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REVIEW 4 major objections 4 minor 59 references

DD-DeepONet: Domain decomposition and DeepONet for solving partial differential equations in three application scenarios

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper claims that decomposing a complex computational domain into simple rectangles or cuboids and training a separate DeepONet on each subdomain reduces training difficulty, cuts per-network data and memory needs, and speeds up…

desk verdict Incremental but relevant engineering contribution; the central efficiency claim hinges on an unvalidated stitching assumption that the referee must check. read the letter →

arxiv 2508.02717 v1 pith:GERMTL6M submitted 2025-07-31 math.NA cs.NA

Bo Yang , Xingquan Li , Jie Zhao , Ying Jiang This is my paper · ORCID
classification math.NAcs.NA MSC 65N5568T07
keywords DomaindecompositionDeepONetOperatorlearningPartialdifferentialequationsShape-dependentStretchingtransformationRepetitivesimulationNavier-Stokes
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

DD-DeepONet is a framework for training DeepONet operator networks piecewise, rather than on an entire complex domain at once. The central move is to decompose a complicated geometry—primarily assemblies of rectangles and cuboids—into simple subdomains, train an independent DeepONet on each, and combine the subdomain outputs to solve a PDE; stretching transformations extend the same idea to families of simple shapes. The paper applies this divide-and-conquer strategy to three repetitive-simulation scenarios (geometry fixed with varying boundary conditions/parameters, geometry varying, and all varying) on Laplace, Poisson, Navier–Stokes, and drift–diffusion equations. The empirical claim is that this decomposition reduces training difficulty, needs smaller datasets and less GPU memory per network, and accelerates obtaining solutions, which would matter for engineering settings that require many rapid solves.

What carries the argument

The central machinery is the pairing of two established ideas: DeepONet, an operator network trained to map input functions (boundary conditions, parameters, or geometry descriptors) to solution functions, and the Schwarz-style domain decomposition of a complex region into elementary rectangles or cuboids. Each subdomain is assigned its own DeepONet, so the global geometry never has to be encoded in a single network; a stretching transformation handles shape-dependent families by mapping simple reference shapes onto the target shapes. This division is what is claimed to cut per-network training cost and memory while enabling reuse of the same subdomain net across many solves.

What would settle it

Solve a Poisson equation on an L-shaped domain with DD-DeepONet using two rectangular subdomains, and compare the solution along the shared interface to a high-resolution finite-element reference; if the interface jump does not shrink as each subdomain network's training loss is driven down, the decomposition's accuracy claim fails.

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Extended reading notes

Core claim

The paper proposes that the universal-approximation power of DeepONet can be made scalable and geometry-friendly by combining it with classical domain decomposition. Instead of encoding the full irregular domain into one network, DD-DeepONet trains several small DeepONets, each responsible for a rectangle or cuboid subdomain, then assembles their outputs into a global solution; for shape-dependent problems, a stretching map pulls a simple reference geometry onto each target shape. The authors demonstrate this on prototypical Laplace, Poisson, Navier–Stokes, and drift–diffusion equations across the three scenarios, and report that per-network dataset size and VRAM shrink and solution acquisition speeds up. In the paper's framing, the decomposition is reversible—complex geometries are built from simple pieces via composition of subdomain operators.

Load-bearing premise

The framework assumes that breaking a complex domain into rectangles or cuboids (or stretching simple shapes) and training independent per-subdomain DeepONets produces a combined solution that remains accurate, without a demonstrated error analysis for the stitching at subdomain interfaces.

Editorial extensions

If this is right

  • Training one global operator on an irregular 3D geometry is replaced by training several smaller networks on simple subdomains, reducing per-network dataset and VRAM requirements.
  • The per-subdomain networks are reusable: the same trained subdomain net can be applied to a family of geometries obtained by stretching the same base shape.
  • The method's scalability is tied to the number of subdomains, so larger or more detailed geometries can be handled by adding more small DeepONets rather than expanding a single one.
  • Across the three tested scenarios, the same decomposition pipeline handles variation in boundary conditions, parameters, and geometry, so a single framework covers the three repetitive-simulation cases.
  • Demonstrations on Laplace, Poisson, Navier–Stokes, and drift–diffusion equations indicate the decomposition strategy is not specific to one PDE family.

Reading between the lines

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

  • One inference beyond the paper's experiments: the stitching of subdomain solutions resembles classical Schwarz alternating methods, so explicitly enforcing interface continuity (for example by iterating between subdomain networks) could reduce the accuracy penalty that the paper does not analyze.
  • The decomposition into rectangles and cuboids could be extended to simplex meshes or merged with adaptive refinement, placing subdomain boundaries where the solution varies most and potentially tightening the overall error bound.
  • The claimed per-network memory and data savings suggest a natural stress test: hold the total training budget fixed and split a domain into more subdomains; if error grows with the number of interfaces, the savings come at a cost that should be quantified.
  • Because per-subdomain networks are independent, distributed training across multiple GPUs is a direct corollary, although the paper reports only per-network VRAM, not end-to-end wall-clock scaling.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. DD-DeepONet combines the classical domain decomposition method (DDM) with the DeepONet operator-learning architecture to address three repetitive-PDE-solving scenarios: fixed geometry with varying boundary conditions and parameters (S1), varying geometry with fixed BCs and parameters (S2), and geometry, BCs, and parameters all varying (S3). The proposed framework decomposes complex geometries composed of rectangles and cuboids into simpler subdomains, trains a DeepONet per subdomain, and combines the subdomain solutions; stretching transformations are proposed for shape-dependent problems. The abstract claims that the method reduces training difficulty, requires smaller datasets and less VRAM per network, and accelerates solution acquisition. The text available for review comprises the abstract, the first part of the introduction, the references, and little else; the method description, numerical experiments, error analyses, and any comparison with baselines are not present in the provided text.

Significance. If the claimed efficiencies are demonstrated with accuracy comparable to a single global DeepONet, the paper would offer a practically useful scalability mechanism for operator learning on complex and varying geometries, with clear engineering relevance in areas such as VLSI parasitic extraction, shape optimization, and clinical prognostication. The taxonomy of three application scenarios is clear and well motivated, and the combination of DDM's divide-and-conquer rationale with operator learning is a natural and potentially fruitful idea. The paper's most valuable intended contributions—quantitative evidence on dataset size, VRAM, training cost, and inference speed relative to established geometry-handling baselines—cannot, however, be credited on the basis of the visible text, because the experimental section and the stitching analysis are absent. The visible portion provides no machine-checked proofs, reproducible code, or error tables; the assessment therefore rests entirely on abstract-level claims that the full manuscript must substantiate.

major comments (4)
  1. [Abstract; §1 (Introduction)] The central efficiency claims of the abstract—reduced training difficulty, smaller datasets, lower VRAM per network, and faster solution acquisition—are asserted without any supporting experimental evidence in the text provided for review: there are no error tables, no baselines, and no experimental setup. This is load-bearing, since the paper's contribution is presented as an empirical demonstration. The authors must supply the full experimental section, including accuracy comparisons (e.g., relative L2 error over the whole domain) against a single global DeepONet and against established geometry-handling operators such as Geo-FNO [33], MIONet [34,50], and point-cloud or geometry-adaptive operators [21,22,40,42,43,45,46], together with the dataset sizes, VRAM usage, and wall-clock times that substantiate each abstract claim.
  2. [Abstract; §1 (Introduction)] The core mechanism of the method—training subdomain DeepONets independently and then combining their solutions—is stated, but no stitching analysis appears anywhere in the visible text. There are no interface conditions, no overlap or non-overlap treatment, no data-transmission scheme between subdomains, and no bound on the interface mismatch relative to the global approximation error. In classical DDM, independent subdomain solves require explicit coupling through interface data (see [48,49,52,53]); the authors should describe their stitching procedure and verify numerically (and, where possible, analytically) that the combined solution is continuous and satisfies the correct interface conditions. If the subdomain networks are trained on the global solution restricted to each subdomain, the paper must still demonstrate that the assembled solution retains the accuracy of a single global DeepONet; otherwise the claimed cost savings amount to an unquantified accuracy-for-efficiency trade-off rather than a genuine acceleration.
  3. [Abstract] The abstract introduces 'stretching transformations' as a means of solving shape-dependent problems on simple geometries, but the visible text never defines these transformations. Because a coordinate transformation changes the PDE through Jacobian and metric terms, and also transforms the boundary conditions, the authors must state the transformation explicitly, specify the class of geometries to which it applies, and show that solving the transformed problem reproduces the solution of the original problem. This is load-bearing for scenarios S2 and S3, where geometry variation is the central difficulty.
  4. [§1 (Introduction)] The introduction asserts that operator learning 'remains ineffective' for scenarios S2 and S3, yet the reference list contains numerous recent methods designed precisely for varying geometries (e.g., Geo-FNO [33], MIONet [34,50], and geometry-adaptive or point-cloud operators [21,22,40,42,43,45,46]). The claim is too strong unless qualified to standard fixed-geometry DeepONet, and the promised two-category taxonomy of existing solutions breaks off at 'The approaches broadly fall into two categories:' on page 2. The authors should complete the related-work discussion and position DD-DeepONet against these baselines, which are also the natural comparison targets for the experiments.
minor comments (4)
  1. [Abstract] The abstract contains typographical and extraction errors: 'Poission' should be 'Poisson', 'N-S' should be expanded as 'Navier-Stokes' at first use, and the abstract displays missing spaces in phrases such as 'smallerdatasetsandVRAMpernetwork' and 'solveshape-dependentproblems'. The final version should be cleaned.
  2. [§1 (Introduction)] The statement that traditional numerical methods' 'computational demands tend to increase exponentially with the increase in dimensionality and the refinement of the discretization' is imprecise: for standard FEM/FDM, error reduction by mesh refinement typically incurs polynomial, not exponential, growth in cost. A more careful complexity statement would strengthen the motivation.
  3. [References] References [31]–[59] are not cited in the visible text; the authors should ensure that all references are cited in the completed manuscript, particularly the domain-decomposition and interface-related references [48,49,52,53] that are listed but not discussed in the visible portion.
  4. [§1 (Introduction)] The text uses 'residual neural networks (RNN)', but the standard abbreviation for residual networks is 'ResNet', while 'RNN' conventionally denotes recurrent neural networks. This should be disambiguated.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found in the visible text; the efficiency claims are empirical and no load-bearing step reduces to its own inputs.

full rationale

The visible portion of the manuscript (title, abstract, introduction, and references) presents DD-DeepONet as a combination of domain decomposition with DeepONet and reports experimental outcomes; it contains no derivation that defines a predicted quantity in terms of the same quantity being fitted. The claimed reductions in training difficulty, dataset size, and VRAM are empirical comparisons rather than quantities constructed from the model output itself, so no fitted input is renamed as a prediction. The skeptical concern about the stitching of independently trained subdomain solutions is a possible missing-analysis or correctness risk, not a circularity that can be exhibited by quoting an equation where X is defined in terms of Y or where a fitted parameter is called a prediction. The only reference sharing an author of the present paper is [32] (Meng, Lu, and Jiang, with Y. Jiang as a co-author), but it is not cited anywhere in the visible text and no visible argument relies on it, so it is not load-bearing. Under the hard rules requiring a quoted reduction for a circularity finding, the appropriate verdict is no significant circularity with score 0.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

Only the abstract and introduction are available, so the ledger is inferred from those. No fitted constants or invented entities are visible; the main assumptions concern domain decomposability and stretching validity.

free parameters (2)
  • Number of subdomains in the domain decomposition
    The abstract does not specify how the geometry is partitioned; this is a hand-chosen hyperparameter that directly affects the size of each subproblem and the stitching quality.
  • Stretching transformation coefficients
    The abstract states stretching transformations are applied to simple geometries, but the parameter values are not given; these would depend on the target shape.
assumptions (3)
  • domain assumption The PDEs are well-posed on each rectangular/cuboid subdomain, and stitching subdomain solutions yields a correct global solution.
    This is the core premise of the domain decomposition approach; no proof is given in the visible text.
  • ad hoc to paper Stretching transformations map shape-dependent problems into simple geometries without altering the PDE in a way that ruins the solution.
    The abstract asserts this transformation as the mechanism for shape-dependent problems, but the derivation is not visible.
  • domain assumption The target applications have geometries that can be represented as unions of rectangles and cuboids.
    The abstract explicitly restricts to such geometries, which limits generality but is acceptable for some engineering applications.

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Cite this review

Pith. "Pith review of DD-DeepONet: Domain decomposition and DeepONet for solving partial differential equations in three application scenarios." pith.science (2026). https://pith.science/paper/GERMTL6M

@misc{pith2026250802717,
  author       = {Pith},
  title        = {Pith review of: DD-DeepONet: Domain decomposition and DeepONet for solving partial differential equations in three application scenarios},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GERMTL6M}},
  note         = {Machine review of arXiv:2508.02717}
}
read the original abstract

In certain practical engineering applications, there is an urgent need to perform repetitive solving of partial differential equations (PDEs) in a short period. This paper primarily considers three scenarios requiring extensive repetitive simulations. These three scenarios are categorized based on whether the geometry, boundary conditions(BCs), or parameters vary. We introduce the DD-DeepONet, a framework with strong scalability, whose core concept involves decomposing complex geometries into simple structures and vice versa. We primarily study complex geometries composed of rectangles and cuboids, which have numerous practical applications. Simultaneously, stretching transformations are applied to simple geometries to solve shape-dependent problems. This work solves several prototypical PDEs in three scenarios, including Laplace, Poission, N-S, and drift-diffusion equations, demonstrating DD-DeepONet's computational potential. Experimental results demonstrate that DD-DeepONet reduces training difficulty, requires smaller datasets andVRAMper network, and accelerates solution acquisition.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

59 extracted references · 55 canonical work pages

  1. [33]

    Fourier neural operator with learned deformations for pdes on general geometries.J

    Zongyi Li, Daniel Zhengyu Huang, Burigede Liu, and Anima Anandkumar. Fourier neural operator with learned deformations for pdes on general geometries.J. Mach. Learn. Res., 24(388):1–26, 2023

    2023

  2. [1]

    Zaky, Ahmed S

    Mahmoud A. Zaky, Ahmed S. Hendy, and Jorge E. Macias-Diaz. Semi-implicit galerkin-legendre spectral schemes for nonlinear time-space fractional diffusion-reaction equations with smooth and nonsmooth solutions.J. Sci. Comput., 82(1), JAN 9 2020

    work page 2020

  3. [2]

    Computational Methods for Electromagnetic Phenomena: Electrostatics in Solvation, Scattering, and Electron Transport.Cambridge University Press, 2013

    WeiCai. Computational Methods for Electromagnetic Phenomena: Electrostatics in Solvation, Scattering, and Electron Transport.Cambridge University Press, 2013

    work page 2013

  4. [3]

    Parallel modeling of cell suspension flow in complex micro-networks with inflow/outflow boundary conditions

    Guansheng Li, Ting Ye, and Xuejin Li. Parallel modeling of cell suspension flow in complex micro-networks with inflow/outflow boundary conditions. J. Comput. Phys., 401, JAN 15 2020

    work page 2020

  5. [4]

    Physics-informed neural networks combined with polynomial interpolation to solve nonlinear partial differential equations.Comput

    Siping Tang, Xinlong Feng, Wei Wu, and Hui Xu. Physics-informed neural networks combined with polynomial interpolation to solve nonlinear partial differential equations.Comput. Math. Appl., 132:48–62, FEB 15 2023

    work page 2023

  6. [5]

    Fuhg and Nikolaos Bouklas

    Jan N. Fuhg and Nikolaos Bouklas. The mixed deep energy method for resolving concentration features in finite strain hyperelasticity.J. Comput. Phys., 451, FEB 15 2022

    work page 2022

  7. [6]

    Finite element methods for the viscous incompressible fluid.Appl

    Wen-Jing Yan and Yi-Chen Ma. Finite element methods for the viscous incompressible fluid.Appl. Math. Comput., 185(1):547–553, FEB 1 2007

    work page 2007

  8. [7]

    J. Li, F. Liu, L. Feng, and I. Turner. A novel finite volume method for the riesz space distributed-order diffusion equation.Comput. Math. Appl., 74(4):772–783, AUG 15 2017

    work page 2017

Show all 59 references
  1. [8]

    Gray and N.M

    W.H. Gray and N.M. Schnurr. A comparison of the finite element and finite difference methods for the analysis of steady two dimensional heat conduction problems.Comput. Methods Appl. Mech. Eng., 6:243–5, Sept. 1975

    1975

  2. [9]

    O. P. Stoyanovskaya, O. A. Burmistrova, M. S. Arendarenko, and T. V. Markelova. Dispersion analysis of sph for parabolic equations: High-order kernels against tensile instability.J. Comput. Appl. Math., 457, MAR 15 2025

    2025

  3. [10]

    Classicalandreactivemoleculardynamics:Principles and applications in combustion and energy systems.Prog

    QianMao,MuyeFeng,XiZhuoJiang,YihuaRen,KaiH.Luo,andAdriC.T.vanDuin. Classicalandreactivemoleculardynamics:Principles and applications in combustion and energy systems.Prog. Energy Combust. Sci., 97, JUL 2023

    2023

  4. [11]

    Pep Espanol and Patrick B. Warren. Perspective: Dissipative particle dynamics.J. Chem. Phys., 146(15), APR 21 2017

    2017

  5. [12]

    Spectral Methods: Algorithms, Analysis and Applications.SpringerPublishingCompany,Incorporated, 1st edition, 2011

    JieShen,TaoTang,andLi-LianWang. Spectral Methods: Algorithms, Analysis and Applications.SpringerPublishingCompany,Incorporated, 1st edition, 2011

    2011

  6. [13]

    Iterative Methods for Sparse Linear Systems

    Yousef Saad. Iterative Methods for Sparse Linear Systems. Society for Industrial and Applied Mathematics, second edition, 2003

    2003

  7. [14]

    Widlund.Domain decomposition methods-algorithms and theory

    Andrea Toselli and Olof B. Widlund.Domain decomposition methods-algorithms and theory. Springer Berlin, Heidelberg, 2010

    2010

  8. [15]

    Lecun, L

    Y. Lecun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-based learning applied to document recognition.Proc. IEEE, 86(11):2278–2324, 1998

    1998

  9. [16]

    Attention is all you need

    AshishVaswani,NoamShazeer,NikiParmar,JakobUszkoreit,LlionJones,AidanN.Gomez,ŁukaszKaiser,andIlliaPolosukhin. Attention is all you need. InProceedings of the 31st International Conference on Neural Information Processing Systems , NIPS’17, California, 2017

    2017

  10. [17]

    Dgm: A deep learning algorithm for solving partial differential equations.J

    Justin Sirignano and Konstantinos Spiliopoulos. Dgm: A deep learning algorithm for solving partial differential equations.J. Comput. Phys., 375:1339–1364, DEC 15 2018

    2018

  11. [18]

    Thedeepritzmethod:Adeeplearning-basednumericalalgorithmforsolvingvariationalproblems

    E.WeinanandBingYu. Thedeepritzmethod:Adeeplearning-basednumericalalgorithmforsolvingvariationalproblems. Commun. Math. Stat., 6(1):1–12, MAR 2018

    2018

  12. [19]

    Physics-informedneuralnetworks:Adeeplearningframeworkforsolvingforwardandinverse problems involving nonlinear partial differential equations.J

    M.Raissi,P.Perdikaris,andG.E.Karniadakis. Physics-informedneuralnetworks:Adeeplearningframeworkforsolvingforwardandinverse problems involving nonlinear partial differential equations.J. Comput. Phys., 378:686–707, FEB 1 2019

    2019

  13. [20]

    Neural operator: Learning maps between function spaces with applications to pdes.J

    NikolaKovachki,ZongyiLi,BurigedeLiu,KamyarAzizzadenesheli,KaushikBhattacharya,AndrewStuart,andAnimaAnandkumar. Neural operator: Learning maps between function spaces with applications to pdes.J. Mach. Learn. Res., 24, 2023

    2023

  14. [21]

    Ascalableframeworkforlearningthegeometry- dependent solution operators of partial differential equations.Nat

    MinglangYin,NicolasCharon,RyanBrody,LuLu,NataliaTrayanova,andMauroMaggioni. Ascalableframeworkforlearningthegeometry- dependent solution operators of partial differential equations.Nat. Comput. Sci., 4(12), DEC 2024

    2024

  15. [22]

    Geom-deeponet: A point-cloud-based deep operator network for field predictions on 3d parameterized geometries.Comput

    Junyan He, Seid Koric, Diab Abueidda, Ali Najafi, and Iwona Jasiuk. Geom-deeponet: A point-cloud-based deep operator network for field predictions on 3d parameterized geometries.Comput. Methods Appl. Mech. Eng., 429, SEP 1 2024

    2024

  16. [23]

    Multilayer feedforward networks are universal approximators.Neural Netw., 2(5):359–366, 1989

    Kurt Hornik, Maxwell Stinchcombe, and Halbert White. Multilayer feedforward networks are universal approximators.Neural Netw., 2(5):359–366, 1989

    1989

  17. [24]

    Universalapproximationtononlinearoperatorsbyneuralnetworkswitharbitraryactivationfunctionsandits application to dynamical systems.IEEE Trans

    TianpingChenandHongChen. Universalapproximationtononlinearoperatorsbyneuralnetworkswitharbitraryactivationfunctionsandits application to dynamical systems.IEEE Trans. Neural Netw., 6(4):911–917, 1995

    1995

  18. [25]

    Nonlocal kernel network (nkn): A stable and resolution-independent deep neural network.J

    Huaiqian You, Yue Yu, Marta D’Elia, Tian Gao, and Stewart Silling. Nonlocal kernel network (nkn): A stable and resolution-independent deep neural network.J. Comput. Phys., 469:111536, 2022

    2022

  19. [26]

    Koopman neural operator as a mesh-free solver of non-linear partial differential equations.J

    Wei Xiong, Xiaomeng Huang, Ziyang Zhang, Ruixuan Deng, Pei Sun, and Yang Tian. Koopman neural operator as a mesh-free solver of non-linear partial differential equations.J. Comput. Phys., 513:113194, 2024

    2024

  20. [27]

    Seidman, Leonardo Ferreira Guilhoto, Victor M

    Georgios Kissas, Jacob H. Seidman, Leonardo Ferreira Guilhoto, Victor M. Preciado, George J. Pappas, and Paris Perdikaris. Learning operators with coupled attention.J. Mach. Learn. Res., 23(215):1–63, 2022

    2022

  21. [28]

    Fourierneuraloperatorforparametricpartialdifferentialequations

    ZongyiLi,NikolaBorislavovKovachki,KamyarAzizzadenesheli,Burigedeliu,KaushikBhattacharya,AndrewStuart,andAnimaAnandku- mar. Fourierneuraloperatorforparametricpartialdifferentialequations. In International Conference on Learning Representations,ICLR’21, Vienna, 2021

    2021

  22. [29]

    Learningnonlinearoperatorsviadeeponetbasedonthe universal approximation theorem of operators.Nat

    LuLu,PengzhanJin,GuofeiPang,ZhongqiangZhang,andGeorgeEmKarniadakis. Learningnonlinearoperatorsviadeeponetbasedonthe universal approximation theorem of operators.Nat. Mach. Intell., 3(3):218+, MAR 2021

    2021

  23. [30]

    Kevrekidis, and Michael D

    Bahador Bahmani, Somdatta Goswami, Ioannis G. Kevrekidis, and Michael D. Shields. A resolution independent neural operator.Comput. Methods Appl. Mech. Eng., 444:118113, 2025

    2025

  24. [31]

    Phygeonet: Physics-informed geometry-adaptive convolutional neural networks for solving parameterized steady-state pdes on irregular domain.J

    Han Gao, Luning Sun, and Jian-Xun Wang. Phygeonet: Physics-informed geometry-adaptive convolutional neural networks for solving parameterized steady-state pdes on irregular domain.J. Comput. Phys., 428, MAR 1 2021

    2021

  25. [32]

    Three operator learning models for solving boundary integral equations in 2d connected domains

    Bin Meng, Yutong Lu, and Ying Jiang. Three operator learning models for solving boundary integral equations in 2d connected domains. Appl. Math. Model., 143, JUL 2025. Yang et al. :Preprint submitted to Elsevier Page 23 of 24 DD-DeepONet

    2025

  26. [34]

    Learning solution operators of pdes defined on varying domains via mionet

    Shanshan Xiao, Pengzhan Jin, and Yifa Tang. Learning solution operators of pdes defined on varying domains via mionet. ArXiv, abs/2402.15097, 2024

    2024 arXiv

  27. [35]

    Physics-informed pointnet: A deep learning solver for steady-state incompressible flows and thermal fields on multiple sets of irregular geometries.J

    Ali Kashefi and Tapan Mukerji. Physics-informed pointnet: A deep learning solver for steady-state incompressible flows and thermal fields on multiple sets of irregular geometries.J. Comput. Phys., 468:111510, 2022

    2022

  28. [36]

    Nonparametric boundary geometry in physics informed deep learning

    Scott Cameron, Arnu Pretorius, and Stephen Roberts. Nonparametric boundary geometry in physics informed deep learning. InProceedings of the 37th International Conference on Neural Information Processing Systems , NIPS ’23, New Orleans, 2023

    2023

  29. [37]

    Novel deeponet architecture to predict stresses in elastoplastic structures with variable complex geometries and loads.Comput

    Junyan He, Seid Koric, Shashank Kushwaha, Jaewan Park, Diab Abueidda, and Iwona Jasiuk. Novel deeponet architecture to predict stresses in elastoplastic structures with variable complex geometries and loads.Comput. Methods Appl. Mech. Eng., 415, OCT 1 2023

    2023

  30. [38]

    Factorized fourier neural operators

    Alasdair Tran, Alexander Mathews, Lexing Xie, and Cheng Soon Ong. Factorized fourier neural operators. InThe Eleventh International Conference on Learning Representations, ICLR’23, Kigali Rwanda, 2023

    2023

  31. [39]

    Unisolver: Pde-conditional transformers towards universal neural pde solvers

    H Zhou, Y Ma, H Wu, H Wang, and M Long. Unisolver: Pde-conditional transformers towards universal neural pde solvers. InProceedings of the 42th International Conference on Machine Learning , ICML’25, Vancouver, 2025

    2025

  32. [40]

    Operator learning with neural fields: tackling pdes on general geometries

    Louis Serrano, Lise Le Boudec, Armand Kassaï Koupaï, Thomas X Wang, Yuan Yin, Jean-Noel Vittaut, and Patrick Gallinari. Operator learning with neural fields: tackling pdes on general geometries. InThirty-seventh Conference on Neural Information Processing Systems , NIPS’23, Ne...

    2023

  33. [41]

    Engsig-Karup, George Em Karniadakis, and Cheol-Ho Jeong

    Nikolas Borrel-Jensen, Somdatta Goswami, Allan P. Engsig-Karup, George Em Karniadakis, and Cheol-Ho Jeong. Sound propagation in realistic interactive 3d scenes with parameterized sources using deep neural operators.Proc. Natl. Acad. Sci., 121(2), JAN 9 2024

    2024

  34. [42]

    Point cloud neural operator for parametric pdes on complex and variable geometries.Comput

    Chenyu Zeng, Yanshu Zhang, Jiayi Zhou, Yuhan Wang, Zilin Wang, Yuhao Liu, Lei Wu, and Daniel Zhengyu Huang. Point cloud neural operator for parametric pdes on complex and variable geometries.Comput. Methods Appl. Mech. Eng., 443, AUG 1 2025

    2025

  35. [43]

    Geometry-informed neural operator for large-scale 3d pdes

    Zongyi Li, Nikola Borislavov Kovachki, Chris Choy, Boyi Li, Jean Kossaifi, Shourya Prakash Otta, Mohammad Amin Nabian, Maximilian Stadler, Christian Hundt, Kamyar Azizzadenesheli, and Anima Anandkumar. Geometry-informed neural operator for large-scale 3d pdes. In Proceedings o...

    2023

  36. [44]

    Gnot:ageneral neuraloperatortransformerforoperatorlearning

    ZhongkaiHao,ZhengyiWang,HangSu,ChengyangYing,YinpengDong,SongmingLiu,ZeCheng,JianSong,andJunZhu. Gnot:ageneral neuraloperatortransformerforoperatorlearning. In Proceedings of the 40th International Conference on Machine Learning,ICML’23,2023

    2023

  37. [45]

    Domain agnostic fourier neural operators

    Ning Liu, Siavash Jafarzadeh, and Yue Yu. Domain agnostic fourier neural operators. InThirty-seventh Conference on Neural Information Processing Systems, NIPS’23, Vancouver, 2023

    2023

  38. [46]

    Transolver++: An accurate neural solver for pdes on million-scale geometries

    H Luo, H Wu, H Zhou, L Xing, Y Di, J Wang, and M Long. Transolver++: An accurate neural solver for pdes on million-scale geometries. In Proceedings of the 42th International Conference on Machine Learning , ICML’25, Vancouver, 2025

    2025

  39. [47]

    Interfacing finite elements with deep neural operators for fast multiscale modeling of mechanics problems.Comput

    Minglang Yin, Enrui Zhang, Yue Yu, and George Em Karniadakis. Interfacing finite elements with deep neural operators for fast multiscale modeling of mechanics problems.Comput. Methods Appl. Mech. Eng., 402(SI), DEC 1 2022

    2022

  40. [48]

    Afictitiousdomainapproachtothedirectnumericalsimulationofincompressible viscous flow past moving rigid bodies: Application to particulate flow.J

    RGlowinski,TWPan,TIHesla,DDJoseph,andJPériaux. Afictitiousdomainapproachtothedirectnumericalsimulationofincompressible viscous flow past moving rigid bodies: Application to particulate flow.J. Comput. Phys., 169(2):363–426, MAY 20 2001

    2001

  41. [49]

    On schwarz alternating methods for nonlinear elliptic pdes.SIAM J

    SH Lui. On schwarz alternating methods for nonlinear elliptic pdes.SIAM J. Sci. Comput, 21(4):1506–1523, APR 27 2000

    2000

  42. [50]

    Mionet: Learning multiple-input operators via tensor product.SIAM J

    Pengzhan Jin, Shuai Meng, and Lu Lu. Mionet: Learning multiple-input operators via tensor product.SIAM J. Sci. Comput , 44(6):A3490– A3514, 2022

    2022

  43. [51]

    Enhanced deeponet for modeling partial differential operators considering multiple input functions.ArXiv, abs/2202.08942, 2022

    Lesley Tan and Liang Chen. Enhanced deeponet for modeling partial differential operators considering multiple input functions.ArXiv, abs/2202.08942, 2022

    2022 arXiv

  44. [52]

    On accelerated convergence of nonoverlapping schwarz methods.J

    SH Lui. On accelerated convergence of nonoverlapping schwarz methods.J. Comput. Appl. Math., 130(1-2):309–321, MAY 1 2001

    2001

  45. [53]

    An analysis for a nonoverlapping domain decomposition iterative procedure.SIAM J

    Qingping Deng. An analysis for a nonoverlapping domain decomposition iterative procedure.SIAM J. Sci. Comput, 18:1517–25, Sept. 1997

    1997

  46. [54]

    DeepXDE: A deep learning library for solving differential equations.SIAM Rev., 63(1):208–228, 2021

    Lu Lu, Xuhui Meng, Zhiping Mao, and George Em Karniadakis. DeepXDE: A deep learning library for solving differential equations.SIAM Rev., 63(1):208–228, 2021

    2021

  47. [55]

    Advanced Field-Solver Techniques for RC Extraction of Integrated Circuits

    XirenWangandWenjianYu. Advanced Field-Solver Techniques for RC Extraction of Integrated Circuits. SpringerBerlin,Heidelberg,2014

    2014

  48. [56]

    Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations

    Maziar Raissi, Alireza Yazdani, and George Em Karniadakis. Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations. SCIENCE, 367(6481):1026+, FEB 28 2020

    2020

  49. [57]

    Figueiredo, Julian L

    Romulo Aparecido de Paula Jr, Ivan Aldaya, Tiago Sutili, Rafael C. Figueiredo, Julian L. Pita, and Yesica R. R. Bustamante. Design of a silicon mach-zehnder modulator via deep learning and evolutionary algorithms.Sci. Rep., 13(1), SEP 5 2023

    2023

  50. [58]

    A class of finite element methods with averaging techniques for solving the three-dimensional drift-diffusion model in semiconductor device simulations.J

    Qianru Zhang, Qin Wang, Linbo Zhang, and Benzhuo Lu. A class of finite element methods with averaging techniques for solving the three-dimensional drift-diffusion model in semiconductor device simulations.J. Comput. Phys., 458, JUN 1 2022

    2022

  51. [59]

    Deepoheat: Operator learning-based ultra-fast thermal simulation in 3d-ic design

    Ziyue Liu, Yixing Li, Jing Hu, Xinling Yu, Shinyu Shiau, Xin Ai, Zhiyu Zeng, and Zheng Zhang. Deepoheat: Operator learning-based ultra-fast thermal simulation in 3d-ic design. In2023 60th ACM/IEEE Design Automation Conference, DAC’23, 2023. Yang et al. :Preprint submitted to E...

    2023

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.