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REVIEW 3 major objections 5 minor 1 cited by

Finite Element Neural Network Interpolation. Part I: Interpretable and Adaptive Discretization for Solving PDEs

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2412.05719 v1 pith:EVEOY2SB submitted 2024-12-07 math.NA cs.LGcs.NAphysics.comp-ph

Kateřina Škardová , Alexandre Daby-Seesaram , Martin Genet This is my paper · ORCID
classification math.NAcs.LGcs.NAphysics.comp-ph MSC 65N3065N5068T0774S05
keywords finiteelementneuralnetworksphysics-informedmachinelearningrh-adaptivityreferencemultigridtraininglinearelasticityweakformulationlossGaussianquadrature
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

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.

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 (3)
  1. [§3.2.3, Eq. (43), and §4.5] 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.
  2. [§4.3.2, Eq. (62), and Figure 11] 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.
  3. [§4.2, Figure 9] 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.
minor comments (5)
  1. [§3.2.2, Eq. (34)] 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.
  2. [§3.2.2, Eq. (41)] 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.
  3. [§1] In the introduction, "we propose a new EFFEN framework named FENNI" should read "EFENN".
  4. [§3.2.2] The sentence "a propriety that will be used" should read "a property that will be used".
  5. [§3.2.3] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: FENNI's central claims are benchmarked against independent analytical and FEM reference solutions, and the only self-citations are motivational or implementation-related, not load-bearing.

full rationale

The paper's derivation chain is self-contained with respect to its claimed contributions. The FENNI architecture is built from standard reference-element shape functions and a sparse neural-network parameterization; the loss functions (potential energy, residual, weak form) are standard and are minimized without fitting any parameter to the reference solutions. The accuracy claims are assessed against an analytical solution in 1D and an independently computed fine-mesh FEM solution in 2D, so the results are not forced by construction. The h-adaptivity criterion in Eq. 43 is a heuristic based on relative Jacobian decrease, but it is not equivalent to the output it aims to improve; it is a methodological choice whose validity is tested empirically on the benchmark problems, not a fitted parameter disguised as a prediction. The multigrid training strategy initializes finer models from coarser trained models, and its speed-up is reported as measured computational time, not derived from the claim itself. Self-citations appear only as forward references to Part II for surrogate modeling, links to the authors' released code and demos, and a passing mention that a continuous equilibrium-gap formulation 'could be used here as well' (Genet, 2023); none of these carries the burden of the paper's central results. No uniqueness theorem is imported from prior work, and no ansatz is smuggled in via citation. The Jacobian-based refinement rule could be questioned as heuristic or insufficiently general, but that is a correctness/robustness concern, not circularity. Overall, the derivation and validation chain does not reduce to its own inputs.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced; FENNI is a numerical architecture, not a new force, particle, or conserved quantity. The central claims rest on standard FEM theory plus a heuristic refinement criterion and unverified training assumptions.

free parameters (5)
  • h-adaptivity threshold tΔJ = not stated
    Elements are split when relative Jacobian change exceeds this threshold (Eq. 43); the value is chosen by hand and no sensitivity study is reported.
  • residual loss weights λ1, λ2 = λ1 = L, λ2 = 1
    Set once in Eq. 62 for the residual loss; the paper notes only one set was tested and balancing remains problematic.
  • maximum number of element splits = 1
    Used in Section 4.5 to limit h-refinement level; a user-chosen hyperparameter that directly controls mesh size.
  • multigrid scaling factor = 2
    The sequence of mesh resolutions is defined by scaling factor 2 in Section 4.4; this schedule is not justified beyond the chosen test cases.
  • Gauss quadrature order = 1 point in 2D, 2 to 5 points in 1D
    Integration accuracy is tuned per experiment; the choice affects the loss evaluation and is part of the method's setup.
assumptions (5)
  • standard math Reference-element shape functions and Gaussian quadrature provide a valid discretization for linear elastostatics.
    Invoked throughout Section 3.2; the method relies on standard finite element theory for interpolation and integration.
  • domain assumption Minimizing the potential energy or weak-form loss yields the solution of the strong-form PDE.
    Section 2.2 constructs losses from the elastostatics equations, assuming the equivalence of weak, energy, and strong forms for the test problems.
  • ad hoc to paper The relative change of the Jacobian determinant identifies regions needing h-refinement.
    Eq. 43 in Section 3.2.3 is introduced without error analysis; the whole rh-adaptivity method depends on this heuristic.
  • domain assumption A trained coarse-mesh solution is a good initialization for a finer mesh.
    Section 3.3.3 uses this to transfer parameters in multigrid training; convergence is not guaranteed for nonconvex losses.
  • domain assumption L-BFGS and Adam converge to a useful minimum of the nonconvex training loss.
    Training relies on these optimizers (Section 4.2), yet the paper reports divergence for some coarse r-adaptive runs, so success depends on initialization and problem.

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Cite this review

Pith. "Pith review of Finite Element Neural Network Interpolation. Part I: Interpretable and Adaptive Discretization for Solving PDEs." pith.science (2026). https://pith.science/paper/EVEOY2SB

@misc{pith2026241205719,
  author       = {Pith},
  title        = {Pith review of: Finite Element Neural Network Interpolation. Part I: Interpretable and Adaptive Discretization for Solving PDEs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EVEOY2SB}},
  note         = {Machine review of arXiv:2412.05719}
}
read the original abstract

We present the Finite Element Neural Network Interpolation (FENNI) framework, a sparse neural network architecture extending previous work on Embedded Finite Element Neural Networks (EFENN) introduced with the Hierarchical Deep-learning Neural Networks (HiDeNN). Due to their mesh-based structure, EFENN requires significantly fewer trainable parameters than fully connected neural networks, with individual weights and biases having a clear interpretation. Our FENNI framework, within the EFENN framework, brings improvements to the HiDeNN approach. First, we propose a reference element-based architecture where shape functions are defined on a reference element, enabling variability in interpolation functions and straightforward use of Gaussian quadrature rules for evaluating the loss function. Second, we propose a pragmatic multigrid training strategy based on the framework's interpretability. Third, HiDeNN's combined rh-adaptivity is extended from 1D to 2D, with a new Jacobian-based criterion for adding nodes combining h- and r-adaptivity. From a deep learning perspective, adaptive mesh behavior through rh-adaptivity and the multigrid approach correspond to transfer learning, enabling FENNI to optimize the network's architecture dynamically during training. The framework's capabilities are demonstrated on 1D and 2D test cases, where its accuracy and computational cost are compared against an analytical solution and a classical FEM solver. On these cases, the multigrid training strategy drastically improves the training stage's efficiency and robustness. Finally, we introduce a variational loss within the EFENN framework, showing that it performs as well as energy-based losses and outperforms residual-based losses. This framework is extended to surrogate modeling over the parametric space in Part II.

Figures

Figures reproduced from arXiv: 2412.05719 by the authors.

Figure 1
Figure 1. Reference problem The governing equations read: ∇ · σ (u) + f = 0 in Ω, (1) σ T = σ in Ω, (2) n · σ (u) = t on ∂ΩN , (3) u = uD on ∂Ωd, (4) where u is displacement, σ is the Cauchy stress tensor, f is the prescribed body force, uD and t are the displacement and surface force prescribed on the respective segments of the domain boundary. With the assumption of linear elasticity, the stress-strain relation is σ = C : ε… view at source ↗
Figure 2
Figure 2. The construction of linear shape functions. The global linear shape functions (b) are assembled by combining [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. The global quadratic shape functions (b) are assembled by combining the auxiliary function for all elements. [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: The effect of trainable nodal values and nodal coordinates in 1D. [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Contrary to the interpolation-layer-based implementation, the shape functions are defined on the reference [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: Two versions of the model implementation, defined for domain [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: The red-green mesh refinement strategy [ [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: 2D problem setting. The displacement u, obtained as the output of the model, is used to compute the stress tensor σ and von Mises stress σ r V M, defined as σV M = q3 2 σ dev : σ dev, where σ dev is the stress deviator tensor σ dev = σ − 1 3 tr(σ)I. 4.1.3 Error metrics…
Figure 9
Figure 9. Figure 9: Normalized displacement (a,b) and strain (c,d) error. Two methods for evaluating the potential energy loss [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: The results on 14-nodes mesh obtained using the 5-point Gaussian quadrature (a-c) and trapezoidal rule with [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 11
Figure 11. Figure 11: Comparison of residual loss, weak formulation loss, and potential energy loss. For the finest tested mesh [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]
Figure 12
Figure 12. Figure 12: Convergence to reference solution for train [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]
Figure 13
Figure 13. Figure 13: The effect of r-adaptivity with increasing mesh resolution. A multigrid training strategy with scaling factor 2 [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]
Figure 14
Figure 14. Figure 14: The reference von Mises stress σ r V M and σV M computed based on displacement obtained from the NN model with fixed mesh and r-adaptivity. The L-BFGS optimizer was used in both cases. Scale 1 Scale 2 Scale 3 Scale 4 Scale 5 44 nodes 144 nodes 484 nodes 1 804 nodes 7 …
Figure 15
Figure 15. Figure 15: The effect of rh-adaptivity on the accuracy of [PITH_FULL_IMAGE:figures/full_fig_p021_15.png]

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Forward citations

Cited by 1 Pith paper

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