
    purpose: Hybrid acceleration layer that supplies learned initial guesses to classical iterative solvers.
    New named strategy introduced to combine operator learning with existing numerical infrastructure.

pith-pipeline@v0.9.0 · 5732 in / 1349 out tokens · 34172 ms · 2026-05-18T01:30:10.511393+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    NOWS employs a neural operator to generate high-quality initial guesses that sharply reduce the initial residual, thereby lowering the iteration count required for full convergence... preserving the stability, interpretability, and rigorous convergence guarantees of the underlying numerical method

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

55 extracted references · 55 canonical work pages · 2 internal anchors

  1. [1]

    & V alli, A

    Quarteroni, A. & V alli, A. Numerical approximation of partial differential equations (Springer, 1994)

    work page 1994

  2. [2]

    & Laxmi, A

    Menghal, P . & Laxmi, A. J. Real time simulation: Recent progress & challenges. In 2012 International Conference on Power , Signals, Controls and Computation, 1–6 (IEEE, 2012)

    work page 2012

  3. [3]

    Biegler, L. T. Nonlinear programming: concepts, algorithms, and applications to chemical processes (SIAM, 2010)

    work page 2010

  4. [4]

    Smith, R. C. Uncertainty quantification: theory, implementation, and applications (SIAM, 2024)

    work page 2024

  5. [5]

    & Barlow, C

    Fuller, A., Fan, Z., Day, C. & Barlow, C. Digital twin: enabling technologies, challenges and open research. IEEE access 8, 108952–108971 (2020)

    work page 2020

  6. [6]

    S., Anitescu, C

    Es-haghi, M. S., Anitescu, C. & Rabczuk, T. Methods for enabling real-time analysis in digital twins: A literature review. Computers & Structures 297, 107342 (2024)

    work page 2024

  7. [7]

    Hageman, L. A. & Y oung, D. M. Applied iterative methods (Courier Corporation, 2012)

    work page 2012

  8. [8]

    Iterative methods for solving linear systems (SIAM, 1997)

    Greenbaum, A. Iterative methods for solving linear systems (SIAM, 1997)

    work page 1997

  9. [9]

    A.Iterative Krylov methods for large linear systems

    V an der V orst, H. A.Iterative Krylov methods for large linear systems . 13 (Cambridge University Press, 2003)

    work page 2003

  10. [10]

    Iterative methods by space decomposition and subspace correction

    Xu, J. Iterative methods by space decomposition and subspace correction. SIAM Review 34, 581–613 (1992) . Available at https://doi.org/10.1137/1034116. https://doi.org/10.1137/1034116

    work page doi:10.1137/1034116 1992

  11. [11]

    & Nalcioglu, O

    Kawata, S. & Nalcioglu, O. Constrained iterative reconstruction by the conjugate gradient method. IEEE Transactions on Medical Imaging 4, 65–71 (1985)

    work page 1985

  12. [12]

    Kershaw, D. S. The incomplete cholesky—conjugate gradient method for the iterative solution of systems of linear equations. Journal of Computational Physics 26, 43–65 (1978) . Available at https://www.sciencedirect.com/ science/article/pii/0021999178900980

    work page arXiv 1978

  13. [13]

    & Lidauer, M

    Strandén, I. & Lidauer, M. Solving large mixed linear models using preconditioned conjugate gradient iteration. Journal of Dairy Science 82, 2779–2787 (1999)

    work page 1999

  14. [14]

    Solving sparse linear systems via flexible gmres with in-memory analog preconditioning

    Kalantzis, V .et al. Solving sparse linear systems via flexible gmres with in-memory analog preconditioning. In 2023 IEEE High Performance Extreme Computing Conference (HPEC), 1–7 (2023)

    work page 2023

  15. [15]

    & Dongarra, J

    Lindquist, N., Luszczek, P . & Dongarra, J. Accelerating restarted gmres with mixed precision arithmetic. IEEE Transactions on Parallel and Distributed Systems 33, 1027–1037 (2022)

    work page 2022

  16. [16]

    & ´Swirydowicz, K

    Thomas, S., Carson, E., Rozložník, M., Carr, A. & ´Swirydowicz, K. Iterated gauss–seidel gmres. SIAM Journal on Scientific Computing 46, S254–S279 (2024) . Available at https://doi.org/10.1137/22M1491241. https: //doi.org/10.1137/22M1491241. 11/15

    work page doi:10.1137/22m1491241 2024

  17. [17]

    Amestoy, A

    Amestoy, P .et al. Five-precision gmres-based iterative refinement. SIAM Journal on Matrix Analysis and Applications 45, 529–552 (2024) . Available at https://doi.org/10.1137/23M1549079. https://doi.org/10.1137/ 23M1549079

    work page doi:10.1137/23m1549079 2024

  18. [18]

    B., V an Wingerden, J.-W., V erhaegen, M

    Qiu, Y ., V an Gijzen, M. B., V an Wingerden, J.-W., V erhaegen, M. & Vuik, C. Efficient preconditioners for pde-constrained optimization problems with a multi-level sequentially semi-separable matrix structure. Electronic Transactions on Numerical Analysis 44, 3 (2015)

    work page 2015

  19. [19]

    & Moin, P

    Mahesh, K., Constantinescu, G. & Moin, P . A numerical method for large-eddy simulation in complex geometries. Journal of Computational Physics 197, 215–240 (2004)

    work page 2004

  20. [20]

    & Kormann, K

    Kronbichler, M. & Kormann, K. A generic interface for parallel cell-based finite element operator application. Computers & Fluids 63, 135–147 (2012)

    work page 2012

  21. [21]

    Li, Z. et al. Neural operator: Graph kernel network for partial differential equations. arXiv preprint arXiv:2003.03485 (2020)

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  22. [22]

    & Karniadakis, G

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

    work page 2021

  23. [23]

    Li, Z. et al. Fourier neural operator for parametric partial differential equations. arXiv preprint arXiv:2010.08895 (2020)

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  24. [24]

    Eshaghi, M. S. et al. V ariational physics-informed neural operator (vino) for solving partial differential equations.Computer Methods in Applied Mechanics and Engineering 437, 117785 (2025)

    work page 2025

  25. [25]

    Hao, Z. et al. GNOT: A general neural operator transformer for operator learning. In Krause, A. et al. (eds.) Proceedings of the 40th International Conference on Machine Learning , vol. 202 of Proceedings of Machine Learning Research , 12556–12569 (PMLR, 2023). Available at https://proceedings.mlr.press/v202/hao23c.html

    work page 2023

  26. [26]

    & Karniadakis, G

    Shih, B., Peyvan, A., Zhang, Z. & Karniadakis, G. E. Transformers as neural operators for solutions of differential equations with finite regularity. Computer Methods in Applied Mechanics and Engineering 434, 117560 (2025) . Available at https://www.sciencedirect.com/science/article/pii/S0045782524008144

    work page 2025

  27. [27]

    Guibas, J. et al. Adaptive fourier neural operators: Efficient token mixers for transformers. arXiv preprint arXiv:2111.13587 (2021)

    work page arXiv 2021

  28. [28]

    Xu, M. et al. Equivariant graph neural operator for modeling 3d dynamics. arXiv preprint arXiv:2401.11037 (2024)

    work page arXiv 2024

  29. [29]

    Fu, X. et al. Spatio-temporal neural operator on complex geometries. Computer Physics Communica- tions 315, 109754 (2025) . Available at https://www.sciencedirect.com/science/article/pii/ S0010465525002565

    work page 2025

  30. [30]

    & Lin, G

    Zheng, H. & Lin, G. Muti-fidelity prediction and uncertainty quantification with laplace neural operators for parametric partial differential equations. arXiv preprint arXiv:2502.00550 (2025)

    work page arXiv 2025

  31. [31]

    & Tang, H

    Li, S., Wang, T., Sun, Y . & Tang, H. Multi-physics simulations via coupled fourier neural operator. arXiv preprint arXiv:2501.17296 (2025)

    work page arXiv 2025

  32. [32]

    R., Holl, P

    Um, K., Brand, R., Fei, Y . R., Holl, P . & Thuerey, N. Solver-in-the-loop: Learning from differentiable physics to interact with iterative pde-solvers. Advances in neural information processing systems 33, 6111–6122 (2020)

    work page 2020

  33. [33]

    Learning Neural PDE Solvers with Convergence Guarantees

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

    work page arXiv 1906

  34. [34]

    He, J. & Xu, J. Mgnet: A unified framework of multigrid and convolutional neural network. Science china mathematics 62, 1331–1354 (2019)

    work page 2019

  35. [35]

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

    work page 2022

  36. [36]

    & Y ang, H

    Huang, J., Wang, H. & Y ang, H. Int-deep: A deep learning initialized iterative method for nonlinear problems. Journal of computational physics 419, 109675 (2020)

    work page 2020

  37. [37]

    & Y avneh, I

    Luz, I., Galun, M., Maron, H., Basri, R. & Y avneh, I. Learning algebraic multigrid using graph neural networks. In III, H. D. & Singh, A. (eds.) Proceedings of the 37th International Conference on Machine Learning , vol. 119 of Proceedings of Machine Learning Research , 6489–6499 (PMLR, 2020). Available at https://proceedings.mlr.press/ v119/luz20a.html. 12/15

    work page 2020

  38. [38]

    & Kimmel, R

    Greenfeld, D., Galun, M., Basri, R., Y avneh, I. & Kimmel, R. Learning to optimize multigrid PDE solvers. In Chaudhuri, K. & Salakhutdinov, R. (eds.) Proceedings of the 36th International Conference on Machine Learning , vol. 97 of Proceedings of Machine Learning Research , 2415–2423 (PMLR, 2019). Available at https://proceedings.mlr.press/ v97/greenfeld19a.html

    work page 2019

  39. [39]

    & Treister, E

    Azulay, Y . & Treister, E. Multigrid-augmented deep learning preconditioners for the helmholtz equation. SIAM Journal on Scientific Computing 45, S127–S151 (2022)

    work page 2022

  40. [40]

    & Rackauckas, C

    Tan, S., Miao, K., Edelman, A. & Rackauckas, C. Scalable higher-order nonlinear solvers via higher-order automatic differentiation. arXiv preprint arXiv:2501.16895 (2025)

    work page arXiv 2025

  41. [41]

    Fast meta-solvers for 3d complex-shape scatterers using neural operators trained on a non-scattering problem

    Lee, Y .et al. Fast meta-solvers for 3d complex-shape scatterers using neural operators trained on a non-scattering problem. Computer Methods in Applied Mechanics and Engineering 446, 118231 (2025)

    work page 2025

  42. [42]

    & Fritzen, F

    Herb, J. & Fritzen, F. Accelerating conjugate gradient solvers for homogenization problems with unitary neural operators. arXiv preprint arXiv:2508.02681 (2025)

    work page arXiv 2025

  43. [43]

    & Xiang, Y

    Giraud, L., Kruse, C., Mycek, P ., Shpakovych, M. & Xiang, Y . Neural network preconditioning: a case study for the solution of the parametric Helmholtz equation . Ph.D. thesis, Inria Centre at the University of Bordeaux, France (2025). Available at https://hal.science/hal-05157038

    work page 2025

  44. [44]

    Zhang, E. et al. Blending neural operators and relaxation methods in pde numerical solvers. Nature Machine Intelligence 6, 1303–1313 (2024)

    work page 2024

  45. [45]

    & Qin, H

    Han, X., Hou, F. & Qin, H. Ugrid: An efficient-and-rigorous neural multigrid solver for linear pdes. arXiv preprint arXiv:2408.04846 (2024)

    work page arXiv 2024

  46. [46]

    Huang, R., Chang, K., He, H., Li, R. & Xi, Y . Reducing operator complexity in algebraic multigrid with machine learning approaches. arXiv preprint arXiv:2307.07695 (2023)

    work page arXiv 2023

  47. [47]

    Leveraging Operator Learning to Acceler- ate Convergence of the Preconditioned Conjugate Gradient Method

    Kopaniˇcáková, A., Lee, Y . & Karniadakis, G. E. Leveraging operator learning to accelerate convergence of the precondi- tioned conjugate gradient method. arXiv preprint arXiv:2508.00101 (2025)

    work page arXiv 2025

  48. [48]

    & Hernández, J

    Rubio, R., Ferrer, A. & Hernández, J. Preconditioning iterative solvers via the empirical interscale finite element method (eifem). Computer Methods in Applied Mechanics and Engineering 446, 118257 (2025)

    work page 2025

  49. [49]

    & Zhang, W

    Song, J., Cao, W. & Zhang, W. A matrix preconditioning framework for physics-informed neural networks based on adjoint method. arXiv preprint arXiv:2508.03421 (2025)

    work page arXiv 2025

  50. [50]

    Zhou, X.-H. et al. Neural operator-based super-fidelity: A warm-start approach for accelerating steady-state simulations. Journal of Computational Physics 529, 113871 (2025)

    work page 2025

  51. [51]

    Eshaghi, M. S. et al. Multi-head neural operator for modelling interfacial dynamics. arXiv preprint arXiv:2507.17763 (2025). Acknowledgement The authors would like to acknowledge the support provided by the German Academic Exchange Service (DAAD) through a scholarship awarded to Mohammad Sadegh Eshaghi during this research, as well as the Compute Servers ...

    work page arXiv 2025

  52. [52]

    Sample a coefficient function a(x) from a prescribed distribution

    work page

  53. [53]

    Solve the PDE numerically to obtain the corresponding solution u(x)

    work page

  54. [54]

    Evaluate the neural operator on a(x) and compute the predicted ˆu(x)

    work page

  55. [55]

    Once trained, the same model can be evaluated on unseen meshes, finer resolutions, or new geometries, owing to its continuous and mesh-independent formulation

    Minimize a loss function such as the relative L2 error, L (θ ) = ∥u − ˆu∥L2(D) ∥u∥L2(D) . Once trained, the same model can be evaluated on unseen meshes, finer resolutions, or new geometries, owing to its continuous and mesh-independent formulation. Interpretation and Applications. Neural operators provide a unifying framework for learning solution operato...

    work page 2000