FE-PINNs: finite-element-based physics-informed neural networks for surrogate modeling

Pranav Sunil; Ryan B. Sills

REVIEW 2 major objections 7 minor 27 references

FE-PINNs: finite-element-based physics-informed neural networks for surrogate modeling

T0 review · 2 major / 7 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Stencil convolution lets physics-informed neural networks trained on finite-element meshes generalize to unseen geometries, with testing error decreasing as training geometries grow.

desk verdict A genuinely new stencil-convolution operator for FE-mesh CNNs, with a plausible generalization story that Fig. 8 weakens by only plotting selected best models. read the letter →

arxiv 2412.07126 v1 pith:HH7ABYLD submitted 2024-12-10 physics.comp-ph cond-mat.mtrl-sci

Pranav Sunil , Ryan B. Sills This is my paper · ORCID

classification physics.comp-phcond-mat.mtrl-sci

keywords physics-informedneuralnetworksfiniteelementmethodsurrogatemodelingstencilconvolutioninverseisoparametricmapconvolutionalunstructuredmesheslinearelasticity

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

This paper introduces FE-PINNs, a way to train physics-informed neural networks as surrogate models for boundary-value problems. Rather than requiring a uniform grid or precomputed solution data, the network is trained on the residual of the finite-element weak form, and a new 'stencil convolution' operation samples fields at stencil points through the inverse isoparametric map of the mesh. The authors show that networks trained on a small set of finite-element geometries predict displacement fields for similar but unseen geometries with reasonable accuracy, and that testing error systematically decreases as the number of training geometries grows. If this holds, FE-PINNs provide a route to surrogate models that inherit the flexibility of existing finite-element codes and need no labeled dataset.

What carries the argument

The load-bearing object is the stencil convolution tensor $S_{klmn}$, which stores the shape-function values that map a node's field values to the stencil points around it: $S_{klmn} = N^{n,e}(\xi_{ikl}(x_{ikl}))$ if node $n$ belongs to the element containing stencil point $(k,l)$, and zero otherwise. It is built once per mesh by locating the element for each stencil point via a spatial join and inverting the isoparametric map analytically, then reused as a sparse tensor contraction during training. This operation is what lets the CNN run on irregular meshes: the field is always stored on nodes, and the stencil weights read it through piecewise polynomial shape functions, so the network sees a grid-like receptive field without requiring a grid.

What would settle it

Inspect the FE-PINN code's element lookup at stencil points: if out-of-domain stencil points are silently skipped or zero-filled, then a test mesh with many nodes near a concave boundary should show a sharp jump in prediction error at those boundary nodes, contradicting the claimed mesh-insensitive generalization.

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Extended reading notes

Core claim

The central claim is that stencil convolution—defined by a stencil tensor of points around each node, with field values obtained by finite-element shape functions via the inverse isoparametric map—makes convolutional PINNs practical on arbitrary, unstructured finite-element meshes. With this operator, a CNN can be trained by minimizing the norm of the finite-element residual, so boundary conditions enter through the external force vector and need no soft-constraint penalty. The paper reports that networks trained on several geometries generalize to untrained geometries that are similar, and that increasing the number of training geometries systematically reduces testing loss while training loss stays flat. This is taken as evidence that the network learns the underlying operator rather than memorizing isolated solutions.

Load-bearing premise

The load-bearing premise is that stencil points that fall outside the finite-element domain near boundaries can be handled without corrupting the convolutional features, yet the paper never specifies what happens when the element search returns no element for such a point.

Editorial extensions

If this is right

A FE-PINN trained on a handful of geometries can replace a full finite-element solve for new, similar geometries, giving near-instant predictions with physics-based accuracy.
No labeled solution data is required: the physics loss is the finite-element residual, and boundary conditions enter automatically through the external force vector.
Testing error decreases systematically as the training set grows, indicating that the network learns a generalizable operator rather than memorizing training cases.
The same machinery is expected to extend to nonlinear and three-dimensional problems, and to variable boundary conditions, material properties, and body forces by adding input channels.

Reading between the lines

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

The paper leaves unspecified how stencil points that fall outside the domain are treated; if the element lookup returns nothing, boundary-node features could be distorted, which may explain why the 0-degree wedge block resists prediction from nearby geometries.
A testable extension would be to change element size after training; the authors note discretization invariance is plausible because inputs and outputs are functions, but it has not been demonstrated.
Non-uniform or multi-length-scale stencil arrangements, which the paper explicitly floats as possibilities, could outperform the single uniform spacing used here.
If the observed scaling with training-set size continues, training on hundreds of finite-element models should yield a high-fidelity operator surrogate; the authors mention this as a large-scale target but do not test it.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things to know. The stencil convolution is a real new operator: it lets a CNN run directly on finite-element meshes by evaluating fields at stencil points via the inverse isoparametric map, and the tensor formulation is clean and efficient. That is a contribution worth taking seriously. The second thing is that the headline claim—that testing errors systematically decrease as more training geometries are added—is probably true, but the figure meant to prove it, Fig. 8, is weaker than it looks because it only plots best-performing models.

The paper does a lot right. The physics loss is the FE residual, so BCs enter through the Galerkin form rather than a penalty term, which is a principled choice. The writing is clear, the method is generalizable in principle, and code plus trained weights are provided. The full results in Figs. 4–7 also support the generalization trend: most three-geometry combinations beat most single-geometry models, and the displacement fields look sensible.

Now the soft spots, in proportion. First, the stress-test note is correct about Fig. 8. The paper keeps only the best of three seeds, and for n=2 and n=3 it appears to select the best among all geometry combinations as well. The minimum over a larger pool of models tends to decrease even if no model actually generalizes better. The paper never reports the mean, median, or spread of losses across seeds or combinations, so the monotone line in Fig. 8 is not by itself evidence of systematic improvement. The broader figures rescue the trend, but the reporting should be fixed.

Second, there is a reproducibility gap in the stencil definition. Sec. 2.2.2 says stencil points are evaluated using the element containing each point, but it never says what happens when a stencil point falls outside the mesh. The GeoPandas sjoin would find no element; there must be a fallback (probably zero-padding) but it is undocumented. That matters for boundary nodes and for anyone trying to reimplement the method.

Third, there are no baselines. Comparing against PhyGeoNet, a graph-convolution PINN, or even a standard uniformly-gridded CNN on the same test problems would make the practical advantage of stencil convolution concrete. Only two families of 2D linear problems are tested, so the scope is honestly small.

Not a circularity problem: the test metric is the FE residual on held-out geometries, and no parameters are fit to the test set. The missing error bars and selection details are the real issues, not the evaluation protocol itself.

Who is this for? Researchers building mesh-adaptive PINN surrogates for solid mechanics and related fields. It deserves a serious referee; the core operator is novel and the paper is honest about limitations, but it needs a revision that reports distributions, documents the out-of-domain stencil handling, and adds at least one baseline comparison.

Referee Report

2 major / 7 minor

Summary. This paper introduces FE-PINNs, a physics-informed neural network framework in which the finite-element weak-form residual is used as the training loss and a custom 'stencil convolution' operator performs convolutional operations directly on unstructured FE meshes. The method is demonstrated on two families of 2D linear-elastic boundary-value problems (wedged blocks and blocks with holes), training on one, two, or three geometries and testing on unseen geometries. The authors report that testing errors decrease as the number of training geometries grows, and they position the method as a step toward general-purpose surrogate models that leverage existing FE codes.

Significance. If the central claims hold, the stencil-convolution idea is a meaningful contribution: it offers a mesh-adaptive CNN-style operator that works on arbitrary FE discretizations, avoids labeled data by using the FE residual as the physics loss, and enforces boundary conditions through the weak Galerkin form. The paper also ships code and trained networks in a public repository, which strengthens reproducibility. However, the quantitative evidence for the headline generalization claim is weakened by the exclusive use of best-of-three and best-of-combination selections without reporting any distribution, and the stencil-convolution construction is incompletely specified for stencil points outside the domain. Both issues are fixable with additional reporting and targeted experiments, so the work is worth considering after major revision.

major comments (2)

[Sec. 2.2.2, Eqs. (14)-(17)] The stencil-convolution construction is incomplete for stencil points that lie outside the FE domain. For any non-zero stencil spacing Delta, stencil points associated with nodes near the boundary will fall outside the union of elements; the element search via the GeoPandas sjoin described in Sec. 2.3 then returns no element, and the manuscript states no fallback rule. Since the stencil point values are formed by evaluating shape functions at the containing element, an undefined or inconsistent choice here would corrupt the convolutional features at boundary nodes and could explain the difficulty reported for the 0-degree wedge block in Sec. 4. Please specify the out-of-domain handling (e.g., zero-padding, clamping to the nearest element, or discarding such stencil points) and confirm that all reported results use the same rule.
[Sec. 3.2 and Fig. 8] The claim in the abstract and Sec. 4 that testing errors systematically decrease with the number of training geometries is based on losses from 'best performing FE-PINNs'. For the wedged-block family, the n=1, n=2, and n=3 minima are selected from 7, 21, and 35 training-set combinations, respectively, with an additional best-of-three-seeds selection per combination. The minimum over a growing pool of models can decrease even if no typical model improves, and the manuscript reports no mean, median, or spread across seeds or across combinations. Please report the distribution of testing losses (e.g., all combinations, or at least median and range) or otherwise demonstrate that the systematic decrease holds beyond the best-case selection.

minor comments (7)

[Sec. 2.3] The word 'implemeneted' should be 'implemented', and 'ansjoin' should be 'an sjoin' (or 'a spatial join').
[Sec. 2.4] The phrase 'the training geometery' should be 'the training geometry'.
[Eq. (15)] The bias term is written as b_r \bar\delta_{mm}, which is not a standard scalar bias; it should presumably be simply b_r. Please correct the notation.
[Eq. (14)] The notation \bar\delta_{kk} = \delta_{kk}/2 is unusual and appears to be a typo; if a Kronecker delta is intended, use the standard symbol and define the coordinate shift accordingly.
[Fig. 8] The linear regression fits are drawn without reporting slopes, confidence intervals, or p-values; adding these (or replacing the fits with median trends) would make the claimed systematic decrease more quantitative.
[Sec. 3.2.1, Fig. 4(b)] The phrase 'lowest average loss' is not defined; specify whether the average is over the test geometries and over the three seeds.
[Sec. 4] The phrase 'in principal' should be 'in principle'.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the held-out test metric is the FE residual, an external physics benchmark, and no tested geometry or test loss enters training; the best-of-selection summary in Fig. 8 is a statistical caveat, not a definitional reduction.

full rationale

The paper's derivation chain is self-contained and does not reduce a prediction to its inputs by construction. The physics loss in Eq. (12), L(d̃)=||R(d̃)||, is computed from the weak-form FE residual; for testing, R is evaluated on unseen geometries, so the reported errors are external measures of PDE satisfaction, not quantities fitted during training. The stencil convolution tensor S_klmn in Eq. (17) is precomputed from the mesh and isoparametric shape functions, but the NN weights are optimized against the residual loss; no test-geometry loss or test-solution label is used to set weights. The paper contains no load-bearing self-citations, uniqueness imports, or ansatz-by-citation: the cited works are contextual (e.g., PhyGeoNet, graph Galerkin networks) and the stencil operation is defined directly from FE concepts. The principal weakness is statistical, not circular: Sec. 2.3 reports only the best of three random seeds, and Fig. 8 plots 'best performing FE-PINNs' across training scenarios, so the decreasing testing-loss trend may be partly an artifact of selecting minima over a growing set of candidate models. Although this warrants a caveat on the generalization claim, it is a model-selection/validation issue rather than a circular derivation, and per the hard rules it does not raise the circularity score beyond minor.

Assumptions & free parameters 3 free parameters · 4 assumptions · 1 invented entities

The FE-PINN method rests on the FE formulation as the source of both the training signal and the test metric. The network learns to minimize the FE residual on training meshes; evaluation on unseen meshes uses the same residual. No physical constants are fit beyond the stiffness matrices. The main ad hoc choices are the stencil geometry (spacing and kernel size) and the unstated rule for out-of-domain stencil points.

free parameters (3)

Stencil spacing Delta/H = 0.025
Chosen by sweeping Delta and picking the value minimizing training loss (Fig. 3b); this is a hyperparameter, not fit to test data.
Kernel size w = 9
Fixed architectural choice; reported without sensitivity study across w.
Learning rate and training epochs = 0.001, 1000
Standard Adam settings; epochs found to yield convergence. These are training hyperparameters, not physics parameters.

assumptions (4)

domain assumption The FE discretization with bilinear quadrilaterals and 2x2 quadrature yields a sufficiently accurate solution for the test problems.
The FE-PINN is trained to reproduce the FE solution; any FE discretization error is inherited, as the paper acknowledges in Discussion ('any numerical error from the FE solution will be baked into the FE-PINN').
domain assumption The inverse isoparametric map is analytic and correctly implemented for all meshes used.
The construction of the stencil convolution tensor S_klmn relies on evaluating shape functions at stencil points via the inverse map; the paper cites Hua [24] but gives no implementation verification.
ad hoc to paper Stencil points outside the FE domain are handled benignly (not specified).
Sec. 2.2.2 states each stencil point is associated with an element containing it, but does not state the rule when no such element exists; the results imply some default (zero contribution) is used.
standard math The FE residual R(d_tilde) = K d_tilde - Fe is a valid training loss that enforces both equilibrium and essential BCs.
Standard FE residual; the paper correctly argues BCs enter through Fe and are therefore weakly enforced in the loss.

invented entities (1)

Stencil convolution operator (stencil tensor s_ikl and stencil convolution tensor S_klmn) independent evidence
purpose: Enables convolutional NN operations on unstructured FE meshes by sampling fields at stencil points via the FE shape functions.
The method is validated on the paper's numerical experiments and its implementation is provided in the public repository; it is a computational operator, not a physical entity.

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Cite this review

Pith. "Pith review of FE-PINNs: finite-element-based physics-informed neural networks for surrogate modeling." pith.science (2026). https://pith.science/paper/HH7ABYLD

@misc{pith2026241207126,
  author       = {Pith},
  title        = {Pith review of: FE-PINNs: finite-element-based physics-informed neural networks for surrogate modeling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HH7ABYLD}},
  note         = {Machine review of arXiv:2412.07126}
}

read the original abstract

We present a method whereby the finite element method is used to train physics-informed neural networks that are suitable for surrogate modeling. The method is based on a custom convolutional operation called stencil convolution which leverages the inverse isoparametric map of the finite element method. We demonstrate the performance of the method in several training and testing scenarios with linear boundary-value problems of varying geometries. The resulting neural networks show reasonable accuracy when tested on unseen geometries that are similar to those used for training. Furthermore, when the number of training geometries is increased the testing errors systematically decrease, demonstrating that the neural networks learn how to generalize as the training set becomes larger. Further extending the method to allow for variable boundary conditions, properties, and body forces will lead to a general-purpose surrogate modeling framework that can leverage existing finite element codes for training.

Figures

Figures reproduced from arXiv: 2412.07126 by the authors.

**Figure 2.** (a) NN architecture used in this work. (b) FE geometries used for NN training and testing. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Training convergence for three different geometries. (a) Training loss versus training epoch number. (b) [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: (a) Losses for all “wedged block” geometries using FE-PINNs trained on different single geometries. (b) [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: (a) Losses for all “block with hole” geometries using FE-PINNs trained on different single geometries. (b) [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: (a,b) Losses for all “wedged block” geometries using FE-PINNs trained on different combinations of three [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

**Figure 7.** Figure 7: (a) Losses for all “block with hole” geometries using FE-PINNs trained on different combinations of three [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

**Figure 8.** Figure 8: Training and testing losses from best performing FE-PINNs as a function of number of training geometries [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

R. K. Tripathy, I. Bilionis, Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification, Journal of Computational Physics 375 (2018) 565–588.doi:10.1016/j.jcp.2018. 08.036

doi:10.1016/j.jcp.2018 2018

[2] [2]

Kovachki, Z

N. Kovachki, Z. Li, B. Liu, K. Azizzadenesheli, K. Bhattacharya, A. Stuart, Neural Operator: Learning Maps Between Function Spaces With Applications to PDEs, Journal of Machine Learning Research 24 (2023) 1–97

work page 2023

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Lagaris, A

I. Lagaris, A. Likas, D. Fotiadis, Artificial neural networks for solving ordinary and partial differential equations, IEEE Transactions on Neural Networks 9 (5) (Sept./1998) 987–1000. doi:10.1109/72.712178

doi:10.1109/72.712178 1998

[4] [4]

Raissi, P

M. Raissi, P. Perdikaris, G. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Computational Physics 378 (2019) 686–707. doi:10.1016/j.jcp.2018.10.045

doi:10.1016/j.jcp.2018.10.045 2019

[5] [5]

Sirignano, K

J. Sirignano, K. Spiliopoulos, DGM: A deep learning algorithm for solving partial differential equations, Journal of Computational Physics 375 (2018) 1339–1364. doi:10.1016/j.jcp.2018.08.029

doi:10.1016/j.jcp.2018.08.029 2018

[6] [6]

Haghighat, M

E. Haghighat, M. Raissi, A. Moure, H. Gomez, R. Juanes, A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics, Computer Methods in Applied Mechanics and Engineering 379 (2021) 113741. doi:10.1016/j.cma.2021.113741

arXiv 2021

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C. Rao, H. Sun, Y . Liu, Physics-Informed Deep Learning for Computational Elastodynamics without Labeled Data, Journal of Engineering Mechanics 147 (8) (2021) 04021043. doi:10.1061/(ASCE)EM.1943-7889. 0001947. 16 A PREPRINT - D ECEMBER 11, 2024

work page doi:10.1061/(asce)em.1943-7889 2021

[8] [8]

S. Cai, Z. Mao, Z. Wang, M. Yin, G. E. Karniadakis, Physics-informed neural networks (PINNs) for fluid me- chanics: A review, Acta Mechanica Sinica 37 (12) (2021) 1727–1738. doi:10.1007/s10409-021-01148-1

doi:10.1007/s10409-021-01148-1 2021