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REVIEW 5 major objections 5 minor 33 references

Phi-FEM-FNO: a new approach to train a Neural Operator as a fast PDE solver for variable geometries

T0 review · 5 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that a Fourier Neural Operator fed with a level-set description of the domain, and trained on φ-FEM solutions, predicts PDE solutions on variable geometries with accuracy close to finite elements and speeds 100 to 1000…

desk verdict A solid, well-documented engineering combination of phi-FEM and FNO; the in-distribution speed/accuracy claims hold up, but out-of-distribution generalization is a real limitation that needs to be stated. read the letter →

arxiv 2502.10033 v1 pith:2FXAPSK3 submitted 2025-02-14 math.NA cs.NA

Michel Duprez , Vanessa Lleras , Alexei Lozinski , Vincent Vigon , Killian Vuillemot This is my paper
classification math.NAcs.NA MSC 65N3068T07
keywords neuraloperatorFourierphi-FEMlevel-setmethodvariablegeometryPoissonequationhyperelasticitysurrogatePDEsolver
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 is trying to establish that the level-set function defining a geometry can be used directly as an input channel for a Fourier Neural Operator (FNO), so a single trained network can solve a PDE on many different shapes without deforming the mesh or mapping the geometry to a rectangle. The network is trained offline on solutions produced by φ-FEM, an immersed-boundary finite element method that lives on the same Cartesian grid the FNO needs for its fast Fourier transforms. Across three test cases, the trained operator reaches accuracy comparable to finite element solvers while being about 100 times faster for Poisson problems and about 1000 times faster for a hyperelastic plate with holes. The payoff is near-instantaneous prediction for families of geometries after a one-time training phase.

What carries the argument

The central object is the level-set function φ, with Ω = {φ < 0}, which is simultaneously the input φ-FEM uses to represent the geometry and an image-like channel fed into the FNO. φ-FEM solves the PDE on a fixed Cartesian mesh cut by the boundary, representing the solution as u = φ w + g so the Dirichlet boundary condition is enforced exactly, and that same Cartesian grid is exactly what the FNO needs for its fast Fourier transforms. The FNO approximates the ground-truth operator G† with four Fourier layers, truncating high frequencies, and the prediction w_θ is converted to a physical solution by u_θ = φ_h w_θ + g_h. The loss is an H1-type error computed only on pixels belonging to the computational domain, which keeps the network focused on the region where the solution is meaningful.

What would settle it

Train the operator only on the ellipse family, then evaluate it on shapes from a different family, such as polygons or domains with reentrant corners; if the relative L2 error jumps far above the finite-element reference error, the claim that geometry can be treated as just another input channel fails. A milder check is to extend the paper's own Fig. 13-style analysis: if error keeps growing with Hausdorff distance to the closest training shape and does not plateau, the method is extrapolating rather than generalizing.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the geometry, encoded as a level-set function φ, can be treated as an ordinary input channel of an FNO alongside the forcing term and boundary data. The operator learned is G†: (f_h, φ_h, g_h) ↦ w_h, and the physical solution is recovered as u = φ_h w_h + g_h, so the Dirichlet condition is enforced by construction rather than learned. In the Poisson experiments on ellipses and on Gaussian-sum shapes, the resulting φ-FEM-FNO matches the relative L2 accuracy of standard FEM and φ-FEM, is about twice as accurate as an FNO trained on standard FEM solutions interpolated onto the same grid, and about ten times more accurate than Geo-FNO, while being roughly 100 times faster than the FEM solvers. For the five-hole hyperelastic plate, the paper introduces a new φ-FEM scheme that weakly imposes homogeneous Neumann conditions on the holes, and the trained operator produces displacements within a factor of about 10 of fine FEM accuracy while computing about 1000 times faster.

Load-bearing premise

The claimed accuracy is only demonstrated for geometries sampled from the same parametric families as the training data, so a genuinely new shape family could fall outside what the learned operator can handle.

Editorial extensions

If this is right

  • A user who needs solutions for new geometries from a trained family can replace an expensive finite element solve with one forward pass of the network, which is the source of the reported 100- to 1000-fold speedup.
  • Because φ-FEM and the FNO share the same Cartesian grid, the pipeline avoids the interpolation error that appears when standard FEM solutions on body-fitted meshes are transferred to a grid for the neural operator.
  • The same combination extends to nonlinear problems: the validated φ-FEM scheme for hyperelastic materials shows that level-set-based training data can carry a neural operator into nonlinear elasticity.
  • The number of FNO parameters is independent of the grid resolution, so a trained operator can in principle be evaluated at different resolutions without retraining.
  • In the paper's comparisons, the level-set-as-channel design is more accurate per unit of computation time than Geo-FNO and than a U-Net trained on the same φ-FEM data, suggesting the FNO is a better fit for this input representation.

Reading between the lines

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

  • The paper leaves implicit that its 'variable geometries' claim is an interpolation claim: all training, validation, and test shapes come from the same parametric families, and its own Fig. 13 shows error increasing with Hausdorff distance to the closest training shape, so extrapolation to genuinely new shape families is unvalidated.
  • The idea of using the level set as a data channel is not tied to FNO in principle; any neural architecture that consumes regular grids could be trained on φ-FEM data, so the level-set input representation may transfer to other operators where Fourier convolutions are not desirable.
  • For real-world applications such as patient-specific biomechanics, the level-set channel would likely come from medical image segmentation, and the practical question is whether a network trained on synthetic hole configurations retains accuracy on anatomical shapes; that is a testable distribution-shift question.
  • The paper's closing suggestion to train on Gaussian forces and sum predictions for a complex force is compositional, but it implicitly assumes the PDE solution depends on the force in a way that respects superposition, which holds for the linear Poisson problem but not for the hyperelastic one.
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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 / 5 minor

Summary. This paper proposes a machine-learning surrogate for PDEs on variable geometries, combining the immersed finite element method φ-FEM with the Fourier Neural Operator (FNO). The geometry, encoded by a level-set function, is supplied as an input channel to an FNO that is trained offline on φ-FEM solutions on Cartesian grids; the output is post-processed with the level-set function to enforce boundary conditions. The method is tested on three problems: the Poisson equation on random ellipses, the Poisson equation on more complex Gaussian-sum level-set domains, and a hyperelastic plate with five holes. In the last case, the authors also introduce a new φ-FEM formulation for hyperelastic materials with Neumann conditions on holes. The reported results, presented graphically, indicate that φ-FEM-FNO achieves accuracy close to FEM-based solvers while being much faster, and that it outperforms the considered baselines (Standard-FEM-FNO, Geo-FNO, U-Net) in the first test case. The paper provides code and datasets.

Significance. The potential significance is moderate-to-high for the computational mechanics and scientific machine learning communities: if the speed/accuracy trade-off holds, the method offers a practical real-time surrogate for parametric families of geometries without the need for body-fitted meshes or learned coordinate deformations. The paper's strengths include comparisons against several baselines including Geo-FNO and Standard-FEM-FNO, validation against fine finite element references via the E2 metric, and the introduction of a new φ-FEM scheme for hyperelasticity with a numerical convergence study. The code and data are made publicly available, which supports reproducibility. However, the central claim about 'variable geometries' is currently only demonstrated in-distribution, and the paper's own Fig. 13 suggests degradation for shapes far from the training set. This limits the significance until either out-of-distribution performance is assessed or the claims are appropriately qualified.

major comments (5)
  1. [Section 4.2 / Fig. 13 / Section 5] All reported accuracies are obtained on test shapes drawn from the same parametric distribution as the training set; the paper never evaluates a geometry outside that family. The paper's own Fig. 13 shows a roughly linear increase of the relative L2 error E2 with the Hausdorff distance from the test shape to the closest training shape, which indicates that accuracy degrades for shapes unlike those seen in training. Section 5 nonetheless concludes that the method handles 'big variations of geometries' without qualification. This is a load-bearing issue for the headline claim: either demonstrate out-of-distribution generalization on shapes not sampled from the training family, or explicitly restrict the claims to in-distribution interpolation and add this as a stated limitation.
  2. [Sections 4.1-4.3] The central quantitative claims — 'precision comparable to FEM-based methods', 'approximately 100 times faster', and 'at most 10 times more precise' for FEM in the hyperelastic case — are made without reporting actual numerical values in the text; they appear only in figures. Please provide explicit mean/median/standard deviation values for E2 and computation times for all methods, in tables or in the text, so the reader can verify and reproduce the trade-off claims.
  3. [Section 4.1, Geo-FNO comparison] The comparison with Geo-FNO may not be fully fair. Geo-FNO is given 2600 input points (average cell size 0.02) while φ-FEM-FNO uses a 64×64 Cartesian grid (cell size ≈0.022), and the text states that all ML methods were trained with the same hyperparameters; however, Geo-FNO has its own architectural choices (latent deformation, number of modes, etc.) that may require different tuning. Please report the Geo-FNO architecture and hyperparameters used, and discuss how the comparison was made fair.
  4. [Section 4.3.1, Eq. (11)] The new φ-FEM scheme for hyperelastic materials is introduced with only a numerical convergence study for a single configuration (Fig. 15). No analysis of well-posedness, stability, or consistency of the variational formulation (11) is provided, and no sensitivity study is given for the stabilization constants γp, γu, γdiv, σN. Since the paper claims this as a new scheme, please either add a formal analysis or substantially broaden the numerical validation (e.g., multiple loads, different material parameters, various hole configurations).
  5. [Section 4 (general)] All reported results are from a single training run for each method; no statistics over random initializations are given. Since the comparison between methods rests on accuracy and timing numbers, the absence of seed variability makes it impossible to assess whether the observed differences are within run-to-run noise. Please retrain the main compared methods with several seeds and report the spread (e.g., mean ± std) for the headline metrics.
minor comments (5)
  1. [Sections 2 and 4] The symbol E1 is used both for the H1-seminorm component of the loss in Eq. (4) and for the relative L2 error metric in Eq. (5). Please use different symbols to avoid confusion.
  2. [Eq. (6)] The projection operator Π_Ωref in Eq. (6) is not precisely defined; please specify how the L2 projection onto the reference finite element mesh is computed.
  3. [Section 4.3.2] The 'relative Hausdorff error' mentioned in the discussion of Fig. 17 (right) is never defined. Please define it, or replace it with a more standard error measure.
  4. [Appendix A.2] The remark on learning-rate calibration states that 'extensive testing was conducted' but does not report the range of learning rates or schedules tried; adding this information would improve reproducibility.
  5. [Whole paper] The text frequently refers to numerical results only through figures; for accessibility, consider adding tables with the key error and timing statistics, especially for the comparisons in Figs. 9, 12, and 17.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: φ-FEM-FNO is trained on φ-FEM data and evaluated against an independent fine FEM reference; self-citations are background convergence results, not load-bearing inputs.

full rationale

The central chain is: (i) choose φ-FEM as the data generator, (ii) define the ground-truth operator G† in Eq. (3) as the map from (fh, φh, gh) to the φ-FEM solution wh, (iii) train the FNO Gθ by minimizing the H1 loss in Eq. (4), and (iv) evaluate the result with E2 in Eq. (6) against fine standard FEM solutions on conforming reference meshes. Step (ii) is an explicit definition of the target mapping, not a hidden restatement of the FNO output; the network must still learn the map, and the loss compares network outputs with independently computed φ-FEM solutions. The construction uθ = φh wθ + gh (Figure 2) enforces the Dirichlet condition by construction, but this only constrains the representation and does not determine wθ or the PDE solution. The main accuracy claims in Figures 9, 12, and 17 rest on E2, an external fine-FEM reference, and on measured computation times, so no fitted parameter is renamed as a prediction. The self-citations to φ-FEM convergence results ([10], [7], [8], [9]) support the choice of the data-generation solver with published convergence theorems; they do not by themselves imply the FNO's accuracy, and the new hyperelastic φ-FEM scheme is independently validated in Figure 15 against a standard FEM reference. No uniqueness theorem is invoked, and no ansatz is adopted solely through a self-citation. The growing E2 error with Hausdorff distance in Figure 13 is a genuine generalization/scope limitation for out-of-distribution shapes and is not discussed as such in the paper; it is a correctness risk for the unqualified 'big variations of geometries' claim in Section 5, but it is not circular. Overall, the numerical claims are self-contained empirical benchmarks against external references.

Assumptions & free parameters 12 free parameters · 5 assumptions · 1 invented entities

The central claims rest on a set of hand-chosen architecture and PDE constants, on the convergence of phi-FEM from prior work, and on the assumption that training and test geometries come from the same distribution. The only new postulated objects are the auxiliary variables in the hyperelastic scheme.

free parameters (12)
  • FNO width nd = 20
    Set in Section 4; chosen by the authors without reported ablation.
  • projection width nQ = 128
    Set in Section 4; chosen without reported ablation.
  • number of low Fourier modes m = 10
    Set in Section 4; low-pass truncation in each Fourier layer.
  • number of Fourier layers = 4
    Architecture choice following the FNO literature; no ablation reported.
  • batch size = 32 for test 1, 8 for tests 2 and 3
    Reported in Sections 4.1 to 4.3.
  • training epochs = 2000
    Fixed for all models.
  • learning rate = 0.0005 with validation-based scheduler
    Tuned by validation loss; the authors note extensive testing but give no details.
  • L2 regularization weight lambda = not specified
    Appears in Algorithm 2 but no value is given in the text; affects the trained operator.
  • phi-FEM stabilization sigma_D = 1
    Used in test cases 1 and 2; h-independent parameter in the phi-FEM formulation of Section 2.2.
  • hyperelastic phi-FEM constants = gamma_u=0.001, gamma_p=gamma_div=sigma_N=0.01
    Set in Section 4.3.1 for the new hyperelastic scheme.
  • grid resolution = 64x64
    Specified for test case 1 and assumed for the other test cases.
  • geometry sampling distributions = U([0.2,0.8]) for ellipse centers, U([0.2,0.45]) for axes, U([0,pi]) for angle; Gaussian-sum parameters via Latin…
    Data generation choices in Equations (7) to (10) and Section 4.3; the generalization claim is conditioned on these distributions.
assumptions (5)
  • standard math phi-FEM converges optimally for the Poisson problem and related equations
    The paper relies on convergence results from prior papers, cited in Section 2.2, for generating training data.
  • domain assumption The FNO can approximate the ground-truth operator G mapping (f_h, phi_h, g_h) to w_h
    The paper relies on the operator approximation properties of FNO from [18, 15] without proving error bounds for this specific operator class.
  • domain assumption Level-set representation with phi_h as Lagrange interpolant captures the geometry accurately enough
    The pipeline is built on this representation; the boundary is approximated at order h, and no high-order geometry correction is used.
  • ad hoc to paper The variational formulation (11) for hyperelastic phi-FEM is stable and convergent for the chosen parameters
    This scheme is introduced in the paper and validated only numerically on a five-hole plate; no stability analysis is provided.
  • domain assumption Train, validation, and test data are generated from the same distribution
    The reported accuracy is a statistical claim about in-distribution generalization, not about arbitrary geometries.
invented entities (1)
  • auxiliary tensor field y_h and pressure-like field p_h
    purpose: Enforce homogeneous Neumann conditions on the holes in the new hyperelastic phi-FEM scheme of Section 4.3.1
    These are not physical quantities; they are introduced for the variational treatment of boundary conditions, and their validity rests only on the numerical convergence study in Fig. 15.

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Cite this review

Pith. "Pith review of Phi-FEM-FNO: a new approach to train a Neural Operator as a fast PDE solver for variable geometries." pith.science (2026). https://pith.science/paper/2FXAPSK3

@misc{pith2026250210033,
  author       = {Pith},
  title        = {Pith review of: Phi-FEM-FNO: a new approach to train a Neural Operator as a fast PDE solver for variable geometries},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2FXAPSK3}},
  note         = {Machine review of arXiv:2502.10033}
}
abstract

In this paper, we propose a way to solve partial differential equations (PDEs) by combining machine learning techniques and the finite element method called Phi-FEM. For that, we use the Fourier Neural Operator (FNO), a learning mapping operator. The purpose of this paper is to provide numerical evidence to show the effectiveness of this technique. We will focus here on the resolution of two equations: the Poisson-Dirichlet equation and the non-linear elasticity equations. The key idea of our method is to address the challenging scenario of varying domains, where each problem is solved on a different geometry. The considered domains are defined by level-set functions due to the use of the Phi-FEM approach. We will first recall the idea of $\varphi$-FEM and of the Fourier Neural Operator. Then, we will explain how to combine these two methods. We will finally illustrate the efficiency of this combination with some numerical results on three test cases. In addition, in the last test case, we propose a new numerical scheme for hyperelastic materials following the Phi-FEM paradigm.

Figures

Figures reproduced from arXiv: 2502.10033 by the authors.

Figure 1
Figure 1. Left: example of φ-FEM meshes. In red, the exact boundary Γ of an ellipse Ω, in white T O h , in gray T Γ h and in blue, Th \ T Γ h . Right: Convergence curves of φ-FEM and a standard finite element method, to solve (1) for 5 combinations of domain, force, and boundary conditions. where Γ is the boundary of Ω. Let T O h be a triangular cartesian mesh of O = [0, 1]2 composed of nx−1 and ny −1 squares divided into tri… view at source ↗
Figure 2
Figure 2. Construction of a prediction of φ-FEM-FNO to solve (1). that approximates the "ground truth" mapping G † (3). We predict the φ-FEM representa￾tion wh with wθ and uθ = φhwθ +gh will be an approximation of uh = φhwh +gh as described in [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. The φ-FEM-FNO pipeline to solve (1). Illustration based on the representation of [18]. The upper part represents the entire pipeline, and the lower part is a zoom on a Fourier layer. The red circles correspond to the inputs provided by the user and the output returned by our φ-FEM-FNO. We represent the inputs and outputs seen by the FNO in purple, where X = (fh, φh, gh). In orange, Pθ and Qθ are two transformations … view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: In red, the real boundary of an example domain. In blue and gray, the set [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: Test case 1. On the left-hand side (resp. right-hand side), we represent the evolution of the cost function L (resp. the relative L 2 error) on a subset of the training set and on the validation set. in [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: Test case 1. Example of result among the validation sample with an error in the norm (5) of 2.5 × 10−3 , corresponding to the median error. 50 100 500 1000 1250 1500 1750 2000 1989 Training epochs 10 3 10 2 10 1 L 2 r elativ e e r r o r E 1 [PITH_FULL_IMAGE:figures/fu…
Figure 7
Figure 7. Figure 7: Test case 1. Evolution of the (5) errors on 2500 test data at different steps of the training. The optimal model is represented in red. Validation of the model of a first test dataset. We now address a second crucial aspect: evaluating the error of the models on a test…
Figure 8
Figure 8. Figure 8: Test case 1. Reference solution (uref). Outputs of standard FEM (ustd) , φ-FEM (uφ), and (φ-FEM-FNO uθ). The presented test case corresponds to the median one among the FNO L 2 relative errors. -FEM Std FEM -FEM-FNO -FEM-FNO 2 Std-FEM-FNO Geo-FNO -FEM-UNET 10 3 10 2 10…
Figure 9
Figure 9. Figure 9: Test case 1. Left: Relative L 2 errors of the methods. Right: Relative L 2 errors, with respect to the computation times. In [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 10
Figure 10. Figure 10: Test case 2. Examples of considered level-set functions and domains, using (10). The black crosses mark the centers of the Gaussian functions [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]
Figure 11
Figure 11. Figure 11: Test case 2. Reference solution (uref). Outputs of standard FEM (ustd) , φ-FEM (uφ), and (φ-FEM-FNO uθ). φ are defined as: φ(x, y) = −ψ(x, y) + 0.5 max (x,y)∈[0,1]2 ψ(x, y), (10) with ψ(x, y) = X 3 k=1 exp  − (x − xk) 2 2σk − (y − yk) 2 2γk  , where the parameters x…
Figure 12
Figure 12. Figure 12: Test case 2. Left: comparison of the three methods on 300 new data. Right: relative L 2 error against computation time. Finally, [PITH_FULL_IMAGE:figures/full_fig_p018_12.png]
Figure 13
Figure 13. Figure 13: Test case 2. Left: Examples of test geometries. Each represented training shape is the closest one to the considered test shape, in the sense of Hausdorff distance. Right: L 2 relative errors against the Hausdorff distance to the closest training shape.    …
Figure 14
Figure 14. Figure 14: Test case 3. Left: Representation of the considered situation for the third test case. Center: example of deformed geometry. Right: Representation of the considered variations of the holes for the data generation. The black dashed squares correspond to the bounds of t…
Figure 15
Figure 15. Figure 15: Test case 3. Convergence of the two finite element methods, with respect to the number of degrees of freedom (left) and the maximal size of cell (right). Remark. The third term in (11) is the differential at (uh, yh) in the direction (vh, zh) of Z Ω Γ,i h (yh + P (F(u…
Figure 16
Figure 16. Figure 16: Test case 3. Example of output of the different methods, corresponding to the median of the relative L 2 errors of φ-FEM-FNO among the 300 problems. -FEM Standard FEM -FEM-FNO 10 3 10 2 R elativ e L 2 e r r o r 10 2 10 1 10 0 Computation time (s) 10 2 R elativ e L 2 e…
Figure 17
Figure 17. Figure 17: Test case 3. Left: L 2 errors of the methods. Right: relative L 2 errors of the methods against computation time. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_17.png]

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