{"id":"580a4c78-996f-4a83-b8a2-4b0ea3d51358","arxiv_id":"2501.11937","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"MeshONet uses a two-branch, shared-trunk operator network to map boundary functions to interior grid coordinates for structured mesh generation.","lead":"MeshONet is a neural network that learns to generate structured meshes, the grids used in computer simulations, directly from boundary shapes. The authors report large speedups over traditional mesh generation and say the network can handle new shapes without retraining.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (5) decouples the x-field from u2 and the y-field from u1, while the elliptic system (1)-(2) couples them; the architecture has no input channel to represent the true coupled operator, and the paper never states whether training targets are TFI or PDE meshes.","rationale":"The reader's weakest assumption matches my read: Eq. (5) is load-bearing and unsupported. The central novelty is stated as an operator G(u1,u2) with multiple input functions, and generalization across unseen geometries; if the operator is actually separable, then MeshONet is essentially two independent DeepONets with a shared trunk, and the phrase 'multivariable mapping' loses its content. For the elliptic equations explicitly written in Section III.A, alpha, beta, and gamma are functions of both x and y derivatives, so the x and y solutions are coupled. The architecture has no mechanism for coupling beyond a shared trunk that depends only on computational coordinates. Consequently, Eq. (5) is either a false description of PDE mesh generation or a statement that training data are TFI-like; the paper does not say which. Because the loss in Eqs. (13)-(14) uses target values Gx(u1,u2), Gy(u1,u2), but the approximations in Eqs. (10)-(11) use Branch_x(u1) and Branch_y(u2), there is an internal inconsistency unless the target generator is separable. My proposed test isolates the separability assumption: keep u1 fixed, vary u2, solve the PDE, and check whether x targets vary. If they do, the model class cannot represent the target operator. This is not a matter of tuning or data volume; it is a representational limitation. The speed advantage in Table II is not affected by this concern, and I would not dispute the timing measurements. But the advertised generalization and quality claims rest on the operator formulation, so the reader's REJECT verdict remains appropriate.","tokens_in":13304,"tokens_out":5241,"duration_ms":60488,"concrete_test":"Construct a parametric family with fixed u1 (same boundary x-coordinates) and varying u2 (boundary y), e.g., ducts with identical vertical side positions and varying top/bottom y-profiles. Solve Eqs. (1)-(2) with a standard elliptic grid solver to obtain target meshes. Train MeshONet exactly as described. If u2 variations induce any change in the interior target x-coordinates, the model, whose x path receives only u1, cannot reproduce those changes; report the resulting Linterior. A nonzero target variation (or a failure to fit) settles the concern. If the targets are instead TFI meshes, the same test should show x independent of u2, confirming the paper's implicit target choice but limiting the claim accordingly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"MeshONet's central claim is that it approximates G(u1,u2)(xi,eta) in Eq. (4), with u1,u2 the boundary x/y functions. The proof of concept depends on Eq. (5): G(u1,u2) = (Gx(u1), Gy(u2)). But for the target PDE in Eqs. (1)-(2), alpha, beta, and gamma mix x and y derivatives, so the interior x-field is a functional of both boundary functions, and likewise for y. The network reinforces the decoupling: Eq. (10) computes x from Branch_x(u1) dot Trunk(xi,eta) only, and Eq. (11) computes y from Branch_y(u2) dot Trunk(xi,eta) only; Trunk depends only on (xi,eta). Thus, if training targets are generated by the elliptic PDE, the model class cannot represent the target operator no matter how it is trained, because the u2 channel is absent from the x path and the u1 channel is absent from the y path. If the targets are instead TFI meshes, Eq. (5) is exact, but then the comparisons to PDE-based quality in Figures 6-15 do not test the claimed operator. The paper never specifies the target generator used in Linterior/Lboundary (Eqs. 13-14), and the notation there writes hat Gx(u1,u2) while Eq. (10) only takes u1. This ambiguity is load-bearing: without knowing the target generator, the reported generalization and quality results cannot be attributed to a sound operator-learning formulation. The cited MIONet guarantee [30] applies to genuinely multi-input operators, not to two independent single-input branches sharing a trunk.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes MeshONet, a dual-branch shared-trunk neural architecture for operator learning, and applies it to structured mesh generation. The stated goal is to learn the operator G(u1, u2)(xi, eta) = (Gx(u1)(xi, eta), Gy(u2)(xi, eta)) that maps boundary coordinate functions u1 and u2 to interior mesh coordinates, so that meshes for previously unseen geometries can be generated without retraining. The paper claims speedups of up to four orders of magnitude over traditional TFI and PDE-based mesh generation, and reports interpolation and extrapolation experiments on six parametric shape families. The experimental section compares MeshONet with modified DeepONet, POD-DeepONet, FNO1D, and FNO2D, and uses visual colormaps of the maximum included angle as the main evidence of mesh quality.","tokens_in":13720,"tokens_out":4633,"duration_ms":51759,"significance":"If the core operator-learning formulation were valid and the experimental claims reproducible, MeshONet would address a real bottleneck: physics-informed mesh generators require retraining for each new geometry, whereas a retraining-free operator model could be practically valuable. The paper also addresses a genuinely useful modeling question, namely how to adapt operator learning to multi-input, multi-output mesh mapping problems. However, the significance is currently not established: the central decomposition in Eq. (5) is inconsistent with the PDE model in Eqs. (1)-(2), the training data generation is never specified, no quantitative mesh-quality metrics or statistical variation are reported, and the promised appendix with hyperparameters and implementation details is absent. The manuscript therefore does not yet provide a sound basis for the claimed generalization and efficiency results.","major_comments":[{"comment":"The decomposition G(u1,u2)(xi,eta) = (Gx(u1)(xi,eta), Gy(u2)(xi,eta)) is not valid for the elliptic grid equations in Eqs. (1)-(2). The coefficients alpha, beta, and gamma mix x and y derivatives, so the interior x-coordinate is a functional of both boundary functions u1 and u2, and likewise for y. Equations (10)-(11) implement exactly the decoupled form: x depends only on Branchx(u1) and y only on Branchy(u2), with the shared trunk depending only on (xi,eta). Consequently, if the training targets are generated by solving the PDE system, the model class cannot represent the target operator regardless of training. If the targets are instead generated by TFI, the decoupling holds but the comparison to PDE-based mesh quality in Figures 6-15 is not a comparison of the same operator. The manuscript never states which target generator is used for L_interior and L_boundary, so this ambiguity is load-bearing for the central claim.","section":"Section III.A, Eqs. (1)-(5), (10)-(11)"},{"comment":"The reproducibility of the experiments is not established. The text repeatedly refers to an appendix for network parameter settings, modified operator-learning baselines, and further experimental details, but no appendix is present in the manuscript. The training data generation procedure, number of training samples, boundary sensor count m, Lift-Layer degree q, loss weights alpha and beta, training iterations, optimizer, learning-rate schedule, and random-seed handling are not reported. Without these details, the reported loss curves and timing comparisons cannot be reproduced or independently verified.","section":"Section IV.A and Section IV.C"},{"comment":"The central generalization claim rests on visual comparisons of colormaps showing the maximum included angle. No quantitative mesh-quality statistics are reported: there are no aggregate values or distributions of included angles, orthogonality, skewness, or aspect ratio, and no comparison of those statistics between MeshONet, TFI, and PDE methods. Likewise, no standard deviations, confidence intervals, or multiple-seed experiments are reported anywhere in the results. The paper's qualitative statements such as 'our method excels' or 'performs exceptionally well' are therefore not supported by measurable evidence.","section":"Section IV.D-E, Figures 6-15"},{"comment":"The loss function notation is inconsistent with the network definition. Equations (13)-(14) write the predictors as Gx(u1, u2) and Gy(u1, u2), whereas Eqs. (10)-(11) define them as Gx(u1) and Gy(u2), and the architecture has no input channel that carries u2 into the x-branch or u1 into the y-branch. This inconsistency needs to be resolved, because it obscures whether the authors intend a truly coupled multi-input operator or two independent single-input operators.","section":"Section III.B.2, Eqs. (13)-(14)"},{"comment":"The universal approximation guarantee from MIONet [30] is invoked for the multi-input operator G(u1,u2), but the proposed architecture is not a MIONet-style multi-input network: it consists of two separate single-input branches, each seeing only one boundary function, that share a trunk. The cited theorem therefore does not justify the approximation in Eq. (6). To use that theorem, the architecture would need to provide each output branch with both boundary inputs, or the authors should explicitly state a different theoretical basis.","section":"Section III.A, Eq. (6) and reference [30]"}],"minor_comments":[{"comment":"The sentence 'The operator G takes three inputs: the computational domain coordinates (xi, eta), and two boundary functions u1 and u2' conflicts with the formal definition G(u1,u2)(xi,eta), where (xi,eta) is an evaluation argument rather than a third function input; please clarify the distinction.","section":"Section I, after Eq. (4)"},{"comment":"There is a grammatical error in 'This methods generate the corresponding mesh...'; it should read 'These methods generate...'.","section":"Section II"},{"comment":"The timing results do not state how many repeated runs were performed, whether the timings are wall-clock or solver-only, or what hardware and solver tolerances were used for the PDE baseline; the '-' entries for the PDE method are also undefined in terms of the timeout threshold.","section":"Section IV.F, Table II"},{"comment":"The four datasets shown in Figure 17 are not described in the text; the reader cannot tell which geometry families or boundary-parameter ranges these datasets correspond to.","section":"Section IV.F, Figure 17"},{"comment":"The definition beta = x_xi x_eta + y_xi y_eta is written with a comma in the displayed equation; this is a typesetting error and should be corrected to a product.","section":"Eq. (2)"}],"recommendation":"reject","confidential_remarks":"For the editor: the manuscript appears to be an incomplete submission rather than a merely stylistically rough paper, since the appendix repeatedly referred to in Section IV is entirely missing from the provided text. Beyond the missing appendix, the central mathematical formulation in Eq. (5) is inconsistent with the PDE model in Eqs. (1)-(2), and the experimental evaluation does not contain the quantitative evidence needed to support the generalization and quality claims. This is not a case where minor local corrections would suffice; the architecture, the data-generation protocol, and the evaluation metrics all need to be reworked and re-reported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"MeshONet is the first application of operator learning to structured mesh generation, and that is a real contribution. The dual-branch shared-trunk architecture is simple, the experiments cover six parametric shape families, and the speedup over iterative PDE solvers is believable. But the paper's central formulation has a load-bearing flaw: Eq. (5) assumes the x-coordinate depends only on u1 and the y-coordinate only on u2, while the elliptic system in Eqs. (1)-(2) couples x and y through alpha, beta, gamma. The network's x path sees only u1 and the y path only u2, so if the training targets are generated by a PDE solver, the model class cannot represent the true operator no matter how it is trained. If the targets are TFI meshes, Eq. (5) is exact, but then comparing against PDE quality does not test the claimed operator. The paper never specifies which generator produced the training targets, and the loss notation in Eqs. (13)-(14) writes hat Gx(u1,u2) while Eq. (10) only takes u1. This is not a minor omission; it determines whether the method is sound.\n\nThe evaluation is also thin: no quantitative mesh quality numbers, no error bars or seed variations, visual comparisons only, and the appendix with hyperparameters is absent. Generalization is demonstrated within each shape family, not across families. The speedup claim is plausible but expected for a feedforward network. On the positive side, the paper is clearly written, cites the relevant operator-learning literature, and honestly notes its own limitations about boundary-sampling sensitivity and 3D extension.\n\nIn sum: the contribution is a new application and a reasonable baseline architecture, but the central claim is unsupported as written. I would not accept it. It does deserve a serious referee, though, if the venue wants to force the authors to either justify the decoupling or fix the architecture; a referee could turn this into a solid paper with significant changes.","headline":"A genuine first application of operator learning to mesh generation, undone by a decoupling assumption that contradicts its own PDE model.","tokens_in":14177,"tokens_out":4352,"would_cite":false,"duration_ms":43076,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65N50","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"MeshONet learns a mesh-generation operator that maps boundary curves to structured grids, aiming at unseen geometries without retraining.","keywords":["structured mesh generation","operator learning","dual-branch neural network","transfinite interpolation","elliptic PDE mesh generation","generalization","mesh refinement","multivariable mapping"],"falsifier":"Take a geometry family, fix the $y$-boundary function $u_2$, and vary the $x$-boundary function $u_1$ while generating training meshes with the elliptic solver of Eqs. (1)-(2). If the interior $y$-coordinates of the resulting meshes change measurably with $u_1$, then the mapping $G_y(u_2)$ alone cannot represent the data; a MeshONet trained on this data should fail to reproduce those $y$-variations, and observing such failure would refute the decoupling assumption.","tokens_in":13113,"feed_emoji":"📐","tokens_out":9717,"duration_ms":89873,"temperature":0.7,"pith_summary":"MeshONet treats structured mesh generation as an operator-learning problem: instead of solving elliptic equations or interpolating boundaries for each new shape, a neural network learns the mapping from the two boundary coordinate functions to the interior grid coordinates. The authors claim this is the first generalizable intelligent method for structured meshes, in that a single trained model handles previously unseen geometries without retraining. They report up to four orders of magnitude speedup over traditional TFI and PDE-based solvers while keeping mesh quality comparable to or better than those baselines across six families of test shapes. The practical stake is that engineering workflows that regenerate meshes under shape changes could replace expensive per-geometry solves with one fast feedforward evaluation.","feed_headline":"Operator learning meshes unseen shapes without retraining","feed_subtitle":"MeshONet turns boundary curves into structured grids at up to 10,000x the speed of standard solvers.","key_machinery":"The load-bearing mechanism is the operator decomposition $G(u_1,u_2)(\\xi,\\eta)=(G_x(u_1)(\\xi,\\eta),G_y(u_2)(\\xi,\\eta))$, which turns the two-boundary-function mesh generation task into two coupled single-input sub-operators sharing one trunk network. Branch networks sample the physical boundary at fixed sensors and pass the sampled $x$- or $y$-coordinate values through an adjust layer; the shared trunk receives the computational coordinates $(\\xi,\\eta)$ after a lift layer expands them into $[\\sin\\xi,\\cos\\xi,\\sin\\eta,\\cos\\eta,\\xi,\\eta,\\xi^2,\\eta^2,\\ldots]$; the Output Layer combines branch and trunk features by dot product. This shared trunk is what couples the $x$ and $y$ generations, while the subsampling strategy keeps the input finite-dimensional.","core_discovery":"The paper's central claim is that the multi-input operator $G(u_1,u_2)(\\xi,\\eta)=(G_x(u_1)(\\xi,\\eta),G_y(u_2)(\\xi,\\eta))$ can be approximated by a dual-branch, shared-trunk network trained on input-output pairs of boundary functions and mesh coordinates. The x-coordinate branch consumes sampled x-boundary values; the y-coordinate branch consumes sampled y-boundary values; a shared trunk embeds the computational coordinates $(\\xi,\\eta)$ through a lift layer that mixes polynomial and trigonometric features. The final coordinates are formed by dot products of branch and trunk features. With an interior plus boundary loss, the trained model outputs the full structured mesh at any chosen resolution in a single forward pass, and the reported experiments show this generalizes across outer-boundary and inner-boundary variations in both interpolation and extrapolation settings.","pith_inferences":["Editorial inference: the success of the decoupled form $G_x(u_1),G_y(u_2)$ on the tested families suggests the cross-dependence in the elliptic coefficients is weak there, or the training targets are close to TFI-style interpolation; on families with strong coupling the architecture would need cross-input channels.","Editorial inference: the four-orders-of-magnitude speedup is generation time, not total time including training; the practical win appears when many query geometries share one trained model.","Editorial inference: since the trunk is resolution-independent, the same model could be probed at adaptive or selective point sets, which points toward goal-oriented meshing rather than uniform refinement.","Editorial inference: a direct extension to 3D would add a third branch and a third output block, with boundary-point memory as the main scaling bottleneck."],"forward_implications":["A single trained MeshONet can generate structured meshes for held-out members of a shape family, in both interpolation and extrapolation regimes, without retraining.","Generation time stays nearly constant as mesh resolution grows, whereas TFI and PDE solve times rise sharply; in the reported runs the PDE solver times out at $1600\\times1600$ and $3200\\times3200$.","The operator-learning formulation turns mesh generation into a feedforward evaluation, so real-time or large-scale meshing becomes feasible where PDE iteration would be too slow.","Mesh refinement can be performed by evaluating the same network on finer $(\\xi,\\eta)$ grids after training on low-resolution samples, avoiding repeated PDE solves.","The dual-branch shared-trunk architecture provides a template for other multi-input, multi-output operator problems beyond mesh generation."],"supporting_citations":[{"why":"Supplies the universal approximation theorem for continuous multi-input operators that justifies approximating $G(u_1,u_2)$ by a neural network.","marker":"[30]"},{"why":"DeepONet is the base operator-learning architecture and the first modified baseline in the multivariable-mapping comparison.","marker":"[24]"},{"why":"FNO1D and FNO2D are Fourier-operator baselines modified for the comparison; removing them would remove the main alternative operator-learning competitors.","marker":"[26]"},{"why":"POD-DeepONet is another modified operator-learning baseline used in the loss and mesh-quality comparisons.","marker":"[31]"},{"why":"MGNet is the physics-informed structured-mesh method whose lack of generalization to new geometries motivates MeshONet.","marker":"[8]"},{"why":"3DMeshNet is the physics-informed baseline whose retraining burden under geometry changes this work aims to remove.","marker":"[9]"},{"why":"TFI is the algebraic mesh-generation baseline for both speed and mesh-quality comparisons.","marker":"[11]"},{"why":"PDE-based mesh generation is the traditional high-quality, slow baseline that MeshONet claims to speed up by four orders of magnitude.","marker":"[7]"}],"fun_headline_variants":["MeshONet meshes unseen shapes 10,000x faster, zero retraining","Operator learning generates structured grids for new geometries instantly","One-shot mesh generation for novel boundaries via shared-trunk operator","Generalizable operator method turns boundaries into meshes at 10,000x speed"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the $x$-coordinate of an interior mesh point depends only on the boundary's $x$-function and the $y$-coordinate only on the boundary's $y$-function; the elliptic equations that produce high-quality meshes couple the two through the coefficients $\\alpha,\\beta,\\gamma$, so a PDE-based training set would contain information the network cannot receive.","fun_headline_variants_meta":{"raw":{"variants":["MeshONet meshes unseen shapes 10,000x faster, zero retraining","Operator learning generates structured grids for new geometries instantly","One-shot mesh generation for novel boundaries via shared-trunk operator","Generalizable operator method turns boundaries into meshes at 10,000x speed"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00025,"raw_usage":{"total_tokens":1539,"prompt_tokens":916,"completion_tokens":623,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":532,"completion_tokens_details":{"reasoning_tokens":546}},"tokens_in":532,"tokens_out":623,"duration_ms":6126,"temperature":1.0,"reasoning_tokens":546,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T17:41:51.259078+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a geometry family, fix the $y$-boundary function $u_2$, and vary the $x$-boundary function $u_1$ while generating training meshes with the elliptic solver of Eqs. (1)-(2). If the interior $y$-coordinates of the resulting meshes change measurably with $u_1$, then the mapping $G_y(u_2)$ alone cannot represent the data; a MeshONet trained on this data should fail to reproduce those $y$-variations, and observing such failure would refute the decoupling assumption.","supporting_citations":[{"cited_title":"MIONet: Learning multiple-input operators via tensor product,","cited_arxiv_id":null,"evidence_quote":"Supplies the universal approximation theorem for continuous multi-input operators that justifies approximating $G(u_1,u_2)$ by a neural network."},{"cited_title":"A comprehensive and fair comparison of two neural operators (with practical extensions) based on fair data,","cited_arxiv_id":null,"evidence_quote":"POD-DeepONet is another modified operator-learning baseline used in the loss and mesh-quality comparisons."},{"cited_title":"MGNet: a novel differential mesh generation method based on unsupervised neural networks,","cited_arxiv_id":null,"evidence_quote":"MGNet is the physics-informed structured-mesh method whose lack of generalization to new geometries motivates MeshONet."},{"cited_title":"3DMeshNet: A Three-Dimensional Differential Neural Network for Structured Mesh Generation","cited_arxiv_id":"2407.01560","evidence_quote":"3DMeshNet is the physics-informed baseline whose retraining burden under geometry changes this work aims to remove."},{"cited_title":"On the applications of algebraic grid generation methods based on transfinite interpolation,","cited_arxiv_id":null,"evidence_quote":"TFI is the algebraic mesh-generation baseline for both speed and mesh-quality comparisons."},{"cited_title":"Babuska, J","cited_arxiv_id":null,"evidence_quote":"PDE-based mesh generation is the traditional high-quality, slow baseline that MeshONet claims to speed up by four orders of magnitude."}],"review_version":1}