Pith. sign in

REVIEW 3 major objections 4 minor 33 references

Neural Network Element Method for Partial Differential Equations

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper claims that finite element 'envelop' functions multiplied by local neural networks form a trial space that respects boundary conditions exactly and inherits the neural networks' high approximation accuracy for elliptic PDEs.

desk verdict A plausible NN-FEM hybrid with striking numerics, but its central error estimate is proved for a space that Algorithm 1 does not solve; the gap is real and needs fixing. read the letter →

arxiv 2504.16862 v1 pith:AUTS3DM2 submitted 2025-04-23 math.NA cs.NA

Yifan Wang , Zhongshuo Lin , Hehu Xie This is my paper · ORCID
classification math.NAcs.NA MSC 68T0765L7065N2565B99
keywords neuralnetworkelementfinitemeshmachinelearningpartialdifferentialequationboundaryvalueconditioncomplexgeometrysingularitypartitionofunity
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 proposes a hybrid way to solve elliptic boundary-value problems: take a finite element mesh, and on each local patch build a basis function by multiplying a piecewise-polynomial 'envelop' (a barycentric-coordinate shape function for a vertex, edge, or triangle) by a small neural network supported on that patch. The envelops vanish on the domain boundary and match across element interfaces, so the whole neural network element space lies in $H^1_0(\Omega)$ and satisfies Dirichlet conditions exactly, even on complicated polygonal domains. The paper proves a partition-of-unity error estimate: if each patch's neural network approximates the true solution well in $L^2$ and $H^1$, then the global approximation does too. Combined with the Galerkin optimality property, this says the numerical solution's accuracy is governed by the neural networks' approximation power rather than by the mesh size. A Laplace test problem shows the new method reaching errors orders of magnitude smaller than standard finite elements on the same mesh.

What carries the argument

The central object is the NN element space $$$V^{{\rm NN}}$_h(T_h,\$\theta$)=\operatorname{span}\{\varphi_Z(x)\phi_Z(x,\$\theta$),\ \varphi_E(x)\phi_E(x,\$\theta$),\ \varphi_K(x)\phi_K(x,\$\theta$)\},$$ where the $\varphi$'s are finite element envelop functions built from barycentric coordinates and the $\phi$'s are local neural networks (two hidden layers, 16 neurons, sine activation in the experiments). The argument-carrying identity is the partition of unity $\psi_i(x)=\varphi_i(x)/\sum_j\varphi_j(x)$, satisfying $\sum_i\psi_i\equiv 1$ on $\Omega$; it lets the global error $\sum_i\psi_i(u-v_i)$ be split into local pieces and each piece be bounded by the local network's approximation errors, giving Theorem 2.1. The second load-bearing mechanism is the Galerkin optimality $\|u-u_h\|_a=\inf_{v_h\in V^{\rm NN}_h}\|u-v_h\|_a$, which turns the space's approximation bound into a guarantee about the actual computed solution. Together these two mechanisms are what let the paper claim mesh geometry and neural-network accuracy are decoupled.

What would settle it

Run Algorithm 1 on an L-shaped domain with a known singular solution, using the stated architecture and loss, and compare the $H^1$ error against finite elements at the same mesh sizes. If the error does not drop substantially below the FEM rate or does not approach the accuracy of the local networks trained independently on each patch, then the claim that the NN element space inherits neural-network accuracy fails.

Watch

Extended reading notes

Core claim

On a regular finite element mesh, define local envelop functions from barycentric coordinates: $\varphi_Z$ for a vertex is the linear hat function on its patch, $\varphi_E=\lambda_2\lambda_3$ on the two triangles sharing an edge, and $\varphi_K=\lambda_1\lambda_2\lambda_3$ on a triangle. Multiplying each envelop by its own neural network $\phi_i(x,\theta)$ gives the NN element basis, and the span $V^{\rm NN}_h(T_h,\theta)$ is a conforming subspace of $H^1_0(\Omega)$ that already encodes the homogeneous Dirichlet condition. The paper's main theorem normalizes these envelops into a partition of unity $\psi_i=\varphi_i/\sum_j\varphi_j$ and shows in Theorem 2.1 that the global $L^2$ and $H^1$ approximation errors are bounded by the sums of the local networks' $L^2$ and $H^1$ errors on each patch. Since the Galerkin solution is optimal in the energy norm, the numerical solution inherits exactly that approximation quality. In the numerical section, the P2 and P3 NN-element variants on the unit-square Laplace problem produce $e_{H^1}$ and $e_{L^2}$ values roughly one to three orders of magnitude below the corresponding finite element methods at matching mesh sizes.

Load-bearing premise

The accuracy claim rests on the assumption that the small neural network on each mesh patch can be trained to approximate the exact solution closely, in both the function value and its gradient; the paper does not prove that the chosen training procedure finds such a network.

Editorial extensions

If this is right

  • Boundary conditions are built into the basis, so there is no need to tune a boundary-loss weight or sample the boundary separately for homogeneous Dirichlet problems.
  • Accuracy no longer depends primarily on mesh refinement: on a fixed mesh, updating the local neural networks improves the trial space, giving an adaptive effect similar to moving-mesh methods without moving the mesh.
  • Complex polygonal or curved geometries, where global neural-network trial functions struggle, can be handled by the finite element mesh while the networks handle local solution detail.
  • The optimality property means the energy error of the NN element solution is no worse than the best approximation in the whole NN element space, so any improvement in local network training translates directly into a better numerical solution.
  • Non-homogeneous Dirichlet data can be treated by first solving a boundary NN element approximation and then lifting it into the interior solve, as described in the paper.

Reading between the lines

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

  • Because all basis functions are locally supported on mesh patches, the method is naturally parallelizable: each patch's network can be trained independently and the stiffness matrix assembled element-wise, which could make it scale to large three-dimensional problems more easily than a single global network.
  • The theorem says nothing about which architecture or training procedure realizes the assumed local approximation; a direct test would be to train the local networks on a single patch and compare their achieved $H^1$ error with the $\varepsilon_2(i)$ required by the bound. If they fall short, the global guarantee degrades in a predictable way.
  • The same partition-of-unity template would work with other envelop families (quadrilateral, tetrahedral, or higher-order finite elements) and could even replace neural networks with other local approximators, so the core idea is broader than the specific implementation tested.
  • The reported gains are for a smooth solution on a square; the method's real advantage for singularities, emphasized in the introduction, still needs a demonstration on a domain with a re-entrant corner and a known singular solution.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The manuscript introduces a neural network element method (NNEM) for second-order elliptic boundary value problems. The trial space is spanned by products phi_i(x)*phi_i(x,theta) of finite element envelop functions (vertex, edge, and triangle patches) with local neural networks. The authors claim this space lies in H^1_0(Omega), satisfies homogeneous Dirichlet conditions directly, and, by a partition-of-unity error estimate (Theorem 2.1) combined with the Galerkin optimality property (3.3), approximates the PDE solution with nearly the accuracy of the local networks. Algorithm 1 alternates assembling and solving a Galerkin linear system with gradient-descent updates of the network parameters. Numerical experiments on the unit square with the manufactured solution u = sin(pi x) sin(pi y) report small L2 and H1 errors for P2 and P3 envelop functions.

Significance. If the central claim were fully established, NNEM would be a valuable bridge between FEM and neural-network methods: it offers a conforming trial space on polygonal domains, exact enforcement of Dirichlet conditions, and local NN enrichment. The paper's Theorem 2.1 is a correct-looking conditional partition-of-unity estimate, and the two-step coefficient/network update in Algorithm 1 is a sensible way to avoid a purely Monte-Carlo loss. The numerical tables, for the one smooth example, report excellent accuracy. However, the logical connection between the analyzed space and the implemented space is missing, and the local approximation assumptions are not verified, so the advertised error control is not presently demonstrated.

major comments (3)
  1. [§2.3 (Theorem 2.1) vs §3 (Algorithm 1)] The trial space analyzed in Theorem 2.1 is V = sum_i psi_i V_i with psi_i = phi_i / S and S = sum_j phi_j, whereas Algorithm 1 solves in V_NN^h = span{phi_i(x) phi_i(x,theta)}. Since phi_i = psi_i S, an element of the implemented space can be written as sum_i psi_i (S c_i phi_i). The theorem applies only if S phi_i belongs to the local NN space V_i for each i; the paper neither states nor proves this closure property, and it is generally false for the reported two-hidden-layer, width-16, sine-activated MLPs. Consequently, inequalities (2.20)-(2.21) and the infimum transfer (3.3) do not currently establish approximation for the space actually solved in Algorithm 1.
  2. [§2.3 vs Remark 3.1] The proof of Theorem 2.1 relies on the partition-of-unity identity sum_i psi_i = 1 on all of Omega, which holds only if the envelop functions associated with boundary vertices, edges, and triangles are all present in the sum. However, the numerical method implements homogeneous Dirichlet conditions by modifying the stiffness matrix and right-hand side as described in Remark 3.1, effectively zeroing out the boundary basis functions. For the remaining trial functions, sum_i psi_i is strictly less than 1 in a boundary layer, so the decomposition u = sum_i psi_i u used in the proof fails exactly where the boundary condition is imposed. This is a second, independent obstruction to applying Theorem 2.1 to the implemented method.
  3. [§2.3 and §4] Theorem 2.1 is conditional on the existence of local NNs v_i satisfying the approximation estimates (2.18)-(2.19) with small eps_1(i) and eps_2(i). The manuscript does not verify these assumptions for the concrete architecture (two hidden layers, width 16, sine activation) nor prove that the Adam/L-BFGS training in Algorithm 1 reaches such v_i. The single numerical example is a smooth manufactured solution on the unit square, so it does not provide evidence for the singular or complex-geometry cases that motivate the method. The central high-accuracy claim therefore rests on an untested trainability premise.
minor comments (4)
  1. [§2.2, after Eq. (2.6)] In the sentence defining the local NN functions, 'omega_K(x)' should read 'phi_K(x)'.
  2. [§2.3, Eq. (2.28)] The constant C_x in the penultimate line of the proof of Theorem 2.1 is undefined; it should be the constant C_infinity from (2.16).
  3. [§4] The text contains typographical errors: 'NVIDA' should be 'NVIDIA' and 'stoping' should be 'stopping'.
  4. [Declaration of Competing Interest] The declaration of competing interest refers to a different manuscript ('Solving High-dimensional Partial Differential Equations Using Tensor Neural Network and A Posteriori Error Estimators') and must be corrected to match the present paper.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the error analysis is a conditional partition-of-unity bound, and the numerical validation uses a manufactured solution; self-citations are contextual rather than load-bearing.

full rationale

The paper's central estimate, Theorem 2.1, is a standard partition-of-unity argument: it assumes local neural-network approximation estimates (2.18)-(2.19) on each patch and derives global L2 and H1 bounds (2.20)-(2.21). The global conclusion is not used as an input; it is deduced from smaller, patchwise assumptions, so the derivation is not circular. The numerical experiments compare against the exact manufactured solution u(x,y)=sin(pi x)sin(pi y), so the reported errors are not a fitted parameter renamed as a prediction. The same authors' earlier works (e.g., [18], [30]) are cited for motivation, training heuristics, and tensor-network background, but the construction of V_NN^h and the partition-of-unity analysis do not reduce to those citations; the PU framework is explicitly attributed to Melenk and Babuska [20]. One non-circular concern is that the space analyzed in Theorem 2.1, V = sum_i psi_i V_i, is not shown to coincide with the space actually implemented in Algorithm 1, span{phi_i(x) phi_i(x,theta)}: because phi_i = psi_i * S with S = sum_j phi_j, the theorem would require the fixed multiplier S to be absorbable into each local NN class, which is not stated or proved. This is a correctness or proof gap rather than an input-output equivalence, so it does not raise the circularity score. The competing-interest declaration appears to be pasted from a different manuscript, but that is an editorial defect, not circularity.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The method's global error bound is conditional on local NN approximation and on a shape-regular mesh. The actual accuracy achieved in the numerical test additionally depends on hand-chosen architecture, optimizer, epochs, and quadrature. No new physical entities are introduced.

free parameters (6)
  • Hidden layers per envelop NN = 2
    Chosen by hand in Section 4; the error analysis does not constrain the architecture.
  • Neurons per hidden layer = 16
    Chosen by hand in Section 4.
  • Activation function = sine
    Chosen by hand in Section 4; all test results depend on this choice.
  • Adam learning rate = 0.0003
    Chosen by hand in Section 4; no sensitivity study is reported.
  • Training epochs = 50,000
    Stopping criterion chosen by hand in Section 4.
  • Gauss quadrature points per triangle = 36
    Chosen by hand in Section 4; integration error is not analyzed.
assumptions (4)
  • domain assumption The finite element mesh is regular
    Stated in Section 2.1; used for the Lipschitz partition-of-unity properties in Lemma 2.1.
  • domain assumption Local NN trial spaces approximate u well on each patch (assumptions (2.18)-(2.19))
    This is the key premise of Theorem 2.1; it is asserted from universal approximation, not proved for the specific architecture or found by training.
  • domain assumption Gradient descent (Adam with L-BFGS) optimizes the nonconvex energy loss sufficiently well
    Algorithm 1 Step 5 assumes the update improves the NN element space; no convergence proof is supplied.
  • domain assumption Boundary vertex and edge basis functions are excluded when homogeneous Dirichlet conditions are imposed
    The statement V_NN^h subset H^1_0(Omega) after (2.7) requires this exclusion, but the general construction defines phi_Z for all vertices; only the L-shape example makes the exclusion explicit.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Neural Network Element Method for Partial Differential Equations." pith.science (2026). https://pith.science/paper/AUTS3DM2

@misc{pith2026250416862,
  author       = {Pith},
  title        = {Pith review of: Neural Network Element Method for Partial Differential Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AUTS3DM2}},
  note         = {Machine review of arXiv:2504.16862}
}
read the original abstract

In this paper, based on the combination of finite element mesh and neural network, a novel type of neural network element space and corresponding machine learning method are designed for solving partial differential equations. The application of finite element mesh makes the neural network element space satisfy the boundary value conditions directly on the complex geometric domains. The use of neural networks allows the accuracy of the approximate solution to reach the high level of neural network approximation even for the problems with singularities. We also provide the error analysis of the proposed method for the understanding. The proposed numerical method in this paper provides the way to enable neural network-based machine learning algorithms to solve a broader range of problems arising from engineering applications.

Figures

Figures reproduced from arXiv: 2504.16862 by the authors.

Figure 1
Figure 1. The finite element mesh Th Z [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. The patch ωZ for one node Z For any interior edge E ∈ Eh, the corresponding patch ωE includes two triangles denoted by K1 and K2, i.e. ωE = K1 ∪ K2 (see [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. The patch ωE for one edge E and the corresponding basis λ2λ3 K 3 1 2 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: The triangle K and the corresponding basis λ1λ2λ3 2.2 Neural network element basis This subsection is devoted to introducing the NN element trial space based on the envelop functions defined on the finite element mesh Th and the local NN functions. In order to express …
Figure 5
Figure 5. Figure 5: The mesh and corresponding vertices, edges and triangles for the L shape domain [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: , the envelop functions on the triangle K can be denoted and ordered as follows φ1(x) = φZ1 (x), φ2(x) = φZ2 (x), φ3(x) = φZ3 (x), φ4(x) = φE1 (x), φ5(x) = φE2 (x), φ6(x) = φE3 (x), φ7(x) = φK(x). Based on the local support φi(x) and ψi(x), the associated partition fun…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

33 extracted references · 21 canonical work pages

  1. [1]

    Brenner and R

    S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods, Vol. 15, Springer Science & Business Media, 2007

    work page 2007

  2. [2]

    J. Chen, X. Chi, W. E and Z. Yang, Bridging traditional and machine learning-based algo- rithms for solving PDEs: the random feature method, J. Mach. Learn., 1(3), 268–298, 2022

    work page 2022

  3. [3]

    P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Vol. 40, SIAM, 2002

    work page 2002

  4. [4]

    S. Dong, Z. Li, Local extreme learning machines and domain decomposition for solving lin- ear and nonlinear partial differential equations, Comput. Methods Appl. Mech. Engrg., 387, 114129, 2021

    work page 2021

  5. [5]

    S. Dong, Z. Li, A modified batch intrinsic plasticity method for pretraining the random coef- ficients of extreme learning machines, J. Comput. Phys., 445, 110585, 2021

    work page 2021

  6. [6]

    W. E and B. Yu, The deep Ritz method: a deep-learning based numerical algorithm for solving variational problems, Commun. Math. Stat., 6, 1–12, 2018

    work page 2018

  7. [7]

    L. C. Evans, Partial Differential Equations (Second Edition), Graduate Studies in Mathemat- ics, Vol. 19, AMS, 2010

    work page 2010

  8. [8]

    Grisvard, Elliptic Problems in Nonsmooth Domains, Society for Industrial and Applied Mathematics, 2011

    P. Grisvard, Elliptic Problems in Nonsmooth Domains, Society for Industrial and Applied Mathematics, 2011

    work page 2011

Show all 33 references
  1. [9]

    J. Han, L. Zhang and W. E, Solving many-electron Schr¨ odinger equation using deep neural networks, J. Comput. Phys., 399, 108929, 2019. 16

    2019

  2. [10]

    Z. Hu, K. Shukla, G. E. Karniadakis and K. Kawaguchi, Tackling the curse of dimensionality with physics-informed neural networks, Neural Networks, 176, 106369, 2024

    2024

  3. [11]

    Huang, Q

    G. Huang, Q. Zhu, C. Siew, Extreme learning machine: theory and applications, Neurocom- puting, 70, 489–501, 2006

    2006

  4. [12]

    Huang and R

    W. Huang and R. D. Russell, Adaptive Moving Mesh Methods, Applied Mathematical Sciences, vol 174. Springer, New York, NY, 2010

    2010

  5. [13]

    D. P. Kingma and J. Ba, Adam: A method for stochastic optimization, arXiv:1412.6980, 2014; Published as a conference paper at ICLR 2015

    2014 arXiv

  6. [14]

    I. E. Lagaris, A. Likas and D. I. Fotiadis, Artificial neural networks for solving ordinary and partial differential equations, IEEE Trans. Neural Networks, 9(5), 987–1000, 1998

    1998

  7. [15]

    I. E. Lagaris, A. C. Likas and G. D. Papageorgiou, Neural-network methods for boundary value problems with irregular boundaries, IEEE Trans. Neural Networks, 11(5), 1041–1049, 2000

    2000

  8. [16]

    R. Li, T. Tang and P. Zhang, Moving mesh methods in multiple dimensions based on harmonic maps, J. Comput. Phys., 170(2), 562–588, 2001

    2001

  9. [17]

    Y. Li, F. Wang, Local randomized neural networks methods for interface problems, arXiv:2308.03087, 2023

    2023 arXiv

  10. [18]

    Z. Lin, Y. Wang and H. Xie, Adaptive neural network subspace method for solving partial differential equations with high accuracy, arXiv:2412.02586, 2024

    2024 arXiv

  11. [19]

    P. Liu, Z. Xu and Z. Sheng, Subspace method based on neural networks for solving the partial differential equation in weak form, arXiv:2405.08513, 2024

    2024 arXiv

  12. [20]

    J. M. Melenk and I. Babuˇ ska, The partition of unity finite element method: Basic theory and applications, Comput. Methods Appl. Mech. Engrg., 139, 289–314, 1996

    1996

  13. [21]

    W. F. Mitchell, A collection of 2D elliptic problems for testing adaptive grid refinement algo- rithms, Appl. Math. Comput., 220, 350–364, 2013

    2013

  14. [22]

    Raissi, P

    M. Raissi, P. Perdikaris and G. E. Karniadakis, Physics informed deep learning (part I): Data- driven solutions of nonlinear partial differential equations, arXiv:1711.10561, 2017

    2017 arXiv

  15. [23]

    Shang, F

    Y. Shang, F. Wang and J. Sun, Randomized neural network with Petrov-Galerkin methods for solving linear and nonlinear partial differential equations, Commun. Nonlinear Sci. and Numerical Simulation, 127, 107518, 2023

    2023

  16. [24]

    Sirignano and K

    J. Sirignano and K. Spiliopoulos, DGM: A deep learning algorithm for solving partial differ- ential equations, J. Comput. Phys., 375, 1339–1364, 2018

    2018

  17. [25]

    J. Sun, S. Dong and F. Wang, Local randomized neural networks with discontinuous Galerkin methods for partial differential equations, J. Comput. Appl. Math., 445, 115830, 2024. 17

    2024

  18. [26]

    T. Wang, Z. Hu, K. Kawaguchi, Z. Zhang and G. E. Karniadakis, Neural Networks, 185, 107165. 2025 (arXiv:2404.05615v1, 2024)

    2025 arXiv

  19. [27]

    T. Kao, J. Zhao and L. Zhang, pETNNs: partial evolutionary tensor neural networks for solving time-dependent partial differential equations, arXiv:2403.06084v1, 2024

    2024

  20. [28]

    Y. Wang, P. Jin and H. Xie, Tensor neural network and its numerical integration, J. Comput. Math., 42, 1714–1742, 2024 (arXiv:2207.02754, 2022)

    2024 arXiv

  21. [29]

    Y. Wang, Y. Liao and H. Xie, Solving Schr¨ odinger equation using tensor neural network, arXiv:2209.12572, 2022

    2022 arXiv

  22. [30]

    Y. Wang, Z. Lin, Y. Liao, H. Liu and H. Xie, Solving high dimensional partial differential equations using tensor neural network and a posteriori error estimators, J. Sci. Comput., 101(67), 2024

    2024

  23. [31]

    Wang and H

    Y. Wang and H. Xie, Computing multi-eigenpairs of high-dimensional eigenvalue problems using tensor neural networks, J. Comput. Phys., 506, 112928, 2024 (arXiv:2305.12656, 2023)

    2024 arXiv

  24. [32]

    Xu and Z

    Z. Xu and Z. Sheng, Subspace method based on neural networks for solving the partial differ- ential equation, arXiv:2404.08223, 2024

    2024 arXiv

  25. [33]

    Y. Zang, G. Bao, X. Ye and H. Zhou, Weak adversarial networks for high-dimensional partial differential equations, J. Comput. Phys., 411, 109409, 2020. 18 Declaration of Competing Interest We declare that we have no financial and personal relationships with other people or org...

    2020

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.