{"id":"57ac92fe-7e5e-4211-8c3e-5f348388fe78","arxiv_id":"2412.16644","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A DeepONet-style network learns a Green's function and its boundary gradient, then solves 3D linear PDEs by numerical integration, outperforming four standard neural operator baselines.","lead":"This paper trains a neural network to learn the Green's function of a linear partial differential equation, then uses that learned map to compute solutions for new boundary conditions and source terms by numerical integration. The authors report better accuracy than PINN, DeepONet, PI-DeepONet, and FNO on 3D heat conduction, reaction-diffusion, and Stokes benchmarks.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (5) asserts a scalar-type boundary integral for vector Stokes systems without derivation; for Stokes the Dirichlet boundary term should involve traction/stresslet of the Green's matrix, so Case 3 may not validate boundary-invariant vector Green's functions.","rationale":"I read the paper in good faith. The scalar Green's representation in Eq. (3) is standard for self-adjoint scalar operators, the numerical experiments are extensive, appendices provide useful ablation studies, and the code is available. My concern is specifically the transition from scalar operators to systems in Section 2. The paper's own wording is an assertion, not a derivation: 'can be extended naturally to systems of differential equations' is followed immediately by Eq. (5), which mirrors the scalar boundary integral. For Stokes, pressure and incompressibility couple the velocity components, and the Lorentz reciprocal theorem gives boundary integrals involving the traction of the Stokes Green's function; a normal derivative of a scalar Green's function does not encode that coupling. This is not a disagreement with consensus; it is a mathematical correctness risk in the vector extension. The experiment in Case 3 uses a single fixed boundary condition (no-slip plus u=(1,0,0) on the lid), so the training data can be fit without a correct boundary representation, which means the reported accuracy does not by itself validate Eq. (5). The proposed frozen-network test with zero force and varied boundary data isolates exactly the boundary term and would settle whether the concern lands. Because the reader already identified this same weak spot and assigned CONDITIONAL, my read does not change the verdict.","tokens_in":22866,"tokens_out":4286,"duration_ms":43680,"concrete_test":"Freeze the trained GON from Case 3 and evaluate it on a Stokes problem with zero body force and a varied Dirichlet velocity boundary condition (e.g., top lid velocity changed to u=(0.5,0,0) or a sinusoidal profile), comparing against COMSOL. If GON cannot reproduce the boundary-driven flow, Eq. (5)'s boundary term is demonstrably missing the Stokes traction/pressure contribution and the boundary-invariance claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim for vector systems rests on Eq. (5), which is introduced in Section 2 with the assertion that the scalar method 'can be extended naturally to systems,' but no derivation is given. For the Stokes system (Eq. (17)), the Dirichlet boundary representation of the velocity is not of the scalar form g_i(ξ) a_i(ξ) ∂_n G_{i,j}; the correct boundary term involves the traction (stress) of the Green's matrix integrated against the full boundary velocity vector, with pressure contributions as well. If Eq. (5) is not the Stokes representation, then Case 3's 3×3 Green's matrix is trained with the wrong boundary integral, and the low test error (5.83e-4) may only demonstrate fitting force-to-velocity maps for the single fixed lid-driven boundary condition, not a boundary-invariant Green's function. The paper never compares the learned G to an exact Green's function, so internal consistency of G with Eq. (5) is unverified. This concern does not affect the scalar Cases 0-2, but it directly undermines the vector generalization.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":23143,"tokens_out":5645,"duration_ms":50778,"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":[{"comment":"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.","section":"Section 2, Eq. (5)"},{"comment":"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.","section":"Section 3.1, 3.6 and Eq. (10)"},{"comment":"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.","section":"Section 3.6 and Appendix A.3"},{"comment":"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.","section":"Tables 3-6 and Appendix A.4"}],"minor_comments":[{"comment":"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.","section":"Throughout"},{"comment":"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.","section":"Table 5"},{"comment":"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.","section":"Section 2, Eq. (5)"},{"comment":"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.","section":"Section 3.6"},{"comment":"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.","section":"Section 4.3(a)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a good fit for JCP and the scalar cases, especially the finned-tube and heterogeneous reaction-diffusion benchmarks, are coherent and well supported by the appendices. The main risk is the vector Stokes claim: Eq. (5) is not the standard boundary integral representation for Stokes, and the paper currently provides no derivation or direct Green's-function validation. If the authors can correct the vector formulation and add direct Green's-function checks, the paper could be publishable; if not, the vector generalization should be removed or substantially weakened in the claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper is a solid DeepONet-style framework for learning Green's functions for 3D scalar linear PDEs, and on its benchmarks it beats PINN, DeepONet, PI-DeepONet, and FNO by a clear margin. The interesting part is the vector extension to Stokes, and that is also where the paper is shaky.\n\nWhat's new and good: coupling a Trunk Net for G with a Branch Net for ∇G, using unstructured 3D Gaussian quadrature for the convolution integral, and training on variable boundary/source families generated by Gaussian random fields. The binary-structured neural network for capturing Green's function singularities is a nice technical addition, and the appendix shows it converges faster and more stably than a standard FNN. The code is on GitHub, and the reproducibility habits are good: quadrature-rule studies, H1 semi-norm errors, and per-case hyperparameters.\n\nThe soft spot is Eq. (5). It writes a scalar-type boundary integral for vector systems—g_i a_i ∂_n G_ij—with the sentence that the scalar method \"can be extended naturally to systems.\" For Stokes, the Dirichlet boundary representation of velocity is not that. It involves a double-layer potential with the traction (stress tensor) of the Green's matrix, plus pressure contributions. If Eq. (5) is not the correct Stokes representation, then Case 3's 3×3 Green's matrix is trained with the wrong boundary integral, and the low test error only shows that a learned kernel fits the force-to-velocity map for the fixed lid-driven cavity boundary condition. It does not validate a boundary-invariant Green's function. Note also that in Case 3 the boundary condition is fixed—only the forcing varies—so the claimed boundary-invariance is not actually demonstrated there. The scalar Cases 0–2 do vary boundary conditions or sources and look credible.\n\nA second, smaller issue: the learned G is never compared to an exact or reference Green's function for a problem where one is known (e.g., 2D Poisson on a square). The loss measures solution error only, so G itself could be non-unique. There are also no seed-to-seed error bars, so the margin over FNO in Table 4 could be within variance.\n\nNet: the scalar framework is a genuine contribution and deserves peer review. The Stokes claim needs a proper derivation of the boundary term, an experiment varying the boundary condition, and ideally a check of the learned G against a known Green's function. I would accept it for review with major revisions.\n\nBest,\n[Your name]","headline":"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.","tokens_in":23635,"tokens_out":3099,"would_cite":true,"duration_ms":28519,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Green's function","operator approximation","integral solutions","neural operator","binary structured neural network","Gaussian random fields","Stokes flow","reaction-diffusion equations"],"falsifier":"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.","tokens_in":22683,"feed_emoji":"🧮","tokens_out":14232,"duration_ms":113613,"temperature":0.7,"pith_summary":"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.","feed_headline":"One trained network generalizes across sources and boundary conditions","feed_subtitle":"After one training pass, new sources and boundaries are solved by integration, with lower error than PINN, DeepONet, PI-DeepONet, and FNO.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the DeepONet Trunk/Branch architecture that GON re-purposes to learn Green's functions.","marker":"[4]"},{"why":"Provides the binary structured neural network used in the Trunk and Branch Nets to capture Green's-function singularities.","marker":"[29]"},{"why":"Defines the Gaussian random fields used to generate training sources and boundary data.","marker":"[28]"},{"why":"Prior Green's-function learning method whose Gaussian Dirac surrogate is the accuracy baseline GON improves on.","marker":"[24]"},{"why":"Prior operator approximation via learned Green's functions; a baseline and motivation for the physics-informed loss comparison.","marker":"[25]"},{"why":"Data-driven Green's-function discovery baseline used for the 2D comparison.","marker":"[23]"},{"why":"The PINN baseline whose soft-constraint convergence is compared against GON.","marker":"[16]"},{"why":"The physics-informed DeepONet baseline; provides a hybrid operator-learning comparison.","marker":"[8]"},{"why":"The Fourier neural operator baseline, especially for regular-geometry cases.","marker":"[18]"},{"why":"Supplies the rationale for why Green's functions have diagonal singularities and why diverse training modes are needed.","marker":"[31]"}],"fun_headline_variants":["Neural network learns Green's functions to solve PDEs fast","DeepONet twist: learn the Green's function, not the map","GreensONet: one training pass, any source and boundary","Solve new PDE conditions by integration after one training","Learning Green's functions beats PINN, DeepONet, and FNO"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Neural network learns Green's functions to solve PDEs fast","DeepONet twist: learn the Green's function, not the map","GreensONet: one training pass, any source and boundary","Solve new PDE conditions by integration after one training","Learning Green's functions beats PINN, DeepONet, and FNO"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000209,"raw_usage":{"total_tokens":1469,"prompt_tokens":1071,"completion_tokens":398,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":687,"completion_tokens_details":{"reasoning_tokens":309}},"tokens_in":687,"tokens_out":398,"duration_ms":3486,"temperature":1.0,"reasoning_tokens":309,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T10:23:47.081933+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Learning Green’s functions of linear reaction-diffusion equations with application to fast numerical solver","cited_arxiv_id":null,"evidence_quote":"Prior Green's-function learning method whose Gaussian Dirac surrogate is the accuracy baseline GON improves on."},{"cited_title":"Operator approximation of the wave equation based on deep learning of Green’s function","cited_arxiv_id":null,"evidence_quote":"Prior operator approximation via learned Green's functions; a baseline and motivation for the physics-informed loss comparison."},{"cited_title":"Data-driven discovery of Green’s functions with human-understandable deep learning","cited_arxiv_id":null,"evidence_quote":"Data-driven Green's-function discovery baseline used for the 2D comparison."},{"cited_title":"Learning elliptic partial di fferential equations with randomized linear algebra","cited_arxiv_id":null,"evidence_quote":"Supplies the rationale for why Green's functions have diagonal singularities and why diverse training modes are needed."}],"review_version":1}