{"id":"cb0823e1-6b94-4656-9f17-ecbdbbeaf0ba","arxiv_id":"2412.05719","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"FENNI embeds finite element shape functions in a sparse neural network, enabling r- and h-adaptive meshes and multigrid training for solving linear elasticity problems.","lead":"The authors present FENNI, a neural network that solves elasticity equations by building finite element shape functions directly into its architecture. Its mesh points can move and split during training, and a multigrid strategy is shown to speed up training on small 1D and 2D test problems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Jacobian-based h-refinement criterion (Eq. 43) is not tied to any error estimator; it has only been shown to work on the two included test problems, so the claimed general 2D rh-adaptivity remains unverified.","rationale":"I agree with the reader's weakest-assumption analysis and would keep the verdict conditional. The reference-element architecture is clearly specified, the code and interactive demos are a significant reproducibility asset, and the multigrid training speedup in Table 2 is a genuine positive result. The h-adaptivity criterion is the least secure link in the central novelty chain: it is the only new mesh-adaptation contribution in 2D, yet Eq. 43 has no connection to a posteriori error estimation and no sensitivity analysis of the threshold. The two test problems are favorable cases where node accumulation and true solution features coincide. The proposed benchmark with a known corner singularity would directly test whether the elements flagged by the Jacobian-decrease criterion are actually the elements where refinement reduces error. If the criterion passes such a test, the current conditional verdict could be upgraded; if it fails, the rh-adaptivity claim in the abstract should be weakened to a demonstration on the two included examples. No formal verification is claimed, but the available code reduces the risk that the architecture description is unrealizable.","tokens_in":20720,"tokens_out":6910,"duration_ms":78183,"concrete_test":"Use the released NeuROM code to solve a 2D benchmark with a known analytical solution and a sharp feature, e.g. linear elasticity on an L-shaped domain with corner singularity u ~ r^{2/3}, running the same multigrid + rh-adaptivity protocol as in Section 4.5 (max splits = 1, threshold t_ΔJ chosen to add a comparable number of nodes). At each refinement trigger, compute the overlap (e.g., Jaccard index over the top 10% of elements sorted by ΔJ_i^n and by true local H1 error), and compare final error-per-DOF against r-adaptivity only and uniform refinement. If the overlap is no better than random, or if error-per-DOF does not improve, Eq. 43 is not a reliable general h-refinement driver and the abstract's rh-adaptivity claim should be explicitly restricted to the two demonstrated problems.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central novelty of the 2D results is combined rh-adaptivity, and the h-adaptivity trigger in Eq. 43 is a heuristic: an element is split when its area decreased by more than a threshold between two training iterations. This identifies where r-adaptivity made nodes accumulate, not where the discrete solution is inaccurate. In the two test problems (1D bar with smooth body force, 2D plate with a hole), the high-error regions coincide with regions of node concentration, so the heuristic works in the demonstrations. However, the abstract claims a general extension of HiDeNN's rh-adaptivity to 2D, and that claim rests on the assumption that relative Jacobian decrease is a reliable proxy for local error. No derivation, error estimator, threshold-sensitivity study, or comparison with classical a posteriori indicators is provided. If the optimizer moves nodes for reasons unrelated to interpolation error, or if large errors occur without strong area decrease, the criterion will refine the wrong regions or under-refine important ones. Section 4.5 also uses only one allowed split ('maximal number of element splits was set to 1'), so the evidence for the h-adaptivity component is quite limited. Thus the headline claim that FENNI 'extends combined rh-adaptivity from 1D to 2D' is not yet supported beyond the specific benchmarks.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes FENNI, a reference-element-based EFENN architecture for solving linear elastostatic problems. The method represents the PDE solution via a sparse neural network whose weights and biases are nodal values and nodal coordinates; shape functions are evaluated locally on a reference element, the loss is integrated by Gauss quadrature, nodal coordinates can be trained (r-adaptivity), and a Jacobian-based splitting criterion adds nodes (h-adaptivity). A multigrid-like transfer strategy initializes fine-mesh models from coarse-mesh solutions. Numerical experiments on a 1D bar and a 2D plate-with-hole compare fixed, r-adaptive, and rh-adaptive meshes, several loss functions (potential energy, residual, weak form), and two optimizers, with reference to analytical and fine-FEM solutions. The central claims are that the reference-element architecture improves over the original HiDeNN formulation, that the multigrid strategy improves training efficiency and robustness, that 2D combined rh-adaptivity is achieved, and that a variational loss matches the energy loss and outperforms a residual-based loss.","tokens_in":21041,"tokens_out":4146,"duration_ms":42713,"significance":"If the claims hold, FENNI is a practically useful, interpretable, and adaptive physics-informed neural-network method that combines finite-element structure with trainable meshes. The paper is strengthened by available code (Zenodo, GitHub) and interactive demos, and the benchmarks are independent analytical or fine-FEM references, so the results are not fitted to the target outputs. The reference-element formulation is a clean way to handle unstructured 2D meshes, and the multigrid initialization is a sensible transfer-learning idea. However, the headline 2D rh-adaptivity result rests on a largely heuristic h-adaptivity trigger, the residual-loss comparison is confounded by unequal interpolation orders, and the experiments report no variability across runs. These points are correctable but currently limit the strength of the abstract-level claims.","major_comments":[{"comment":"The h-adaptivity criterion in Eq. (43) is an ad-hoc heuristic: an element is refined when its Jacobian determinant (area) decreases by more than a threshold, on the assumption that shrinking elements indicate regions needing refinement. No link to an a posteriori error estimator is provided, no threshold-sensitivity study is reported, and Section 4.5 restricts the experiment to a single split level (\"maximal number of element splits was set to 1\"). Consequently, the abstract claim that combined rh-adaptivity is extended from 1D to 2D is not yet supported beyond the two benchmarks, where high-error regions coincide with node accumulation. The authors should compare the criterion with a standard residual- or recovery-based error indicator, test at least two thresholds tΔJ and two maximum split levels, and ideally report a problem setup where node concentration and interpolation error are not spatially coincident.","section":"§3.2.3, Eq. (43), and §4.5"},{"comment":"The comparison between the residual loss and the potential-energy loss is confounded by different interpolation orders: the residual model in Eq. (62) uses quadratic shape functions for displacement and linear shape functions for strain, while the potential-energy model uses linear displacement and then differentiates it to obtain piecewise-constant strain. The statement that the residual loss has \"significantly higher\" displacement error and \"lower\" strain error is therefore not attributable to the loss choice alone. The claim in the abstract that the variational loss \"outperforms residual-based losses\" needs a same-order comparison, or an explicit decomposition of interpolation error versus optimization error.","section":"§4.3.2, Eq. (62), and Figure 11"},{"comment":"The conclusion that Gaussian quadrature is more robust than the trapezoidal rule for r-adaptive meshes is based on a single run per configuration, and the reported data include a diverging case on the 10-node mesh and r-adaptive errors that are worse than fixed-mesh errors on several finer resolutions. Without repeated runs, error bars, or at least a statement about sensitivity to initialization and optimizer settings, the robustness comparison is not established. The authors should report variability over a few seeds or show that the observed behavior is deterministic by construction.","section":"§4.2, Figure 9"}],"minor_comments":[{"comment":"Equation (34) appears to contain typos: the third column of the coordinate matrix should be the node c coordinates (xc, yc), but the text shows xb and yb twice.","section":"§3.2.2, Eq. (34)"},{"comment":"The expression for the physical coordinates of a quadrature point uses \"ygi yaj + ygi ybj + ...\" in the second component; this should presumably involve xg_i and yg_i as the barycentric coordinates for the x and y components, respectively.","section":"§3.2.2, Eq. (41)"},{"comment":"In the introduction, \"we propose a new EFFEN framework named FENNI\" should read \"EFENN\".","section":"§1"},{"comment":"The sentence \"a propriety that will be used\" should read \"a property that will be used\".","section":"§3.2.2"},{"comment":"The parameter controlling the maximum number of element splits is introduced in the methods text as \"the maximal split value\" but is not defined until Section 4.5, where it is set to 1; this should be specified when the refinement strategy is described.","section":"§3.2.3"}],"recommendation":"major_revision","confidential_remarks":"The paper is within scope for a computational mechanics or scientific machine learning journal and the code release is commendable. The main risk is overclaiming: the h-adaptivity criterion and the residual-loss comparison are not strong enough to support the abstract as written. Both are fixable within the manuscript's scope, so I do not recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a solid, honest paper about embedding finite element shape functions in a neural network. The reference-element architecture (shape functions evaluated on a reference element via element IDs) is a real improvement over HiDeNN's global shape-function layer: it enables Gaussian quadrature, works on unstructured triangular meshes, and makes the network parameters interpretable. The multigrid training strategy, reusing coarse-mesh parameters to initialize finer meshes, is pragmatic and clearly explained. The experiments are reproducible—code and interactive demos are archived—and the authors are candid about several limitations.\n\nThe soft spots are real but mostly addressable. The h-adaptivity criterion (Eq. 43) splits elements where the Jacobian (area) shrank most during r-adaptivity. That is a heuristic, not an error estimator; no derivation, threshold-sensitivity study, or comparison to standard a posteriori indicators is given, and Section 4.5 caps the number of splits at one. So the claim that FENNI 'extends combined rh-adaptivity from 1D to 2D' is currently supported only for the two benchmarks. The residual-loss comparison is also confounded: the model uses quadratic shape functions for displacement in the residual case and linear elsewhere, which the authors note, but the abstract still says the variational loss outperforms residual loss—that comparison needs equal interpolation order. No error bars or repeated-run statistics are reported anywhere. Two other things jumped out: Table 2 shows multigrid is actually slower than single-level training for several mesh resolutions (speed-ups of 0.48, 0.8, 0.78), so the abstract's 'drastically improves efficiency' is an overstatement; and Section 4.2 contains a contradiction, saying r-adaptivity improves accuracy for 10–21 nodes while also saying the 10-node case diverges and then that other resolutions are unchanged or better. That passage needs rewriting.\n\nNone of this sinks the paper. The core architecture is sound, the multigrid idea works for the hardest cases where single-level training fails, and the code is there for others to build on. The right verdict is major revision: add a proper study of the h-refinement criterion (or at least a threshold sweep), redo the residual comparison at matched interpolation order, run multiple seeds, and align the abstract with what the tables actually show.\n\nWho is it for? Anyone working on FEM-NN hybrids, PINN transfer learning, or adaptive mesh methods for elasticity. It deserves a serious referee, and the issues look fixable.","headline":"FENNI is a clean, reproducible step for FEM-embedded neural networks, but its headline 2D rh-adaptivity and multigrid speed-up claims are thinner than the abstract suggests.","tokens_in":21556,"tokens_out":4158,"would_cite":true,"duration_ms":37962,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65N30","65N50","68T07","74S05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that a finite-element-based neural network whose trainable parameters are nodal values and nodal coordinates can solve linear elasticity problems, adapt its mesh during training, and transfer learned solutions across mesh…","keywords":["finite element neural networks","physics-informed machine learning","rh-adaptivity","reference element","multigrid training","linear elasticity","weak formulation loss","Gaussian quadrature"],"falsifier":"Take a 2D elasticity problem with a known exact solution and arrange, or observe, that the largest solution error sits in elements that do not shrink fastest under r-adaptivity; if the elements flagged by $\\Delta J^n_i > t_{\\Delta J}$ miss those high-error regions, or if rh-adaptivity is no more accurate than r-adaptivity alone, the central claim fails.","tokens_in":20544,"feed_emoji":"📐","tokens_out":9636,"duration_ms":86152,"temperature":0.7,"pith_summary":"The paper's goal is to make finite-element-style neural networks practical PDE solvers by rebuilding the network around a reference element, so the same architecture works on intervals in 1D and unstructured triangular meshes in 2D. It claims three coordinated improvements over the earlier HiDeNN approach: shape functions evaluated on a reference element with Gaussian quadrature for loss integrals, a multigrid training schedule that reuses a coarse-mesh solution to initialize finer meshes, and a Jacobian-based criterion that extends combined rh-adaptivity (moving nodes plus adding nodes) to 2D. On the paper's 1D bar and 2D plate-with-a-hole tests, the multigrid strategy makes fine-mesh training much faster and sometimes prevents divergence, and a weak-form variational loss matches the potential-energy loss while outperforming the residual loss. A sympathetic reader takes away that adaptive mesh refinement can live inside a trainable network with interpretable parameters, which is the groundwork for the parametric surrogate models announced in Part II.","feed_headline":"Finite-element neural nets adapt their own mesh while training","feed_subtitle":"Reference-element shape functions plus coarse-to-fine multigrid training make PDE solving cheaper and more reliable on 1D and 2D tests.","key_machinery":"The carrying object is the reference-element interpolation block. For each evaluation point, the network receives the point's coordinates plus the ID of the element containing it, maps the point to a reference interval $[-1,1]$ or reference triangle via a linear map, evaluates elemental shape functions there, and combines them with the nodal values listed in a connectivity table. For the loss, the same map gives Gaussian quadrature points element by element and a Jacobian determinant equal to twice the element area; the relative decrease of that determinant, $\\Delta J^n_i = (|J^{n-1}_i| - |J^n_i|)/|J^{n-1}_i|$, is the criterion that selects elements for red-green splitting. The multigrid training strategy completes the mechanism: a converged coarse model's nodal values and coordinates are evaluated on a finer mesh and used as that model's initialization, which is why changing discretization does not force training from scratch.","core_discovery":"On its own terms, the paper's central discovery is that a sparse neural network whose parameters are exactly the nodal values and nodal coordinates of a finite element mesh can solve linear elastostatic problems and refine its own discretization while training. Because the shape functions are evaluated on a reference element rather than assembled globally, the network's architecture is fixed by the mesh connectivity, boundary values are imposed by freezing boundary nodal parameters, and the loss integrals use tabulated Gaussian quadrature. Making nodal coordinates trainable gives r-adaptivity; when the relative shrink of an element exceeds a threshold, the element is split by red-green refinement, giving h-adaptivity; and a coarse-to-fine multigrid schedule transfers the converged parameters from one mesh to the next. The paper reports that this combination matches or beats the fixed-mesh baselines in accuracy on its test problems, and that the weak-form loss is as accurate as the energy loss but more generally applicable than either the energy or residual losses.","pith_inferences":["An extension the paper gestures at but does not demonstrate: the same reference-element architecture should carry to 3D tetrahedral meshes and to nonlinear constitutive laws, since only the reference element and the loss integrand would need to change.","The reported speed-up is a combined effect of parameter initialization and the coarse mesh's regularizing influence; a follow-up study could separate the two by comparing warm starts against randomly initialized fine meshes.","Because the paper's CPU timings show a classical FEM solver remains far cheaper for a single solve, the method's practical payoff depends on many-query or parametric settings, which is precisely the setting targeted by Part II."],"forward_implications":["The same network can solve a linear elasticity problem on unstructured triangular meshes and refine the mesh during training without changing the core architecture.","Dirichlet boundary conditions are imposed exactly by freezing boundary nodal values, so the loss contains no penalty term for essential boundary conditions.","The multigrid schedule transfers nodal values and coordinates from coarse to fine meshes, making fine-mesh training converge where single-level training fails in the reported experiments.","On the tested problems, the weak-form loss is as accurate as the potential-energy loss and more accurate than the residual loss for displacement and stress."],"supporting_citations":[{"why":"Introduces the HiDeNN/EFENN architecture and potential-energy loss that FENNI extends and compares against.","marker":"[Zhang et al., 2021]"},{"why":"Demonstrates r-adaptivity and parameter reuse across mesh levels in HiDeNN, the basis for the multigrid transfer strategy.","marker":"[Zhang et al., 2022]"},{"why":"Supplies the red-green refinement procedure used to split elements selected by the Jacobian criterion.","marker":"[Carstensen, 2004]"},{"why":"Defines the residual physics-informed loss that FENNI tests and finds less reliable.","marker":"[Raissi et al., 2019]"},{"why":"Provides the Deep Ritz potential-energy loss used as the primary loss in the experiments.","marker":"[Yu et al., 2018]"},{"why":"Formulates the variational/weak-form loss that FENNI adapts to the embedded finite element setting.","marker":"[Berrone et al., 2022a]"},{"why":"Shows an r-adaptive deep-learning PDE solver with potential-energy loss, a direct comparison point for FENNI's r-adaptivity.","marker":"[Omella & Pardo, 2024]"},{"why":"Generates the initial meshes whose geometry and connectivity define each FENNI model's architecture.","marker":"[Geuzaine & Remacle, 2020]"}],"fun_headline_variants":["Neural net with trainable mesh nodes solves PDEs adaptively","Self-refining neural mesh cuts cost of solving PDEs","Reference-element neural nets adapt meshes during training","Multigrid-trained neural net refines its own FEM mesh","Sparse neural net auto-refines mesh for efficient PDE solving"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the places where elements shrink fastest during node movement are exactly the places where adding new nodes improves the solution; this Jacobian-based refinement rule is a heuristic, and the paper does not derive it from an error estimate.","fun_headline_variants_meta":{"raw":{"variants":["Neural net with trainable mesh nodes solves PDEs adaptively","Self-refining neural mesh cuts cost of solving PDEs","Reference-element neural nets adapt meshes during training","Multigrid-trained neural net refines its own FEM mesh","Sparse neural net auto-refines mesh for efficient PDE solving"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001264,"raw_usage":{"total_tokens":5237,"prompt_tokens":1069,"completion_tokens":4168,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":685,"completion_tokens_details":{"reasoning_tokens":4085}},"tokens_in":685,"tokens_out":4168,"duration_ms":24420,"temperature":1.0,"reasoning_tokens":4085,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T20:25:21.112434+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a 2D elasticity problem with a known exact solution and arrange, or observe, that the largest solution error sits in elements that do not shrink fastest under r-adaptivity; if the elements flagged by $\\Delta J^n_i > t_{\\Delta J}$ miss those high-error regions, or if rh-adaptivity is no more accurate than r-adaptivity alone, the central claim fails.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the red-green refinement procedure used to split elements selected by the Jacobian criterion."}],"review_version":1}