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REVIEW 5 major objections 7 minor 34 references

The Finite Element Neural Network Method: One Dimensional Study

T0 review · 5 major / 7 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A neural network trained on per-element weak-form residuals can solve 1D differential equations with flux and boundary information built into the loss.

desk verdict A worthwhile 1D proof-of-concept for a variational PINN that recovers flux terms via non-vanishing Lagrange test functions, held back by missing baseline comparisons and an overclaimed equivalence to FEM. read the letter →

arxiv 2501.12508 v1 pith:C6EWHD44 submitted 2025-01-21 cs.CE cs.AI

Mohammed Abda , Elsa Piollet , Christopher Blake , Frédérick P. Gosselin This is my paper · ORCID
classification cs.CEcs.AI
keywords finiteelementneuralnetworkmethodweakformulationPetrov-GalerkinLagrangetestfunctionsconvolutionfiltersfluxcontinuityvariationalphysics-informednetworksadaptivemeshrefinement
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 introduces FENNM, a neural-network solver that treats the network as the global trial function in a Petrov-Galerkin finite element scheme. Its key move is to use Lagrange test functions that stay nonzero at element boundaries, so the per-element weak-form residual keeps boundary flux terms that earlier variational PINN variants discard. The loss is formed by squaring these per-element residuals and averaging over elements and test functions, with Gauss quadrature evaluated through convolution filters. The paper claims this embeds flux continuity, forcing terms, and natural boundary conditions directly in the training objective, and demonstrates accuracy on a beam with a point load, a nonlinear pendulum, convection-dominated transport, and steep Poisson layers. A sympathetic reader should care because, if correct, FENNM gets a neural solver to behave like a conventional finite element solver while keeping the network's flexibility for inverse or ill-posed problems.

What carries the argument

The carrying mechanism is the per-element weak-form residual in Eq. (12): for each element, the boundary flux terms $\phi_k x\, d\hat{U}/dx$ evaluated at element endpoints are kept because the Lagrange test functions $\phi_k$ are nonzero there, while the volume integrals are approximated by Gauss quadrature and written as convolutions with fixed filters $W_q \phi_k(\xi_q)$ and $W_q\, d\phi_k/d\xi$. These filters are defined in local coordinates so the same filters work on any element size, and the squared residuals are averaged over all elements and test functions to form the training loss. The nonvanishing boundary values are what let natural boundary conditions and intermediate forcing enter the loss rather than staying in separate penalty terms.

What would settle it

Train FENNM on the manufactured Poisson problem with a smooth exact solution, then measure the jump of the flux $x\,d\hat{U}/dx$ across every internal element boundary after training. If these flux jumps do not decrease with mesh refinement even as pointwise errors decrease, the boundary flux terms in Eq. (12) are not actually constraining the solution, and the paper's claim that nonvanishing Lagrange test functions embed flux information would be falsified.

Watch

Extended reading notes

Core claim

The paper's central claim is that a neural network can serve as the global trial function in a Petrov-Galerkin finite element scheme, provided the test functions are Lagrange polynomials that do not vanish at element boundaries. The resulting per-element weak-form residual, Eq. (12), contains four ingredients: boundary flux terms, lower-order differential operators, the forcing term, and convolution filters built from test functions and quadrature weights. Minimizing the sum of squared residuals over elements and test functions embeds flux continuity, natural boundary conditions, and intermediate point loads directly in the loss, so the network is trained on the same information a classical FEM assembles. The paper supports this by showing error saturation with quadrature order, convergence rates for linear, quadratic, and cubic test functions, and case studies—a cantilever beam with a midspan force, a nonlinear pendulum, convection-dominated transport, and steep Poisson layers—where FENNM reaches pointwise errors near machine precision.

Load-bearing premise

The paper assumes that a sum of squared per-element weak-form residuals—with each internal interface flux counted twice, once per adjacent element—is a faithful training objective that drives the network to the true solution rather than a misleading or overconstrained loss.

Editorial extensions

If this is right

  • Natural boundary conditions and intermediate loads, such as a point force, can be imposed inside the residual loss of a single network, removing penalty terms that compete during training.
  • Per-element convolution filters compute Gauss quadrature sums in parallel, avoiding the loop over elements that makes hp-VPINN expensive.
  • A lower bound on quadrature points follows from the polynomial order of the test functions, and an upper bound is identified by error saturation, so using more points adds cost without accuracy.
  • The strong-form residual is never used in training and can be evaluated afterward as an error proxy to drive adaptive mesh refinement, reducing elements by more than half for sharp-layer Poisson problems.
  • FENNM can be applied to the time dimension as another coordinate, approximating transient solutions without committing to a time-stepping scheme.

Reading between the lines

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

  • Because the loss squares per-element residuals before summing, internal interface fluxes do not cancel as they would in an assembled global weak form; a direct comparison between FENNM and a Galerkin FEM with identical meshes and test functions would reveal whether this least-squares enforcement of flux continuity improves or degrades conditioning.
  • The number of network parameters stays fixed as the mesh is refined, so h-refinement changes only the filters and quadrature points; this opens the door to mesh adaptation without increasing optimization variables, a consequence the paper does not develop.
  • The time-as-coordinate treatment in the pendulum case suggests a full space-time formulation in higher dimensions, but multidimensional convolution filters are needed, and whether the method scales there remains open.
  • A sharper comparison would pit FENNM against least-squares FEM with the same elements and test functions to isolate the network's contribution from the weak-form least-squares structure.
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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

5 major / 7 minor

Summary. The paper proposes FENNM, a Petrov-Galerkin neural solver for one-dimensional differential equations. The neural network supplies the global trial solution, while Lagrange test functions, chosen to be nonvanishing at element boundaries, are used to form per-element weak-form residuals. Gauss quadrature is evaluated with convolution filters following the cv-PINN idea, and the total loss is a mean-squared sum of per-element residuals that includes flux terms, natural boundary conditions, and point or interface forcing terms. The paper presents the derivation, studies the effect of quadrature order and test-function order, reports empirical convergence rates, and demonstrates the method on a cantilever beam with an intermediate point force, damped and undamped pendulums, an advection-diffusion problem with a boundary layer, and Poisson problems with steep layers and discontinuous forcing. It also presents a residual-based adaptive mesh refinement example.

Significance. If substantiated, the method would offer a practical way to combine FEM-like weak-form structure with the flexibility of a single neural network, and the use of convolution for parallel quadrature evaluation is attractive. The derivations in Section 3 are mostly correct, the test suite is broad, and the idea of using the strong-form residual as an a posteriori error proxy is useful. At present, however, the claimed advantages over VPINN, hp-VPINN, and cv-PINN are not demonstrated by direct comparison, the convergence analysis is not a standard discretization convergence analysis, and much of the numerical evidence lacks repeated-run statistics. The manuscript is a promising methods paper but requires substantial strengthening before publication.

major comments (5)
  1. [Sec. 3, Eq. (12)] The loss in Eq. (12) is a sum of squared per-element weak-form residuals. In assembled FEM, interior flux terms cancel before discrete equations are formed, whereas here they are squared element-by-element; this is a legitimate least-squares Petrov-Galerkin formulation, and zero loss would imply the elementwise weak form holds, so I do not regard it as internally inconsistent. However, the paper does not analyze the minimizer of this functional: there is no consistency, coercivity, or convergence argument relating the minimizer of Eq. (12) to the exact solution as the mesh and the number of quadrature points increase. Because the central claim that FENNM is an FEM-like reliable solver rests on this loss, this gap should be addressed, either theoretically or through a careful numerical study in which the trial-space capacity is also varied.
  2. [Sec. 4.2, Eq. (27), Fig. 5] The convergence-rate study is not a standard discretization convergence study. The network parameter count, called DoF in Eq. (27), remains fixed (481 parameters) while Nel increases, so h-refinement does not enrich the trial space; the observed error decrease and plateaus are therefore a mix of optimization, quadrature, and round-off effects. Eq. (27) also counts weights and biases, not degrees of freedom of the output manifold, which is particularly misleading in the FEM comparison of Fig. 11, where FEM DoF grows as pNel+1. The authors should either present a genuine approximation-theoretic convergence analysis or substantially soften the convergence-rate and accuracy claims.
  3. [Abstract, Sec. 4] The paper repeatedly claims advantages over VPINN, hp-VPINN, and cv-PINN (flux information, natural boundary conditions, cost reduction), but no experiment in Section 4 compares FENNM with any of these methods; the numerical comparisons are with analytical solutions, FEM, FDM, and RK45. Direct benchmark comparisons on the same problems, with error and cost or epoch metrics, are required to substantiate the central contribution.
  4. [Table 1, Sec. 4.3.3] The transport-equation experiment is internally inconsistent: Table 1 lists quartic test functions for Section 4.3.3, while the text and Figure 8 describe linear, quadratic, and cubic test functions for 22, 15, and 11 elements. Since the dependence on test-function order is one of the paper's main messages, the authors must reconcile this mismatch and ensure the reported results correspond to the stated test functions.
  5. [Secs. 4.3.1-4.3.4] Except for Figures 5 and 11, the numerical case studies are single training runs without repeated initializations, error bars, or a variability statement. The conclusions about accuracy, training histories, and adaptive mesh refinement are therefore not robustly supported. Please provide multiple-seed statistics, or at least a clear statement of observed variability, for the main case studies.
minor comments (7)
  1. [Secs. 1 and 4.3.1] There are several typos, including 'Neural netowrks' in Section 1 and 'Diriclet' in Section 4.3.1.
  2. [Eq. (15)] The Neumann boundary conditions in Eq. (15) are garbled: 'd3w/dx2' should presumably be 'd3w/dx3', and the condition at x=L/2 is not written consistently with the flux-jump condition used in Eq. (18a).
  3. [Eq. (4)] The definition of LB in Eq. (4) has unbalanced parentheses and an ambiguous factor of 1/NB; please rewrite it for clarity.
  4. [Sec. 4.3.4] The phrase 'introduced a prior noise' should be 'introduced a priori noise' or 'introduced some noise'; the current wording is unclear.
  5. [Sec. 3, Eq. (12)] Please clarify how the 'predefined' convolution filters are applied when Jx varies per element: Eq. (12) places Jx inside Filters2, so those filter weights are not mesh-independent unless additional per-element scaling is performed; the text should describe the implementation used for nonuniform and adaptively refined meshes.
  6. [Sec. 4.3.4] The adaptive mesh refinement rule is described only qualitatively as 'elements with high residuals'; please specify the quantitative criterion, such as a threshold on the DE residual, and how elements are split or coarsened.
  7. [Appendix 7] Appendix 7 counts total network parameters, weights plus biases, in Eq. (27) and calls them degrees of freedom; this terminology should be changed to 'network parameters' or the DoF concept should be defined properly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central loss is the PDE weak-form residual, and all accuracy claims are benchmarked against external reference solutions.

full rationale

The paper's central object is the per-element weak-form residual in Eq. (12). This residual is obtained from the governing differential equation by integration by parts with Lagrange test functions; the network minimizes this residual directly. There is no fitted parameter that is later renamed a prediction, and no quantity in the loss is defined in terms of the solution it claims to predict. The only free choices are test-function order and quadrature point count, which are tuned empirically in Sec. 4.1 but not used to generate the benchmark answers. The verification case studies compare FENNM output against exact manufactured solutions, FEM, FDM, and RK45, so the claimed accuracy is externally anchored. The load-bearing prior work (cv-PINN convolution machinery, hp-VPINN weak formulation) is cited to external authors; the reference list contains no self-citations by the present authors, so there is no self-citation chain forcing the result. The skeptical concern that the convergence study fixes network DoF while refining the mesh is a legitimate limitation of the numerical study, but it is not circularity: it concerns what the measured convergence rates mean, not whether the loss reduces to its inputs. Similarly, the claim that flux information embeds natural BCs follows from the algebra of integration by parts, not from naming. Accordingly, no circular step can be quoted, and the honest finding is no significant circularity (score 0).

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

The method relies on standard FEM weak-form machinery plus a set of hand-chosen training parameters. The most consequential axiom is that the squared per-element residual loss behaves like a FEM assembly; this is plausible but unproved. No new physical entities are introduced.

free parameters (5)
  • Quadrature points per element q = 3 to 15 depending on case and test order
    Selected by hand based on Fig. 4 saturation analysis; affects the computed loss and accuracy.
  • Mesh element count Nel = 1 to 1000 depending on case
    Chosen per experiment; convergence plots sweep this range.
  • Penalty weights τR and τB = Not reported (adaptively updated)
    Eq. (4) and Section 4; update schedule described only qualitatively, affects all reported losses.
  • Network depth ℓ and width N = 2-4 layers, 20 neurons
    Table 1; chosen per case without a stated selection criterion.
  • Activation function = tanh or sin per case
    Table 1; chosen per case without a stated selection criterion.
assumptions (5)
  • standard math Integration by parts produces the flux terms in Eq. (12) exactly as in standard FEM weak forms.
    Used throughout Section 3; standard calculus identity.
  • domain assumption The neural network can represent the solution well enough for the weak-form residual to be minimized to machine precision.
    Universal approximation is cited (Refs. [5,6]) but no approximation or training-convergence guarantee is provided for these problems.
  • ad hoc to paper Minimizing the sum of squared per-element residuals in Eq. (12) is equivalent to satisfying the weak form and does not introduce spurious solutions or harmful overconstraints.
    This is the core loss design; the paper provides no equivalence or consistency proof.
  • ad hoc to paper Predefined convolution filters remain valid on nonuniform adaptive meshes even though the Jacobian Jx appears inside Filters2 in Eq. (12).
    Section 4.3.4 uses adaptive meshes; the paper does not explain how a single fixed filter accommodates element-dependent Jx.
  • standard math Gauss quadrature with q >= (p+1)/2 points exactly integrates polynomial terms of order p.
    Invoked in Section 4.1 to select quadrature counts.

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Cite this review

Pith. "Pith review of The Finite Element Neural Network Method: One Dimensional Study." pith.science (2026). https://pith.science/paper/C6EWHD44

@misc{pith2026250112508,
  author       = {Pith},
  title        = {Pith review of: The Finite Element Neural Network Method: One Dimensional Study},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C6EWHD44}},
  note         = {Machine review of arXiv:2501.12508}
}
read the original abstract

The potential of neural networks (NN) in engineering is rooted in their capacity to understand intricate patterns and complex systems, leveraging their universal nonlinear approximation capabilities and high expressivity. Meanwhile, conventional numerical methods, backed by years of meticulous refinement, continue to be the standard for accuracy and dependability. Bridging these paradigms, this research introduces the finite element neural network method (FENNM) within the framework of the Petrov-Galerkin method using convolution operations to approximate the weighted residual of the differential equations. The NN generates the global trial solution, while the test functions belong to the Lagrange test function space. FENNM introduces several key advantages. Notably, the weak-form of the differential equations introduces flux terms that contribute information to the loss function compared to VPINN, hp-VPINN, and cv-PINN. This enables the integration of forcing terms and natural boundary conditions into the loss function similar to conventional finite element method (FEM) solvers, facilitating its optimization, and extending its applicability to more complex problems, which will ease industrial adoption. This study will elaborate on the derivation of FENNM, highlighting its similarities with FEM. Additionally, it will provide insights into optimal utilization strategies and user guidelines to ensure cost-efficiency. Finally, the study illustrates the robustness and accuracy of FENNM by presenting multiple numerical case studies and applying adaptive mesh refinement techniques.

Figures

Figures reproduced from arXiv: 2501.12508 by the authors.

Figure 1
Figure 1. Schematic of one-dimensional FENNM. The NN outputs formulate the fluxes, differential equation [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Construction of the convolution filters for linear Lagrange test functions with [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. An illustration of a uniform mesh comprising five elements over the domain [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: The average absolute error E(x) = 1/N PN i=1 |UNN (x) − U(x)| in FENNM when increasing the number of the quadrature points for different test function orders. order of test functions [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: The rate of convergence of FENNM using linear, quadratic, and cubic test functions at [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: Solution of a cantilevered beam subjected to an intermediate static force using FENNM: (a) FENNM [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: The solution of damped and undamped nonlinear pendulum using FENNM: (a) FENNM approxi [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 8
Figure 8. Figure 8: The solution of transport equation using FENNM: (a) comparison between the solutions of FEM and [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: The solution of one-dimensional Poisson’s equation using FENNM: (a and d) the solutions of FENNM [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]
Figure 10
Figure 10. Figure 10: The error distribution across various quadrature points using linear, quadratic, cubic, and quartic [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]
Figure 11
Figure 11. Figure 11: CR comparison between FENNM and FEM using linear, and quadratic test functions at [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]
Figure 12
Figure 12. Figure 12: The solution of one-dimensional Poisson’s equation using FENNM: (a and d) the solutions of [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]
Figure 13
Figure 13. Figure 13: The solution of one-dimensional Poisson’s equation using FENNM: (a) the FENNM approximation [PITH_FULL_IMAGE:figures/full_fig_p024_13.png]

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