{"id":"00953e6e-3115-4843-9566-f438ac7c18eb","arxiv_id":"2506.18427","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"NOEM replaces large FEM meshes with pretrained neural-operator elements inside a variational energy-minimization framework, cutting computation time.","lead":"This paper introduces NOEM, a hybrid method that combines finite element meshes with pretrained neural operators to solve PDEs using much coarser discretizations. It reports faster simulations and reusable neural elements across many geometries and coefficient fields.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 3.3 concedes that soft-constraint NOEs violate the C0/C1 regularity on which the variational formulation in Eqs. (6)-(7) depends; until consistency of the resulting broken form is established, NOEM's accuracy and efficiency rest on an unproven empirical regularity.","rationale":"The reader's conditional verdict is appropriate. The paper is transparent about the soft-constraint gap in Section 3.3, and the experiments provide some supporting evidence (e.g., the hard/soft comparisons in Sections 4.2.1 and 4.3.1, and the error-correlation plots). But those comparisons are not a convergence study. The broken variational form is the load-bearing assumption: if it is inconsistent, no amount of training data or network capacity makes the assembled solution converge to the true PDE solution, and the efficiency advantage is irrelevant because the answers are wrong. The proposed test directly targets this assumption by isolating the effect of interface boundary error. If the test passes, the method is a legitimate nonconforming approximation; if it fails, the paper should either add interface penalties, adopt hard-constraint NOEs, or restrict its claims to the demonstrated cases. No change to the reader's conditional verdict is needed; the conditional acceptance should require this consistency check.","tokens_in":17490,"tokens_out":9736,"duration_ms":127548,"concrete_test":"Take the multi-hole benchmark in Section 4.3.2 and train a sequence of NOs with decreasing soft-constraint boundary error (e.g., by increasing training data or by applying a hard-constraint projection to the outputs). Plot the NOEM relative L2 error against the maximum mismatch between the NO output and the input boundary values at NOE interface sensors. If the NOEM error does not approach the FEM solution as this mismatch approaches zero, the soft-constraint broken formulation is inconsistent. To separate interior NO error from interface error, repeat with a fixed high-accuracy NO and artificially perturb only the interface sensor values by a known amount epsilon; if the resulting NOEM error does not vanish as epsilon goes to zero, the variational coupling at interfaces is the limiting factor.","verdict_should_be":"UNCHANGED","load_bearing_attack":"For the central claim—that solving the energy minimization (7) over the mixed ansatz (6) yields a valid replacement for a dense FEM solve—the trial function q must belong to the energy space of the PDE. For the second-order problems treated here, that requires at least C0 continuity across NOE/FE and NOE/NOE interfaces. The paper's own Section 3.3 states that hard-constraint NOs 'should be used for NOEs' but that soft-constraint NOs are used instead because hard constraints are costly. Soft-constraint NOs do not enforce the input boundary values, so q is generally discontinuous at interfaces. Appendix C evaluates the energy as a sum of per-element quadratures—a broken energy—and no DG-style numerical flux or interface penalty is introduced; the statement that this 'resembles the idea of the discontinuous Galerkin approach' is not supported by a consistency argument (e.g., a Strang lemma or an a priori error estimate). The empirical correlation in Figs. 2D and 3D shows that NOEM error tracks NO error, but correlation is not convergence, and no bound shows that the broken-energy minimizer approaches the true solution as the NO's boundary error tends to zero. Until that consistency question is settled, the headline '14x speedup' and the scalability claims are demonstrations for the selected benchmarks rather than established properties of the method.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes the neural-operator element method (NOEM), a hybrid solver that partitions a computational domain into subdomains represented either by classical finite elements or by a single neural-operator element (NOE) built from a pretrained DeepONet, MIONet, or CNN. The trial space is the sum of standard FE shape functions and NO outputs, and the approximate solution is obtained by minimizing the energy functional in Eq. (7). Ten numerical experiments in 1D and 2D cover multiscale coefficients, complex geometries, and Darcy flow with discontinuous and nonlinear permeability fields, reporting relative L2 errors of roughly 0.1% to 2% and a claimed speedup of about 14 times over a fine-mesh FEM baseline for a 100-hole domain.","tokens_in":17842,"tokens_out":5537,"duration_ms":64403,"significance":"The core idea is practically attractive: pretrained neural operators are used as reusable 'super-elements' that absorb fine-scale or geometrically complex behavior, potentially reducing the number of degrees of freedom substantially. The numerical section is broad, and the reuse of a single trained NO across 2000 coefficient samples and across domains with different numbers of holes is a genuine strength that demonstrates the intended reusability. The paper also provides useful details for constructing hard-constraint DeepONets and promises public code. However, the mathematical consistency of the mixed variational formulation is not established, and the efficiency claims are not benchmarked against the nearest existing hybrid FEM/operator-learning methods. If these gaps are closed, NOEM would be a valuable contribution to computational methods; in its current form, the central methodological promise rests on empirical demonstration rather than on a verified variational principle.","major_comments":[{"comment":"The central variational claim is not supported by a consistency argument. As the paper itself states in Section 3.3, the trial functions in Eq. (6) must satisfy C0/C1 regularity for the formulation to be conforming, and that hard-constraint NOs 'should be used' for NOEs; however, the main experiments use soft-constraint NOs, which do not enforce interface boundary values. Appendix C evaluates the energy as a sum of independent element quadratures, so the minimized functional is a broken energy with no interface penalty or numerical flux. Invoking the discontinuous-Galerkin analogy does not replace the missing analysis: no Strang-type lemma, no consistency estimate, and no bound showing that the broken-energy minimizer approaches the true PDE solution as the NO boundary error tends to zero. The correlations in Figs. 2D and 3D show that NOEM error tracks NO error, but correlation is not convergence. This issue is load-bearing because Eq. (7) is the definition of the NOEM solution.","section":"Section 3.3, Appendix C"},{"comment":"The headline efficiency claim--'around 14 times speedup in the computational cost for 100 holes'--is a runtime comparison against a single fine-mesh FEM baseline. The NO training cost is not reported, and no comparison to the cited hybrid methods of Refs. [54] and [55] is provided. Without this information, the reader cannot determine whether the apparent speedup is a property of NOEM or an artifact of the chosen FEM baseline and the exclusion of training and data-generation costs. I ask for (i) an explicit accounting of NO training cost and data-generation cost with the amortization assumptions stated, (ii) a clear definition of the measured solve time (mesh generation, assembly, Newton iterations, tolerances), and (iii) at least one comparison with an established domain-decomposition or reduced-order hybrid method on the same benchmark.","section":"Section 4.3.2"},{"comment":"For the nonlinear examples in Section 4.4, the functional J in Eq. (7) is not specified. In particular, for the nonlinear permeability field K'(x,y) = 1/(|u|+0.1) of Section 4.4.4, no energy functional is written down, and the linear variational framework of Eqs. (3)-(4) does not cover this case. It is not immediate that an energy of the form (1/2) ∫ K(u)|∇u|^2 has the stated PDE as its Euler-Lagrange equation when K depends on u. The manuscript should state the functional being minimized for each nonlinear case and verify the Euler-Lagrange equivalence, or explain why the mismatch is negligible. Without this, the nonlinear-PDE claims are not reproducible.","section":"Section 4.4, Section 3.2"}],"minor_comments":[{"comment":"The Gaussian-process kernel is written as k_l(x,x') = -(1/2)((x-x')/l)^2, which cannot be a valid covariance function; it should presumably be the squared-exponential form exp(-(1/2)((x-x')/l)^2).","section":"Section 4.1.2"},{"comment":"The notation α'_j appears in the paragraph after Eq. (6) without being defined; the comparison between the DOF count for finite elements and the boundary parametrization β_j should be stated more carefully.","section":"Section 3.2"},{"comment":"The paper claims model reusability for varying geometries, but all NOE subdomains share the same rectangular-with-hole geometry; the limitation to a single pre-trained geometry should be stated explicitly in the results section.","section":"Section 4.3"},{"comment":"The 'computational cost' shown in Fig. 6D should specify whether wall-clock time or CPU time is reported, and the FEM solver details (linear solver, tolerance, mesh size) should be given so the speedup can be reproduced.","section":"Section 4.3.2, Fig. 6D"},{"comment":"The nonconvexity of Eq. (7) is acknowledged, but no initialization sensitivity study is presented; a brief experiment varying the initial guess of Newton's method would make the accuracy claims more robust.","section":"Section 3.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the journal's scope and the numerical demonstrations are extensive, but the main methodological claim is currently supported by empirical correlation rather than by a consistency analysis. The missing pieces--a consistency argument for the broken variational form, a specification of the nonlinear energy functional, and fairer efficiency benchmarks--are identifiable and likely addressable, so I do not recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nThe paper's core idea is the new thing: it treats a pretrained neural operator as a single finite element in a global variational energy minimization, mixing classical FEs and \"neural-operator elements\" (NOEs) in one tessellation. That is not in the cited literature; Ref. [54] and [55] use iterative coupling or domain decomposition, and Ref. [52] uses finite operators in a different way. The experimental scope is genuinely broad: ten examples, including 1D multiscale coefficients, 2D heat transfer on domains with holes, Darcy flow with rough and piecewise constant permeability, and a nonlinear permeability field. The reusability claim is demonstrated: a DeepONet trained on a single-hole domain is reused for 2x2 through 10x10 hole arrays with reported relative L2 errors mostly 1-2%, and a 14x speedup at 100 holes. The authors are also candid about their choices: they compare only to a fine FEM baseline, not to the closest hybrid methods, and they flag the soft-constraint BC issue directly.\n\nThe soft spots are in proportion. The main one: the variational form in Eq. (7) assumes C0/C1 regularity across element interfaces, but the soft-constraint NOs used in most experiments do not enforce that, so the assembled solution is generally discontinuous. The paper's \"resembles DG\" sentence is not backed by a consistency analysis, a penalty formulation, or an error estimate. The empirical correlation between NO error and NOEM error (Figs. 2D, 3D) is suggestive but not convergence. This makes NOEM, as currently presented, a heuristic surrogate-in-the-loop scheme rather than a fully justified numerical method. A Strang-type bound or a systematic study of error vs. decreasing NO boundary error would close much of the gap.\n\nTwo smaller issues: the 14x speedup counts solve time only, not the offline NO training, which matters for the reuse claim; and there is no code released yet, only a promise of a GitHub repository after publication. Neither is fatal.\n\nOverall, the paper is a solid, honest contribution to hybrid ML/FEM. It deserves a serious referee: the method is novel, the experiments are extensive, and the authors have correctly identified the theoretical weakness themselves. I would accept it for peer review and ask for a convergence/consistency experiment or analysis, plus a comparison with at least one of the cited hybrid methods. My own recommendation: engage with it; it is citable and will likely be built upon.","headline":"NOEM is a genuinely new hybrid FEM/neural-operator construction with broad experiments; its main weakness is the unproven interface continuity that the authors disclose but do not resolve.","tokens_in":18360,"tokens_out":3415,"would_cite":true,"duration_ms":35997,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65N30","68T07","65M60"],"pacs":[],"model":"deepseek-v4-flash","headline":"NOEM replaces dense finite-element meshes with pretrained neural-operator elements in a variational framework.","keywords":["neural-operator element method","finite element method","neural operators","DeepONet","MIONet","variational formulation","multiscale PDEs","mesh-free simulation"],"falsifier":"Solve a problem with a strong solution gradient or a coefficient discontinuity aligned with the NOE-FE interface—for example, a Darcy flow whose permeability jumps sharply exactly at the interface—using a soft-constraint NOE, and check whether the NOEM solution converges to the FEM reference as the NO is trained to progressively smaller error. If the global error stagnates or the jump in the solution and its flux across the interface fails to vanish as NO training error goes to zero, the empirical regularity assumption is falsified.","tokens_in":17292,"feed_emoji":"📐","tokens_out":6868,"duration_ms":68014,"temperature":0.7,"pith_summary":"The paper tries to establish that a pretrained neural operator can serve as a single finite element—a neural-operator element—over a large subdomain, and that assembling such elements with ordinary finite elements in the standard energy-minimization framework produces accurate PDE solutions without dense meshing. If true, this gives a way to reuse one trained operator across many problems and scales, lowering the cost of multiscale and complex-geometry simulations. The headline quantitative evidence is about a 14x speedup over FEM for a 100-hole heat-conduction problem, with errors staying stable as the domain grows. The method is demonstrated on nonlinear and multiscale PDEs, complex geometries, and discontinuous coefficient fields.","feed_headline":"Neural operators replace dense meshes as reusable finite elements","feed_subtitle":"A single trained network acts as one element, giving about 14x speedup on a 100-hole heat problem.","key_machinery":"The central object is the neural-operator element (NOE), defined as $\\phi_j^{\\mathrm{NOE}}(x;\\beta_j) = G[u|_{\\partial \\Omega_j^{\\mathrm{NOE}}}](x)$: a pretrained neural operator $G$ that maps the discretized Dirichlet boundary values $\\beta_j$ on a subdomain to the PDE solution inside that subdomain. The method's mechanism is the mixed variational representation $q(x;c) = \\sum_i \\phi_i^{\\mathrm{FE}}(x;\\alpha_i) + \\sum_j \\phi_j^{\\mathrm{NOE}}(x;\\beta_j)$, with coefficients $c$ found by minimizing the energy functional $J[q(\\cdot;c)]$ via Newton's method, using automatic differentiation to assemble the gradient and Hessian of the NOE contributions. The NOE carries the argument because it converts a densely meshed region into a single element whose internal degrees of freedom are eliminated, leaving only boundary values as optimization variables.","core_discovery":"The central claim is that the solution of a PDE can be represented as a sum of standard finite-element shape functions and neural-operator elements (NOEs), where each NOE is the output of a pretrained neural operator mapping the Dirichlet data on a subdomain boundary to the solution inside that subdomain. The coefficients—FE nodal values and the boundary data feeding the NOs—are found by minimizing the same energy functional that FEM minimizes. Because one NOE replaces many finite elements, the number of optimization variables no longer grows with the mesh density inside the subdomain. The paper further claims this makes trained operators reusable: the same DeepONet trained on a single hole is reused for domains with many holes, and a single MIONet handles many different coefficient functions.","pith_inferences":["A sharper practical rule suggested by the error-correlation experiments is to certify the subdomain operator's error before assembly, since the global solution error appears unlikely to be much better than the local operator error.","One stress test left implicit is to push NOEM into regimes where the formal $C^0$/$C^1$ continuity requirement matters most—strong solution gradients or higher-order PDEs such as the biharmonic equation—where soft-constraint NOs may not match the accuracy seen in the tested elliptic examples.","Because training a subdomain operator is cheaper than training a full-domain operator, NOEM could be combined with adaptive refinement, retraining or refining NOEs only where an error indicator is large, though adaptivity is not explored in the paper."],"forward_implications":["Pretrained NOs become reusable building blocks: the same NOE trained on one subdomain geometry can be placed into many larger domains without retraining, as demonstrated by reusing the single-hole DeepONet for domains with up to 100 holes.","Computational cost for complex-geometry problems can drop by about an order of magnitude at scale, with the demonstrated 14x speedup over FEM for the 100-hole case.","The mixed FE/NOE variational framework extends to nonlinear PDEs, multiscale coefficients, discontinuous permeability fields, and non-polygonal geometries.","Because NOEs are differentiable, the assembled model supports downstream tasks such as design optimization once trained operators are available.","The accuracy of the global NOEM solution is controlled by the accuracy of the subdomain NO: experiments show a strong linear correlation between NO prediction error and final solution error."],"supporting_citations":[{"why":"Supplies the DeepONet architecture used to build NOEs and the operator-learning formulation the method builds on.","marker":"[26]"},{"why":"Supplies the MIONet multi-input operator architecture used when the NO must also take coefficient or permeability fields as inputs.","marker":"[25]"},{"why":"Provides the universal operator approximation theorem that underlies the neural-operator representation of the Dirichlet-to-solution map.","marker":"[27]"},{"why":"Supplies the prior hybrid FEM/neural-operator domain-decomposition approach that NOEM distinguishes itself from by avoiding tedious iteration.","marker":"[54]"},{"why":"Supplies the convolutional DeepONet variant and the hard-constraint boundary-condition construction used in several NOEM experiments.","marker":"[58]"},{"why":"Supplies the multiscale coefficient benchmark problem used to test NOEM in Section 4.2.1.","marker":"[61]"}],"fun_headline_variants":["Neural operators as reusable elements cut mesh costs","NOEM: One neural net per subdomain, no dense mesh","Reusable neural operators replace thousands of elements","14x speedup: neural-operator elements beat dense meshes","FEM meets deep learning: neural operators as elements"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that a neural operator trained with soft boundary constraints still produces a subdomain solution consistent enough, in the continuity sense implicit in the variational formulation, to assemble with standard finite elements and yield a correct global solution; the paper relies on numerical observation for this, not a proof.","fun_headline_variants_meta":{"raw":{"variants":["Neural operators as reusable elements cut mesh costs","NOEM: One neural net per subdomain, no dense mesh","Reusable neural operators replace thousands of elements","14x speedup: neural-operator elements beat dense meshes","FEM meets deep learning: neural operators as elements"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000247,"raw_usage":{"total_tokens":1524,"prompt_tokens":910,"completion_tokens":614,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":526,"completion_tokens_details":{"reasoning_tokens":535}},"tokens_in":526,"tokens_out":614,"duration_ms":6778,"temperature":1.0,"reasoning_tokens":535,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T23:15:48.208884+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Solve a problem with a strong solution gradient or a coefficient discontinuity aligned with the NOE-FE interface—for example, a Darcy flow whose permeability jumps sharply exactly at the interface—using a soft-constraint NOE, and check whether the NOEM solution converges to the FEM reference as the NO is trained to progressively smaller error. If the global error stagnates or the jump in the solution and its flux across the interface fails to vanish as NO training error goes to zero, the empirical regularity assumption is falsified.","supporting_citations":[{"cited_title":"Interfacing finite elements with deep neural operators for fast multiscale modeling of mechanics problems.Computer methods in applied mechanics and engineering, 402:115027, 2022","cited_arxiv_id":null,"evidence_quote":"Supplies the prior hybrid FEM/neural-operator domain-decomposition approach that NOEM distinguishes itself from by avoiding tedious iteration."},{"cited_title":"Bayesian deep operator learning for homogenized to fine-scale maps for multiscale pde.Multiscale Modeling & Simulation, 22(3):956–972, 2024","cited_arxiv_id":null,"evidence_quote":"Supplies the multiscale coefficient benchmark problem used to test NOEM in Section 4.2.1."}],"review_version":1}