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REVIEW 4 major objections 5 minor 39 references

An explainable operator approximation framework under the guideline of Green's function

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper introduces GreensONet, a neural operator that learns the Green's function of a linear PDE and solves it for new source terms and boundary conditions by direct integration.

desk verdict Solid scalar Green's function learning; the Stokes extension is under-derived and the boundary term looks wrong, so the vector claim should be revised. read the letter →

arxiv 2412.16644 v2 pith:CETUHFSO submitted 2024-12-21 physics.comp-ph cs.LG

Jianghang Gu , Ling Wen , Yuntian Chen , Shiyi Chen This is my paper · ORCID
classification physics.comp-phcs.LG
keywords Green'sfunctionoperatorapproximationintegralsolutionsneuralbinarystructurednetworkGaussianrandomfieldsStokesflowreaction-diffusionequations
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

GreensONet (GON) is a deep operator network that learns the Green's function of a linear PDE once and then obtains solutions for new source terms and Dirichlet boundary data by direct integration, without retraining. The Trunk Net outputs $G(x,\xi)$, the Branch Net outputs $\nabla_\xi G(x,\xi)$, and Gaussian quadrature turns the volume and boundary integrals into matrix operations. On 3D benchmarks - heat conduction in a finned tube, heterogeneous reaction-diffusion on a plate and in a pipe, and lid-driven Stokes flow - GON reports lower testing errors than PINN, DeepONet, PI-DeepONet, and FNO, with inference times near 0.1-0.5 seconds. If the learned object is in fact the Green's function, the framework provides a boundary-invariant and source-term-invariant surrogate solver, which is the claim the paper sets out to establish.

What carries the argument

The load-bearing object is the learned Green's function $G(x,\xi)$ and its gradient $\nabla_\xi G(x,\xi)$. The Trunk Net produces $G$, the Branch Net produces $\nabla_\xi G$, and Gaussian quadrature on a tetrahedral mesh converts the volume and boundary integrals into matrix operations, so one forward pass yields the solution at all query points. The networks are binary-structured neural networks, a mixture-of-experts-style architecture in which neuron blocks branch like a binary tree; the paper uses them because Green's functions are singular near the diagonal $x=\xi$, and BsNNs capture such local features with faster convergence than ordinary feed-forward networks. Domain partitioning makes the per-point Green's-function evaluations independent and parallelizable.

What would settle it

Compute a high-accuracy finite-element reference for a 3D lid-driven Stokes flow under a single smooth body force, and compare GON's predicted velocity with the velocity rebuilt from its learned Green's matrix using the proper traction boundary integral; agreement validates the vector extension, while systematic disagreement would show that the Branch Net is not learning the boundary term the Green's representation requires.

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Extended reading notes

Core claim

The central claim is that a DeepONet-style architecture can learn the Green's function of a linear PDE system rather than a direct input-output map. For the scalar problem, GON approximates $G(x,\xi)$ with the Trunk Net and $\nabla_\xi G(x,\xi)$ with the Branch Net, then reconstructs the solution as $$u(x)=\int_\$\Omega$ f(\xi)G(x,\xi)\,d\xi-\int_{\partial\$\Omega$} g(\xi)a(\xi)(\nabla_\xi G(x,\xi)\cdot n_\xi)\,dS(\xi),$$ so user-supplied source terms and Dirichlet boundary data enter only through the integrals. Training minimizes the squared difference between this integrated prediction and finite-element reference solutions over a family of sources and boundary conditions drawn from Gaussian random fields, and the same trained networks are then invariant to the particular $f$ and $g$. For Stokes flow, the paper learns a $3\times3$ Green's-function matrix one row at a time. On the tested cases, GON's testing $L^2$ errors range from about $2.6\times10^{-5}$ to $5.8\times10^{-4}$, below all compared baselines.

Load-bearing premise

The load-bearing premise is that one learned Green's function, together with the boundary integral of its normal derivative, represents the solution for every source term and boundary condition; for vector flow problems like Stokes this boundary representation is assumed rather than derived, and the true boundary term involves pressure and wall stress, not just a normal derivative of the Green's matrix.

Editorial extensions

If this is right

  • If the central claim holds, a single trained GON solves a whole family of Dirichlet problems with new source terms and boundary data in one forward pass, with no retraining and no iterative solver.
  • The learned Green's function is independent of the particular $f$ and $g$, so the same network can serve as a reusable forward solver in outer loops for inverse problems, optimization, and uncertainty quantification.
  • Because the solution is assembled by explicit integration, the approximation inherits the superposition principle of linear equations, which is the source of its generalization behaviour.
  • On the reported benchmarks, the method yields lower testing $L^2$ and $H^1$ semi-norm errors than PINN, DeepONet, PI-DeepONet, and FNO, including on an irregular finned-tube domain and on a pipe with a discontinuous diffusion coefficient.
  • For Stokes flow, the framework learns a $3\times3$ Green-function matrix, and the paper reports that it captures the main lid-driven-cavity vortex structure that the comparison models miss.

Reading between the lines

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

  • Editorial inference: if the learned object genuinely is the Green's function, the method offers a way to numerically discover Green's functions for variable-coefficient operators on complex domains, which are rarely available analytically; this would make $G$ itself a reusable scientific output, not just a means to a solution.
  • Editorial inference: a natural stress test the paper does not report is to train on only homogeneous-source data ($f=0$) and check whether the learned boundary term alone reproduces the homogeneous solutions; if it does, the boundary-invariance claim is stronger.
  • Editorial inference: because the paper uses fixed Gauss rules, accuracy should improve with adaptive quadrature concentrated near the diagonal singularity of $G$; the reported memory-time trade-off suggests this as a cheap extension.
  • Editorial inference: the Stokes boundary term in Eq. (5) is assumed by analogy; checking it against a traction-based representation, which the paper does not do, would confirm whether Case 3 is a true validation of the vector Green's-function framework or a favorable coincidence.
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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 / 5 minor

Summary. The paper proposes GreensONet (GON), a DeepONet-style architecture in which a Trunk Net outputs entries of a learned Green's function or Green's matrix and a Branch Net outputs the corresponding boundary gradient; predictions are then formed by volume and surface quadrature using the integral representation in Eq. (3) for scalars and Eq. (5) for systems. Training pairs consist of GRF-generated source/boundary conditions and FEM reference solutions. GON is tested on 2D Poisson, 3D steady heat conduction on a finned tube, heterogeneous reaction-diffusion on a flat plate and a pipe, and 3D lid-driven Stokes cavity flow, and the reported L2 errors are lower than those of PINN, DeepONet, PI-DeepONet, and FNO in all 3D cases. The central claim is that GON simultaneously learns boundary-invariant, source-invariant Green's functions for 3D bounded linear PDEs and solves them by convolution.

Significance. If the learned object is genuinely the Green's function of the system, GON would be a valuable interpretable operator-learning method: the integral representation is a real physics constraint, and test-time evaluation on unseen f and g is a genuine convolution rather than a direct fit. The paper also contributes a BsNN backbone that appears well suited to the diagonal singularity of Green's functions, a quadrature-order convergence study in Appendix A.1, H1 semi-norm comparisons in Appendix A.4, and a public code repository. However, the vector generalization in Eq. (5) is asserted without derivation and is questionable for the Stokes system, and the learned G is never compared with an exact or reference Green's function; these issues currently limit the significance of the central 'explainable Green's function' claim.

major comments (4)
  1. [Section 2, Eq. (5)] The scalar-to-vector extension is asserted with the sentence 'can be extended naturally to systems' but is never derived. For the Stokes system in Eq. (17), the correct boundary integral representation of the velocity involves the traction (stresslet) of the Green's matrix integrated against the boundary velocity, together with pressure contributions; it is not g_i(ξ)a_i(ξ)∂_nG_{i,j}. Moreover, a_i(ξ) is not defined for Eq. (17), so Eq. (5) is at least underspecified and at worst the wrong representation. Because Case 3 (Section 4.4) trains the 3×3 Green's matrix using Eq. (5) with a fixed lid-driven boundary condition, the low test error of 5.83e-4 in Table 6 does not establish that a boundary-invariant physical Green's function was learned; it may only fit the force-to-velocity map for that single boundary condition. Please either derive the correct boundary integral formula for Stokes or restrict the vector claim to systems with the same structure as the scalar operator.
  2. [Section 3.1, 3.6 and Eq. (10)] The learned G and ∇G are never compared with an exact or high-accuracy reference Green's function. The loss in Eq. (10) only measures the deviation of the integrated solution u_hat from the FEM solution u, so many non-Green kernels could produce small losses on the training set; the paper's claim that the Trunk Net 'approximates the unknown Green's functions' is therefore not directly validated. Add a pointwise comparison of the learned G and its gradient against an analytic or finely resolved reference Green's function (for example, Poisson on a cube), and test boundary/source invariance on inputs outside the GRF training distribution. Without such checks, the interpretability claim remains unsupported.
  3. [Section 3.6 and Appendix A.3] The Branch Net is not constrained to be the gradient of the Trunk Net's G; the paper explicitly states that no compatibility penalty on ∇G is imposed. Thus the 'auxiliary gradient' is an independently learned function that is only implicitly shaped through the boundary integral term in Eq. (9). This is an admitted gap between the architecture and the claimed identification of the Branch Net output with ∇_ξG. Please add a consistency check (for example, comparing BranchNet output with automatic differentiation of the Trunk Net on test points, or adding a mild compatibility term) and report the discrepancy; otherwise the role of the Branch Net in the boundary integral is not established.
  4. [Tables 3-6 and Appendix A.4] All numerical comparisons are single-run and no seed-to-seed variability is reported. Given random network initialization and random GRF draws, the claimed consistent superiority over FNO, DeepONet, PI-DeepONet, and PINN could be within run-to-run variation. Report mean ± standard deviation over at least five independent seeds for the main comparisons in Cases 1-3, and state the number of seeds used in the experiments.
minor comments (5)
  1. [Throughout] There are numerous typos and formatting issues, including 'adress', 'strucutre', 'V olterra', 'Domputing' in reference [30], and the stray 'mathb f' before the training error in Table 6; please proofread the manuscript carefully.
  2. [Table 5] The caption of Table 5 says 'case of flat plane', but the table reports the pipe case from Section 4.3(b); correct the caption.
  3. [Section 2, Eq. (5)] In Eq. (5), a_i(ξ) is called 'a vector of material parameters', but in the scalar formula Eq. (3) it is a scalar coefficient of the operator, and for the Stokes case in Eq. (17) no such coefficient appears; clarify the notation or remove a_i for systems where it is not defined.
  4. [Section 3.6] The sentence stating that the use of Trunk Net and Branch Net has 'no connection to the universal approximation theorem' is confusing, because DeepONet's trunk/branch decomposition was originally motivated by that theorem; please rephrase to say what the authors intend.
  5. [Section 4.3(a)] The text describes the computational domain as 'a cylinder with a radius of 0.08 m along the x-axis and a height of 0.8 m along the y-axis', which is inconsistent with the usual cylinder coordinate description and with Figure 10; please clarify the coordinate convention and axis orientation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: GON's test-time evaluation on unseen source/boundary data is a genuine convolution with a learned kernel; the physical Green's-function interpretation is under-validated but not definitionally forced.

full rationale

The derivation chain in Sections 2-3 is not circular. Equation (3) is a standard Green's representation for the scalar Dirichlet problem and is stated independently of the network. The trainable quantities G(x,xi) and grad_xi G(x,xi) are fitted by minimizing Eq. (10) against FEM solutions on a training family of GRF source terms and boundary conditions; at inference the model evaluates Eq. (9) with unseen f and g. The test-time map is therefore a genuine convolution with a learned kernel rather than a re-evaluation of the training fit. The paper's identification of the fitted kernel as the physical Green's function is an interpretability claim that is not independently verified (no comparison of G against an analytical Green's function, and the Branch Net is not constrained to equal the Trunk Net's derivative), but this is underdetermination or a validation gap, not a definitional reduction. The self-citations appear only in the literature review and are not load-bearing. The unsupported vector extension of Eq. (5) to Stokes is a mathematical-correctness concern, not a circularity, because the paper does not define the Green's matrix in terms of the solutions it predicts; it asserts the representation. No step reduces to its own input by construction.

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

The ledger shows the central claim is not derived from first principles: the Green's function is a high-dimensional function fitted to FEM data, the GRF lengthscale and quadrature order are user choices that affect what is learned, and the vector extension rests on an unproven boundary representation.

free parameters (4)
  • Trunk Net weights Θ_G = Not reported, learned during training
    Network parameters approximating G(x,ξ), trained by minimizing Eq. (10) against COMSOL solutions.
  • Branch Net weights Θ_∇G = Not reported, learned during training
    Network parameters approximating ∇_ξ G(x,ξ), trained jointly in the same loss.
  • GRF lengthscale ℓ_d = chosen in [0.1, 1] per case
    Sets the correlation and diversity of training f and g; affects which modes of the Green's function are identifiable (Section 3.2).
  • Gaussian quadrature rule = 3-point on triangles, 4-point on tetrahedra
    Integration order chosen as an accuracy and memory trade-off; Appendix A.1 shows errors vary with it.
assumptions (4)
  • standard math The standard Green's function representation Eq. (3) for the scalar self-adjoint operator L = -div(a grad) + r with homogeneous Dirichlet Green's function.
    Textbook result; basis of the volume and boundary integral evaluation.
  • domain assumption The learned network function G_Θ and its gradient, when convolved via Eq. (9), give the PDE solution; equivalently, the optimization recovers a kernel that behaves as the Green's function on the training family.
    The loss supervises only u, not the Green's function equation; no direct check that G_Θ satisfies L G = δ or homogeneous boundary conditions (Sections 3.3 and 3.6).
  • ad hoc to paper The Green's matrix representation Eq. (5) extends the scalar boundary integral formula to systems such as Stokes.
    Presented without derivation; for Stokes the boundary term should involve traction and pressure, not just g_i a_i ∂_n G. This is the basis of Case 3.
  • standard math Gaussian quadrature on tetrahedra and triangles accurately approximates the volume and surface integrals involving the singular Green's function.
    Numerical quadrature assumption; Appendix A.1 tests it for integration order, but no singularity-aware correction is used.

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Cite this review

Pith. "Pith review of An explainable operator approximation framework under the guideline of Green's function." pith.science (2026). https://pith.science/paper/CETUHFSO

@misc{pith2026241216644,
  author       = {Pith},
  title        = {Pith review of: An explainable operator approximation framework under the guideline of Green's function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CETUHFSO}},
  note         = {Machine review of arXiv:2412.16644}
}
read the original abstract

Traditional numerical methods, such as the finite element method and finite volume method, adress partial differential equations (PDEs) by discretizing them into algebraic equations and solving these iteratively. However, this process is often computationally expensive and time-consuming. An alternative approach involves transforming PDEs into integral equations and solving them using Green's functions, which provide analytical solutions. Nevertheless, deriving Green's functions analytically is a challenging and non-trivial task, particularly for complex systems. In this study, we introduce a novel framework, termed GreensONet, which is constructed based on the strucutre of deep operator networks (DeepONet) to learn embedded Green's functions and solve PDEs via Green's integral formulation. Specifically, the Trunk Net within GreensONet is designed to approximate the unknown Green's functions of the system, while the Branch Net are utilized to approximate the auxiliary gradients of the Green's function. These outputs are subsequently employed to perform surface integrals and volume integrals, incorporating user-defined boundary conditions and source terms, respectively. The effectiveness of the proposed framework is demonstrated on three types of PDEs in bounded domains: 3D heat conduction equations, reaction-diffusion equations, and Stokes equations. Comparative results in these cases demonstrate that GreenONet's accuracy and generalization ability surpass those of existing methods, including Physics-Informed Neural Networks (PINN), DeepONet, Physics-Informed DeepONet (PI-DeepONet), and Fourier Neural Operators (FNO).

Figures

Figures reproduced from arXiv: 2412.16644 by the authors.

Figure 1
Figure 1. The framework of GON: (a) Import user-defined free tetrahedral mesh and user-defined physical conditions; (b) Calculate the locations of [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. 3D Gaussian Field with different ℓd: a. 3 slices-Gaussian Field when ℓd = 0.1; b. 3 slices-Gaussian Field when ℓd = 0.5; c. 3 slices￾Gaussian Field when ℓd = 1. 3.3. Numerical quadrature and Loss function The computation of Eq. (3) is central to this method, where Gaussian quadrature is employed to approximate it. Specifically, the computational domain Ω is discretized into a mesh composed of unstructured tetrahedra… view at source ↗
Figure 3
Figure 3. Illustration of numerical quadrature: (a) [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: a. The structure of a BsNN consists of 3 hidden layers, each with 4 neurons (except the first and the last hidden layers, the neuron blocks in each hidden layer are fully connected to two neuron blocks in the next hidden layer). b. The structure of a FNN consists of 3 …
Figure 5
Figure 5. Figure 5: Illustration of point set partition strategy: a partition strategy based on the domain of the regular plat, partition strategy based on the point-set of the regular plat; b partition strategy based on the domain of the pipe, partition strategy based on the point-set of…
Figure 6
Figure 6. Figure 6: Illustration of of (a) structure of DeepONet and (b) structure of GON. [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: Comparison with existing Green’s function-based methods: (a) Results acquired by methods in [23], [24], [25] and our method respec￾tively; (b) Difference between exact solutions and the acquired solutions (|uˆ − u|). The hyperparameter are kept the same for fairness (E…
Figure 8
Figure 8. Figure 8: (a) The simulation setting of the steady heat conduction case on finned tube; (b) sample results with generated GRF conditions. [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: Case study for steady heat conduction equation: temperature distribution from exact solution, and inference of GON, FNO, DeepONet, PI-DeepONet, PINN (from left column to right column, from first row to second row). In this case, DeepONet consists of a Trunk Net with la…
Figure 10
Figure 10. Figure 10: a. The simulation setting of the steady heterogeneous reaction-diffusion case on flat plane. b. The simulation setting of the steady heterogeneous reaction-diffusion case on pipe. (a) Reaction-diffusion equations case on a flat plate. As shown in [PITH_FULL_IMAGE:fig…
Figure 11
Figure 11. Figure 11: Case study for steady heterogeneous reaction-di [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]
Figure 12
Figure 12. Figure 12: Case study for steady heterogeneous reaction-di [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]
Figure 13
Figure 13. Figure 13: a. The simulation setting of the Stokes case; b. An example of the applied body force generated by GRF [PITH_FULL_IMAGE:figures/full_fig_p018_13.png]
Figure 14
Figure 14. Figure 14: Case study for stokes cavity flows: a velocity magnitude from inference of GON, PINN, DeepONet, PI-DeepONet, FNO (from left to right); b Point-wise error of velocity magnitude (|uˆ − u|) by GON, PINN, DeepONet, PI-DeepONet, FNO (from left to right); c Stream traces ca…

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Reference graph

Works this paper leans on

39 extracted references · 26 canonical work pages

  1. [1]

    A general finite di fference method for arbitrary meshes

    Nicholas Perrone and Robert Kao. A general finite di fference method for arbitrary meshes. Computers & Structures, 5(1):45–57, 1975

    work page 1975

  2. [2]

    Finite element methods

    Susanne C Brenner and Carsten Carstensen. Finite element methods. Encyclopedia of Computational Mechanics, 1:73–114, 2004

    work page 2004

  3. [3]

    Finite volume methods

    Robert Eymard, Thierry Gallouët, and Raphaèle Herbin. Finite volume methods. Handbook of Numerical Analysis, 7:713–1018, 2000

    2000

  4. [4]

    Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators

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

    2021

  5. [5]

    Fourier neural operator for parametric partial differential equations

    Zongyi Li, Nikola Borislavov Kovachki, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Andrew Stuart, Anima Anandkumar, et al. Fourier neural operator for parametric partial differential equations. In International Conference on Learning Representations, 2021

    2021

  6. [6]

    Approximation theory of the mlp model in neural networks

    Allan Pinkus. Approximation theory of the mlp model in neural networks. Acta Numerica, 8:143–195, 1999

    work page 1999

  7. [7]

    Physics-informed neural networks (PINNs) for fluid mechanics: A review

    Shengze Cai, Zhiping Mao, Zhicheng Wang, Minglang Yin, and George Em Karniadakis. Physics-informed neural networks (PINNs) for fluid mechanics: A review. Acta Mechanica Sinica, 37(12):1727–1738, 2021

    2021

  8. [8]

    Learning the solution operator of parametric partial di fferential equations with physics- informed DeepONets

    Sifan Wang, Hanwen Wang, and Paris Perdikaris. Learning the solution operator of parametric partial di fferential equations with physics- informed DeepONets. Science Advances, 7(40):eabi8605, 2021

    2021

Show all 39 references
  1. [9]

    Multi-level neural networks for accurate solutions of boundary-value problems

    Ziad Aldirany, Régis Cottereau, Marc Laforest, and Serge Prudhomme. Multi-level neural networks for accurate solutions of boundary-value problems. Computer Methods in Applied Mechanics and Engineering, 419:116666, 2024

    2024

  2. [10]

    Theory-guided hard constraint projection (HCP): A knowledge-based data-driven scientific machine learning method.Journal of Computational Physics, 445:110624, 2021

    Yuntian Chen, Dou Huang, Dongxiao Zhang, Junsheng Zeng, Nanzhe Wang, Haoran Zhang, and Jinyue Yan. Theory-guided hard constraint projection (HCP): A knowledge-based data-driven scientific machine learning method.Journal of Computational Physics, 445:110624, 2021

    2021

  3. [11]

    Filtered partial di fferential equations: a robust surrogate constraint in physics-informed deep learning framework

    Dashan Zhang, Yuntian Chen, and Shiyi Chen. Filtered partial di fferential equations: a robust surrogate constraint in physics-informed deep learning framework. Journal of Fluid Mechanics, 999:A40, 2024

    2024

  4. [12]

    Ensemble of physics-informed neural networks for solving plane elasticity problems with examples

    Aliki D Mouratidou, Georgios A Drosopoulos, and Georgios E Stavroulakis. Ensemble of physics-informed neural networks for solving plane elasticity problems with examples. Acta Mechanica, pages 1–20, 2024

    2024

  5. [13]

    Probabilistic physics-informed neural network for seismic petrophysical inversion

    Peng Li, Mingliang Liu, Motaz Alfarraj, Pejman Tahmasebi, and Dario Grana. Probabilistic physics-informed neural network for seismic petrophysical inversion. Geophysics, 89(2):M17–M32, 2024

    2024

  6. [14]

    Enhanced functional evaluation for the finite element penalty method

    Vikram V Garg and Serge Prudhomme. Enhanced functional evaluation for the finite element penalty method. Computers & Mathematics with Applications, 78(12):3821–3840, 2019

    2019

  7. [15]

    Weak adversarial networks for high-dimensional partial di fferential equations

    Yaohua Zang, Gang Bao, Xiaojing Ye, and Haomin Zhou. Weak adversarial networks for high-dimensional partial di fferential equations. Journal of Computational Physics, 411:109409, 2020

    2020

  8. [16]

    Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

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

    2019

  9. [17]

    The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems

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

    2018

  10. [18]

    Fourier neural operator for parametric partial differential equations

    Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations. arXiv preprint arXiv:2010.08895, 2020

    2010 arXiv

  11. [19]

    Neural operator: Graph kernel network for partial differential equations

    Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Neural operator: Graph kernel network for partial differential equations. arXiv preprint arXiv:2003.03485, 2020

    2003 arXiv

  12. [20]

    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. In 2023 60th ACM/IEEE Design Automation Conference (DAC), pages 1–6. IEEE, 2023

    2023

  13. [21]

    A fast general thermal simulation model based on Multi-Branch Physics-Informed deep operator neural network

    Zibo Lu, Yuanye Zhou, Yanbo Zhang, Xiaoguang Hu, Qiao Zhao, and Xuyang Hu. A fast general thermal simulation model based on Multi-Branch Physics-Informed deep operator neural network. Physics of Fluids, 36(3), 2024

    2024

  14. [22]

    DeepGreen: Deep learning of Green’s functions for nonlinear boundary value problems

    Craig R Gin, Daniel E Shea, Steven L Brunton, and J Nathan Kutz. DeepGreen: Deep learning of Green’s functions for nonlinear boundary value problems. Scientific Reports, 11(1):21614, 2021

    2021

  15. [23]

    Data-driven discovery of Green’s functions with human-understandable deep learning

    Nicolas Boullé, Christopher J Earls, and Alex Townsend. Data-driven discovery of Green’s functions with human-understandable deep learning. Scientific Reports, 12(1):4824, 2022

    2022

  16. [24]

    Learning Green’s functions of linear reaction-diffusion equations with application to fast numerical solver

    Yuankai Teng, Xiaoping Zhang, Zhu Wang, and Lili Ju. Learning Green’s functions of linear reaction-diffusion equations with application to fast numerical solver. In Mathematical and Scientific Machine Learning, pages 1–16. PMLR, 2022

    2022

  17. [25]

    Operator approximation of the wave equation based on deep learning of Green’s function

    Ziad Aldirany, Régis Cottereau, Marc Laforest, and Serge Prudhomme. Operator approximation of the wave equation based on deep learning of Green’s function. Computers & Mathematics with Applications, 159:21–30, 2024

    2024

  18. [26]

    Learning domain-independent Green’s function for elliptic partial differential equations

    Pawan Negi, Maggie Cheng, Mahesh Krishnamurthy, Wenjun Ying, and Shuwang Li. Learning domain-independent Green’s function for elliptic partial differential equations. Computer Methods in Applied Mechanics and Engineering, 421:116779, 2024

    2024

  19. [27]

    Rational neural networks

    Nicolas Boullé, Yuji Nakatsukasa, and Alex Townsend. Rational neural networks. Advances in Neural Information Processing Systems , 33:14243–14253, 2020

    2020

  20. [28]

    Gaussian processes for machine learning

    Matthias Seeger. Gaussian processes for machine learning. International Journal of Neural Systems, 14(02):69–106, 2004

    2004

  21. [29]

    Binary structured physics-informed neural networks for solving equations with rapidly changing solutions

    Yanzhi Liu, Ruifan Wu, and Ying Jiang. Binary structured physics-informed neural networks for solving equations with rapidly changing solutions. arXiv preprint arXiv:2401.12806, 2024. 20

    2024 arXiv

  22. [30]

    PaddlePaddle: An open-source deep learning platform from industrial practice

    Yanjun Ma, Dianhai Yu, Tian Wu, and Haifeng Wang. PaddlePaddle: An open-source deep learning platform from industrial practice. Frontiers of Data and Domputing, 1(1):105–115, 2019

    2019

  23. [31]

    Learning elliptic partial di fferential equations with randomized linear algebra

    Nicolas Boullé and Alex Townsend. Learning elliptic partial di fferential equations with randomized linear algebra. Foundations of Compu- tational Mathematics, 23(2):709–739, 2023

    2023

  24. [32]

    Adaptive mixtures of local experts

    Robert A Jacobs, Michael I Jordan, Steven J Nowlan, and Geo ffrey E Hinton. Adaptive mixtures of local experts. Neural Computation, 3(1):79–87, 1991

    1991

  25. [33]

    Monte Carlo strategies in scientific computing, volume 10

    Jun S Liu and Jun S Liu. Monte Carlo strategies in scientific computing, volume 10. Springer, 2001

    2001

  26. [34]

    Numerical methods for special functions

    Amparo Gil, Javier Segura, and Nico M Temme. Numerical methods for special functions. SIAM, 2007

    2007

  27. [35]

    Pfnn: A penalty-free neural network method for solving a class of second-order boundary-value problems on complex geometries

    Hailong Sheng and Chao Yang. Pfnn: A penalty-free neural network method for solving a class of second-order boundary-value problems on complex geometries. Journal of Computational Physics, 428:110085, 2021

    2021

  28. [36]

    Binet: learning to solve partial di fferential equations with boundary integral networks

    Guochang Lin, Pipi Hu, Fukai Chen, Xiang Chen, Junqing Chen, Jun Wang, and Zuoqiang Shi. Binet: learning to solve partial di fferential equations with boundary integral networks. arXiv preprint arXiv:2110.00352, 2021

    2021 arXiv

  29. [37]

    Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks

    Natarajan Sukumar and Ankit Srivastava. Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks. Computer Methods in Applied Mechanics and Engineering, 389:114333, 2022

    2022

  30. [38]

    Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics

    Howard C Elman, David J Silvester, and Andrew J Wathen. Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics. Oxford university press, 2014

    2014

  31. [39]

    A note on the image system for a stokeslet in a no-slip boundary

    John R Blake. A note on the image system for a stokeslet in a no-slip boundary. InMathematical Proceedings of the Cambridge Philosophical Society, volume 70, pages 303–310. Cambridge University Press, 1971. 21 Appendix A.1 Effect of quadrature rule To investigate the impact of...

    1971

Pith tools

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