{"id":"3d67086a-1424-4aa6-acef-879923f697d3","arxiv_id":"2607.06026","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":7,"one_line_summary":"SplineNet constructs a neural network whose architecture exactly reproduces isogeometric spline basis functions via Bézier extraction, enabling both data-free PDE solving and operator learning for complex shell structures.","lead":"SplineNet embeds CAD spline geometry directly into neural networks, letting deep learning solve shell mechanics problems on complex real-world shapes without converting between design and analysis models. A generalist might read it because it could let engineers skip the slow CAD-to-simulation conversion step in design loops.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"P-SNet optimization error grows sharply with mesh refinement (Table 1: 0.003% on Mesh 1 vs. 0.679% on Mesh 2), raising scalability questions for the data-free solver as control variable count increases.","rationale":"The reader identified O-SNet's approximation capacity as the weakest assumption, but this concern is partially misdirected: since the reference solutions are generated in the same ASUTS spline space that O-SNet uses as its trunk, the spline prior does not restrict the output space relative to the targets. The real question for O-SNet is branch-net capacity for high-dimensional control-variable regression (e.g., 3507 CPs for the B-pillar), which is a standard DeepONet concern, not specific to the spline construction. The reader's valid point is the missing vanilla DeepONet comparison, which is an evaluation gap. My identified concern — P-SNet's non-monotonic optimization error with mesh refinement — is more load-bearing because it directly questions the scalability of the data-free solver, a core part of the contribution. However, neither concern rises to the level of falsifying the central claim: the construction is correct by design (Eq. 16 exactly recovers IGA discretization), P-SNet achieves sub-1% error, and O-SNet achieves 1.7–3.9% test errors on complex geometries with extraordinary points. The CONDITIONAL verdict is appropriate — the method is sound and the examples are relevant, but the scalability question and evaluation gaps (no DeepONet baseline, no error bars, no code) prevent full acceptance. The verdict should remain CONDITIONAL.","tokens_in":18573,"tokens_out":6324,"duration_ms":302087,"concrete_test":"Re-solve the Mesh 2 Scordelis–Lo roof (1624 control points) with P-SNet using L-BFGS instead of Adam, with at least 10,000 iterations and a gradient-norm stopping criterion below 1e-8. If the relative error drops below 0.01%, the issue is optimizer choice and the method scales. If it remains above 0.1%, the scalability concern for energy-based neural solvers at large control-variable counts lands. Additionally, report the energy residual ‖∇Π‖ at the final iterate for all three meshes to quantify convergence quality.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim includes that P-SNet enables data-free energy-based shell analysis. Table 1 compares P-SNet against IGA on the same mesh, so the reported errors are purely optimization residuals, not discretization errors. The error pattern is non-monotonic: 0.027% (Mesh 0, 124 control points), 0.003% (Mesh 1, 432 CPs), then 0.679% (Mesh 2, 1624 CPs). For the linear KL shell, the total potential energy (Eq. 14) is convex and quadratic in U, so a direct solver achieves machine precision. The two-order-of-magnitude degradation on the finest mesh indicates that Adam-based minimization of the energy loss (Eq. 19) struggles as the dimension of U grows. If this trend continues, P-SNet would face a practical accuracy ceiling well before reaching production-scale meshes (tens of thousands of control variables). This is the most load-bearing concern because it directly questions whether the data-free solver mode scales, which is a core part of the claimed contribution. The O-SNet results (1.7–3.9% test errors) are less concerning because the spline trunk space exactly matches the reference data space (both generated by the same ASUTS IGA solver), so the spline prior is not a restriction on approximation capacity — the reader's framing of this as a capacity concern is somewhat misdirected. The valid part of the reader's O-SNet concern is the absence of a vanilla DeepONet comparison, which prevents assessing the accuracy trade-off of the spline prior, but this is an evaluation gap rather than a correctness risk.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"This paper introduces SplineNet, a neural network architecture that embeds analysis-suitable unstructured T-splines (ASUTS) via Bézier extraction, with Bernstein polynomials as activations and extraction coefficients as fixed weights. Two variants are proposed: P-SNet, a data-free energy-based solver for Kirchhoff–Love shells, and O-SNet, which replaces the trunk net of DeepONet with SplineNet for operator learning. The construction is verified on the Scordelis–Lo roof benchmark, and O-SNet is demonstrated on three shell problems of increasing complexity, including a geometrically nonlinear case. The core mathematical construction is sound: Eq. (16) recovers the IGA discretization by construction, and the energy-based loss (Eq. 19) is a standard variational formulation.","tokens_in":18867,"tokens_out":2559,"duration_ms":112819,"significance":"The paper addresses a genuine gap: integrating watertight CAD spline representations into deep learning for shell analysis, including geometries with extraordinary points that go beyond simple multi-patch settings. The SplineNet construction is parameter-free in its trunk (fixed extraction coefficients, Bernstein activations), which is a clean and interpretable design. The application to operator learning on complex real-world geometries (B-pillar, plane nose) with 1.7–3.9% test errors is a useful practical demonstration. The CAD/CAE integration motivation is well-articulated and the approach is technically coherent.","major_comments":[{"comment":"§5.1, Table 1: The P-SNet errors are non-monotonic across mesh refinement: 0.027% (Mesh 0, 124 CPs), 0.003% (Mesh 1, 432 CPs), then 0.679% (Mesh 2, 1624 CPs). For the linear KL shell, the total potential energy (Eq. 14) is convex and quadratic in U, so a direct solver achieves machine precision regardless of problem size. The two-order-of-magnitude degradation on the finest mesh indicates that Adam-based minimization of Eq. (19) struggles as the dimension of U grows. Since P-SNet as a data-free solver is a central contribution, the authors should (a) discuss this scaling behavior explicitly rather than presenting all three results as equally validating, and (b) report whether alternative optimizers (e.g., L-BFGS), preconditioning, or longer training schedules mitigate the degradation. Without this, the practical viability of P-SNet beyond modest mesh sizes remains unclear.","section":null},{"comment":"§4.3 and §5.2: O-SNet is presented as replacing the DeepONet trunk net with fixed spline basis functions, but no comparison against a standard DeepONet with an MLP trunk net is provided on any of the three test problems. This makes it impossible to assess the accuracy trade-off introduced by the spline prior. While the absolute errors (1.7–3.9%) are reasonable, the reader cannot determine whether the spline trunk helps, hurts, or is neutral relative to a learned trunk. A single head-to-head comparison on, e.g., the roof problem would substantially strengthen the contribution claim.","section":null},{"comment":"§4.2, Eq. (19): The boundary condition term uses a penalty formulation with equally spaced boundary points, but the weight on this term relative to the energy is not specified. For the linear problems this appears to work, but the sensitivity of the solution accuracy to the penalty weight and the number of boundary points M is not reported. Since essential boundary condition enforcement is mentioned as a motivation (§1, citing conflicting gradient issues in PINNs), a brief sensitivity study or at least a statement of the chosen weights for each example would be appropriate.","section":null}],"minor_comments":[{"comment":"§5.2.4: The plane nose problem uses the geometrically nonlinear KL shell formulation [3,48], but the energy functional in Eq. (14) and the P-SNet loss in Eq. (19) are stated for the linear case. Since O-SNet uses a data-driven loss (Eq. 20), this is internally consistent, but the paper should clarify that the nonlinear physics enters only through the reference data, not through the network loss.","section":null},{"comment":"§5.2.1: The GRF length scale l=0.2 is stated, but the amplitude factor γ varies per problem (1.5×10^8, 1.5×10^7, 5×10^8). The rationale for these specific values is not given. A brief note on how these were chosen would help reproducibility.","section":null},{"comment":"Table 2 lists shell parameters for the linear cases, but the plane nose uses different values (E=2.06×10^11, t=0.03) stated only in the text. Consolidating all problem parameters into a single table would improve clarity.","section":null},{"comment":"Fig. 13b, 17b, 21b: The y-axis labels 'Relative L2 error' and 'Relative L2 error (%)' are inconsistent between subfigures. Standardizing the format would help.","section":null},{"comment":"§4.1, Remark 1 mentions that training the second-layer weights yields r-adaptivity but is not pursued. A citation to where this has been explored (e.g., HiDeNN r-adaptivity references) would contextualize the remark.","section":null},{"comment":"§5.1: The Scordelis–Lo roof verification uses meshes with 'intentionally introduced' extraordinary points, but the number and valence of these EPs are not specified. This information is relevant for assessing the ASUTS treatment.","section":null}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern about P-SNet scalability is the most substantive issue; it lands because the paper presents P-SNet as a contribution without acknowledging the optimization scaling problem evident in its own Table 1. The O-SNet approximation-capacity concern raised by the reader is somewhat misdirected (the training data comes from the same ASUTS space, so the spline prior is not restrictive for these examples), but the absence of a vanilla DeepONet baseline is a legitimate evaluation gap. The paper is a reasonable fit for the journal's scope given the CAD/CAE integration focus."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for a careful and constructive report. The referee correctly identifies the core mathematical construction as sound and acknowledges the practical motivation. The three major comments are all legitimate: (1) the non-monotonic P-SNet errors on the finest mesh require explicit discussion and investigation of alternative optimizers; (2) the absence of a head-to-head comparison between O-SNet and a standard DeepONet with an MLP trunk is a gap that should be filled; and (3) the penalty weight and boundary point sensitivity for essential boundary condition enforcement should be reported. We agree with all three points and will revise the manuscript accordingly. We provide point-by-point responses below.","responses":[{"response":"The referee is correct on both counts. The non-monotonicity in Table 1 is a real phenomenon, not a typographical error, and we agree that presenting all three results as equally validating without discussion is misleading. The degradation from 0.003% to 0.679% on Mesh 2 is attributable to the scaling behavior of Adam as the dimension of the control variable vector U grows from 432 to 1624. As the referee notes, the energy functional for the linear KL shell is convex and quadratic in U, so a direct solver would achieve machine precision; the observed degradation is entirely an optimizer limitation, not a discretization issue. We will revise the manuscript in two ways. First, we will add an explicit discussion of this scaling behavior in §5.1, acknowledging that Adam-based minimization of Eq. (19) becomes less effective as the problem dimension grows and that the Mesh 2 result should not be interpreted as evidence of discretization error but rather as an indicator of optimization difficulty. Second, we will conduct additional experiments using L-BFGS (both as a standalone optimizer and as a second-stage refinement after Adam) and report whether the finest-mesh accuracy improves. We expect L-BFGS to perform better given the convex quadratic structure of the loss landscape, but we will report the actual results honestly. If L-BFGS does not fully resolve the issue, we will state this plainly and discuss preconditioning and longer training schedules as further remedies to be explored in future work. We will not claim that P-SNet in its current form is practically viable for large-scale problems without these additional results.","revision_made":"yes","referee_comment":"§5.1, Table 1: The P-SNet errors are non-monotonic across mesh refinement: 0.027% (Mesh 0, 124 CPs), 0.003% (Mesh 1, 432 CPs), then 0.679% (Mesh 2, 1624 CPs). For the linear KL shell, the total potential energy (Eq. 14) is convex and quadratic in U, so a direct solver achieves machine precision regardless of problem size. The two-order-of-magnitude degradation on the finest mesh indicates that Adam-based minimization of Eq. (19) struggles as the dimension of U grows. Since P-SNet as a data-free solver is a central contribution, the authors should (a) discuss this scaling behavior explicitly rather than presenting all three results as equally validating, and (b) report whether alternative optimizers (e.g., L-BFGS), preconditioning, or longer training schedules mitigate the degradation. Without this, the practical viability of P-SNet beyond modest mesh sizes remains unclear."},{"response":"We agree that a head-to-head comparison against a standard DeepONet with an MLP trunk is essential to substantiate the claim that the spline prior is beneficial. The current manuscript presents only absolute errors for O-SNet without a baseline, which is insufficient. We will add a comparison on the roof problem (§5.2.2) between O-SNet and a standard DeepONet with an MLP trunk net of comparable parameter count, trained on the same dataset with the same branch net and optimization schedule. We will report relative L2 errors at Gaussian quadrature points and element corners for both architectures. We note that the comparison may not be purely about raw accuracy: the spline trunk provides interpretability (predictions live in the ASUTS spline space, enabling direct CAD/CAE integration) and independence from query point locations, which an MLP trunk does not. However, if the spline trunk also achieves comparable or better accuracy, this strengthens the contribution; if it is worse, we will report this honestly and frame the contribution in terms of interpretability and CAD integration rather than accuracy. We will not selectively report only favorable outcomes.","revision_made":"yes","referee_comment":"§4.3 and §5.2: O-SNet is presented as replacing the DeepONet trunk net with fixed spline basis functions, but no comparison against a standard DeepONet with an MLP trunk net is provided on any of the three test problems. This makes it impossible to assess the accuracy trade-off introduced by the spline prior. While the absolute errors (1.7–3.9%) are reasonable, the reader cannot determine whether the spline trunk helps, hurts, or is neutral relative to a learned trunk. A single head-to-head comparison on, e.g., the roof problem would substantially strengthen the contribution claim."},{"response":"The referee is correct that the penalty weight and the number of boundary points M are not specified in the manuscript, and this omission makes it difficult for readers to assess the robustness of the boundary condition enforcement. We will address this in the revision. First, we will state the specific penalty weights and values of M used in each example. In the current implementation, the boundary condition term in Eq. (19) has a coefficient of 1/M (as written), and the prescribed displacements are additionally imposed directly on the corresponding boundary control variables, which provides a strong enforcement mechanism beyond the penalty term alone. We will clarify this dual enforcement strategy in the revised text. Second, we will add a brief sensitivity study on the roof problem showing how the solution accuracy varies with the penalty weight (e.g., scaling the boundary loss by factors of 10, 1, 0.1 relative to the energy term) and with M. This will allow readers to gauge whether the method is robust to these hyperparameters or requires careful tuning. If the sensitivity study reveals that accuracy depends strongly on these choices, we will state this as a limitation.","revision_made":"yes","referee_comment":"§4.2, Eq. (19): The boundary condition term uses a penalty formulation with equally spaced boundary points, but the weight on this term relative to the energy is not specified. For the linear problems this appears to work, but the sensitivity of the solution accuracy to the penalty weight and the number of boundary points M is not reported. Since essential boundary condition enforcement is mentioned as a motivation (§1, citing conflicting gradient issues in PINNs), a brief sensitivity study or at least a statement of the chosen weights for each example would be appropriate."}],"tokens_in":18422,"tokens_out":1478,"duration_ms":123028,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"Bottom line: SplineNet is a legitimate methodological contribution to isogeometric deep learning. The core idea — using Bézier extraction coefficients as fixed network weights and Bernstein polynomials as activations so that the network exactly reproduces the ASUTS basis — is clean and correct. Eq. 16 recovers the IGA discretization by construction, and the energy-based loss for P-SNet is a standard variational formulation applied to the Kirchhoff-Love shell. The O-SNet variant, which swaps SplineNet in as the DeepONet trunk net, is a natural extension that keeps predictions in the spline space and ties them back to the CAD model. The numerical examples on the Scordelis-Lo roof, B-pillar, and plane nose demonstrate that the method works on geometries with extraordinary points, which is where prior multi-patch approaches struggle. Credit is due for tackling unstructured splines rather than the easier multi-patch setting, and for testing on a geometrically nonlinear case (plane nose) rather than only linear problems. The O-SNet test errors of 1.7–3.9% on complex geometries are reasonable for this type of operator learning. The stress-test concern about P-SNet's optimization error growing with mesh refinement (0.003% on Mesh 1 vs. 0.679% on Mesh 2) is valid and worth flagging to the authors. For a convex quadratic energy, Adam should not be losing two orders of magnitude of accuracy as the control variable count grows from 432 to 1624. This is an optimization scaling issue, not a discretization issue, and the paper does not address it. If the trend continues, P-SNet would hit a practical wall well before production-scale meshes. That said, this is a verification example on a single benchmark — it does not undermine the overall construction. The reader's concern about O-SNet's approximation capacity is somewhat misdirected: since the reference data is generated by the same ASUTS IGA solver, the spline trunk space exactly matches the target space, so the spline prior is not restrictive in these experiments. The real gap is the absence of a vanilla DeepONet baseline (MLP trunk) on the same problems, which would show whether the spline prior helps or hurts. Minor issues: no error bars across training runs, no code or data repository. These are standard expectations but not dealbreakers for a first submission. The paper is for researchers working at the intersection of IGA and deep learning, particularly those interested in CAD/CAE integration. It deserves a serious referee who can push the authors to (1) address or explain the P-SNet scaling behavior, (2) add a DeepONet baseline for O-SNet, and (3) provide code. The central construction is sound enough to warrant that level of engagement.","headline":"SplineNet embeds ASUTS basis functions into a neural network via Bézier extraction, enabling IGA-consistent shell analysis in both solution-learning and operator-learning modes. The construction is sound and the engineering motivation is real, but P-SNet's optimization scalability is untested and O-SNet lacks a baseline comparison.","tokens_in":19474,"tokens_out":694,"would_cite":false,"duration_ms":94328,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Spline Bases Replace Neural Activations in Shell Analysis Network","keywords":["isogeometric analysis","neural networks","Bézier extraction","T-splines","Kirchhoff–Love shells","operator learning","DeepONet","structural analysis"],"falsifier":"Construct a loading function class whose displacement response contains modes outside the ASUTS spline space used by O-SNet; if the operator-learning error does not decrease with additional training data or wider branch nets, the fixed spline prior is the bottleneck.","tokens_in":18628,"feed_emoji":"🔧","tokens_out":1194,"duration_ms":92839,"temperature":0.7,"pith_summary":"SplineNet is a neural network architecture whose weights and activation functions are fixed by the mathematics of CAD spline surfaces, rather than learned freely. The first hidden layer evaluates the 16 bicubic Bernstein polynomials at the input parametric coordinates; the second hidden layer multiplies those values by a precomputed Bézier extraction matrix to produce analysis-suitable unstructured T-spline (ASUTS) basis functions; the output layer multiplies the basis functions by trainable control variables to predict displacement. This exactly recovers the discretization used in isogeometric analysis (IGA), meaning the network's approximation space is identical to that of a conventional spline-based solver. The paper uses this architecture in two modes: P-SNet, which trains control variables by minimizing the total potential energy of a Kirchhoff–Love shell without any precomputed solution data, and O-SNet, which replaces the trunk net of a Deep Operator Network with SplineNet so that the learned load-to-displacement operator lives in the spline space and predictions for new loadings require no re-solving. The paper tests both modes on shell structures including a Scordelis–Lo roof, a B-pillar, and a geometrically nonlinear plane nose, all represented by ASUTS meshes with extraordinary points, reporting displacement errors below 1% for P-SNet and 1.7–3.9% for O-SNet relative to IGA reference solutions.","feed_headline":"Spline bases replace learned activations in shell analysis networks","feed_subtitle":"A neural net whose weights are CAD extraction coefficients solves shell problems data-free and learns load-to-displacement operators without","key_machinery":"Bernstein polynomials as activation functions; Bézier extraction matrix as fixed weights; trainable control variables as output-layer weights; energy-based loss (P-SNet) or data-driven MSE loss (O-SNet); ASUTS basis functions as the network's approximation space.","core_discovery":"The central object is the elementwise SplineNet: a two-hidden-layer fully connected network in which the first layer's activation functions are the 16 bicubic Bernstein polynomials, the second layer's weights are the fixed Bézier extraction coefficients C_e, and the output layer's weights are the trainable control variables U_e. The output u_e(ξ,η) = (U_e)^T C_e B(ξ,η) is algebraically identical to the per-element IGA discretization. The paper shows that this construction can serve as a drop-in neural surrogate for IGA: as a physics-informed solver (P-SNet) it minimizes shell energy with no training data, and as an operator-learning trunk net (O-SNet) it maps loading functions to control变量s,","pith_inferences":["If the ASUTS spline space is not rich enough to represent a particular load-response pattern, O-SNet cannot recover the missing modes by learning, because the trunk net is fixed—its expressiveness is bounded by the spline basis. This sets a hard accuracy ceiling that a vanilla DeepONet with a learned MLP trunk would not face.","The energy-based P-SNet mode could in principle be combined with the data-driven O-SNet mode in a hybrid loss, using sparse experimental data to regularize the physics-informed solve—neither mode alone exploits this possibility.","Because each Bézier element is an independent sub-network with no inter-element connections, the architecture is naturally parallelizable across elements, which could make large-scale shell problems tractable on GPUs in ways that monolithic PINN architectures are not.","Replacing ASUTS with other spline types (e.g., hierarchical B-splines or LR-splines) would require only changing the extraction matrix, suggesting the method generalizes beyond T-splines to any spline technology with a Bézier extraction operator."],"forward_implications":["Engineering design loops that currently require repeated IGA or FEM solves for each new loading or boundary condition could be replaced by a single trained O-SNet forward pass, with predictions staying in the CAD-native spline representation.","Shape optimization can treat control-point positions as differentiable parameters within the same network, since geometry and displacement share the spline basis, enabling end-to-end gradient-based optimization without mesh-to-CAD remapping.","The fixed-basis trunk net removes the dependence of operator-learning accuracy on query-point placement, a known sensitivity in point-sampling-based methods like standard DeepONet.","The architecture extends to geometrically nonlinear shell formulations, as shown by the plane-nose example, suggesting the spline-prior approach is not limited to linear elasticity."],"fun_headline_variants":["Neural net built on Bézier extraction matches isogeometric shell analysis","Bézier polynomials as activations give a net identical to IGA shell elements","SplineNet turns CAD extraction coefficients into neural network weights","One network replaces CAD-to-CAE pipeline for complex shell structures","Bernstein-basis neural net solves Kirchhoff-Love shells without training data"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The paper assumes that using fixed spline basis functions as the trunk net of an operator network does not sacrifice approximation power relative to a network with learned basis functions, but provides no theoretical approximation result and no head-to-head comparison against a standard DeepONet with an MLP trunk on the same problems.","fun_headline_variants_meta":{"raw":{"variants":["Neural net built on Bézier extraction matches isogeometric shell analysis","Bézier polynomials as activations give a net identical to IGA shell elements","SplineNet turns CAD extraction coefficients into neural network weights","One network replaces CAD-to-CAE pipeline for complex shell structures","Bernstein-basis neural net solves Kirchhoff-Love shells without training data"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":738,"prompt_tokens":642,"completion_tokens":96,"prompt_tokens_details":null},"tokens_in":642,"tokens_out":96,"duration_ms":18992,"temperature":1.0,"reasoning_tokens":null,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-08T18:52:04.968344+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"Construct a loading function class whose displacement response contains modes outside the ASUTS spline space used by O-SNet; if the operator-learning error does not decrease with additional training data or wider branch nets, the fixed spline prior is the bottleneck.","supporting_citations":[],"review_version":1}