Operator Inference for Elliptic Eigenvalue Problems

Haoqian Li; Jiguang Sun; Zhiwen Zhang

REVIEW 4 major objections 6 minor 23 references

Operator Inference for Elliptic Eigenvalue Problems

T0 review · 4 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Neural nets map any 2D shape to its first 20 eigenpairs.

desk verdict A useful neural surrogate for Dirichlet eigenpairs on random 2D domains; solid experiments but with a real degeneracy in the PCA alignment step and missing baseline/reproducibility support. read the letter →

arxiv 2504.15733 v1 pith:OSPRQHFO submitted 2025-04-22 math.NA cs.NA

Haoqian Li , Jiguang Sun , Zhiwen Zhang This is my paper · ORCID

classification math.NAcs.NA MSC 35J1565N2568T0765T50

keywords operatorinferenceellipticeigenvalueproblemsDirichleteigenvalueseigenfunctionpredictionFourierNeuralconvolutionalnetworkmain-axisalignmentdetailedpixelization

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 tries to establish that the mapping from an arbitrary two-dimensional domain to its first few Dirichlet eigenpairs can be learned end to end from pixel images. It splits the learning task into two: a convolutional network maps the pixelated domain image to eigenvalues, and a Fourier Neural Operator maps the same image to eigenfunctions. A preprocessing module scales each domain to the unit square, refines boundary pixels with average pooling, and aligns each domain along its principal axes, so the networks see only intrinsic geometry. On 20,000 generated polygonal and smooth domains, the first eigenvalue is predicted to 1.15% mean absolute percentage error and the first eigenfunction to 1.30% relative L1 error at $32\times 32$ resolution, with errors growing for higher modes. If correct, the method is a fast surrogate for repeated eigenvalue queries in shape optimization and inverse spectral problems.

What carries the argument

The central mechanism is the image-based operator representation plus a three-part preprocessing module. Main-axis alignment computes the covariance matrix of boundary sample points, rotates the domain so its principal axes align with the coordinate axes, and resolves the two reflection ambiguities by comparing standard deviations across the left/right and lower/upper halves; this is what turns a rotation-invariant learning problem into a fixed-orientation regression. Detailed pixelization generates a high-resolution mask and average-pools it down to the target grid, so boundary pixels take fractional values that encode the area fraction of the domain inside the pixel; this is what prevents four narrow rectangles from sharing one input image. The eigenvalue CNN uses convolutional blocks with skip connections and a fully connected head, while the eigenfunction FNO uses four Fourier blocks with 16 modes and a masked, sign-aware, normalized L2 loss.

What would settle it

Take a square or disk, a domain whose boundary-point covariance matrix has equal eigenvalues, and run the pipeline twice on the same geometric shape, once with the principal-axis eigenvectors in one numerical orientation and once with them swapped by a tiny perturbation; if the predicted first eigenvalue differs by more than the reported MAPE, the deterministic orientation assumption fails. A second check is to compute test MAPE on a set of shapes with near-degenerate covariance and compare it with the full-set MAPE of 1.15%.

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Extended reading notes

Core claim

The paper's central assertion is that a two-network operator pipeline can directly approximate the Dirichlet eigenvalue maps $\Omega \mapsto (\lambda_1,u_1,\dots,\lambda_{20},u_{20})$ for simply connected Lipschitz domains in the plane, after the domain is represented as a preprocessed image. The authors argue that the decisive ingredient is not the network architecture alone but the preprocessing: scaling enforces the known $k^{-2}$ scaling law of Dirichlet eigenvalues, detailed pixelization keeps distinct narrow rectangles from collapsing to the same binary image, and main-axis alignment removes rotational and flipping ambiguity before training. With these ingredients, the CNN reaches $R^2=0.9988$ and MAPE $1.15\%$ for the first eigenvalue at $32\times 32$, and the FNO reaches PSNR $52.00$ and RelL1 $1.30\%$ for the first eigenfunction; higher eigenfunctions degrade as their oscillation count grows. The paper further shows that relative eigenvalue error stays roughly constant across the first 20 eigenvalues, so the surrogate is not limited to the ground state.

Load-bearing premise

The load-bearing premise is that principal-axis alignment gives every input domain a unique, stable orientation; for shapes whose covariance eigenvectors are degenerate (squares, disks, equilateral triangles), the orientation is arbitrary, so the same shape can be fed to the network in different rotations and the deterministic input-output mapping breaks.

Editorial extensions

If this is right

A trained model answers a new shape query with one forward pass, so repeated eigenvalue evaluations in loops such as shape optimization or real-time simulation no longer require a fresh finite-element solve per candidate.
The same trained networks generalize to unseen polygonal and smooth domains drawn from the same generator, giving a checkable baseline for operator-learning surrogates of spectral problems.
Relative eigenvalue error stays nearly flat across the first 20 eigenvalues, so a single model covers a spectrum rather than only the fundamental mode.
Eigenfunction accuracy falls off with mode number, so the practical envelope is the low- to mid-frequency range; higher modes are captured structurally but not quantitatively.

Reading between the lines

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

An immediate stress test would be to restrict training to shapes with degenerate or near-degenerate covariance, such as squares or near-circles; if the principal-axis alignment is non-unique there, the model should either fail or require orientation augmentation to recover.
The same two-network split should transfer to Neumann or Robin boundary conditions, since only the labeled data and the mask change, not the architecture.
Predicted eigenfunctions could be recycled as initial guesses for a few finite-element refinement iterations, potentially combining the surrogate's speed with classical convergence guarantees.
Because the paper's largest-error samples are elongated domains, a test distribution with more extreme aspect ratios is likely to push MAPE above the reported 1.15%.

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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

Colleague,

The paper is a straightforward piece of applied operator learning: it takes pixelated 2D domains as input and trains a CNN to predict the first 20 Dirichlet eigenvalues and an FNO to predict the corresponding eigenfunctions. On the author-generated test set of 4,000 random polygonal and smooth domains, the first eigenvalue MAPE is 1.15% (32x32) and the first eigenfunction RelL1 is 1.3%. Those are decent numbers, and the ablation tables show their preprocessing—particularly the boundary-aware pixelization—is pulling real weight.

What's new: as far as the citations go, this is the first domain-to-eigenpair operator learning formulation. The divide-and-conquer architecture (CNN for scalars, FNO for functions) is sensible. The paper does not oversell the mathematics; it's a data-driven surrogate, and the held-out split is the right way to evaluate it.

Where I would push back, in increasing order of seriousness.

Minor: equation (3.18) defines an L1 loss but the sentence below says "MSE penalizes large deviations." Harmless typo.

Reproducibility: no code or data. For a paper built entirely on numerical experiments, that's a significant gap. Also, all tables report a single run; no seed variance. I wouldn't be surprised if the 1.15% MAPE varies by 0.2% across seeds. The authors should report mean and std over 3-5 runs.

Baselines: The ablations compare against removing their own modules, but not against any alternative surrogate—say, a single CNN for both eigenvalues and eigenfunctions, or a regression on shape descriptors, or an FNO for eigenvalues as well. Without a baseline, "high accuracy" is only relative to what a trivial predictor does.

The larger issue is the main-axis alignment. The stress-test note is right: for shapes with degenerate covariance (squares, disks, equilateral triangles), the principal axes are arbitrary, and for shapes near that set a tiny perturbation can swap the ordering and rotate the input by 90 degrees. The preprocessing map then isn't continuous at that set. Since the paper claims generality to "arbitrary 2D domains," this is an unsupported overreach. That said, the training and test sets are random polygons and Bezier shapes, which likely sit away from the degeneracy, so the reported numbers stand for that distribution. The paper should either restrict its claim to generic shapes or add some handling for near-symmetric domains.

Bottom line: this is a useful contribution for people building fast surrogates for spectral problems, shape optimization, or inverse queries. It deserves a serious referee. I'd send it out, with the expectation that the authors add code/data, seed variance, at least one meaningful baseline, and a discussion of the PCA degeneracy.

Best,

Referee Report

4 major / 6 minor

Summary. The paper proposes a data-driven operator surrogate for Dirichlet eigenvalue problems on two-dimensional domains. Domains are represented as pixel images and preprocessed by scaling, principal-axis alignment, and boundary-aware (averaged) pixelization. A CNN is used to predict eigenvalues and an FNO to predict eigenfunctions, with separate loss functions and training protocols. The authors report held-out test metrics on 20,000 randomly generated polygonal and smooth domains, including a first-eigenvalue MAPE of 1.15% and a first-eigenfunction RelL1 of 1.30% at 32x32 resolution, together with ablations of the preprocessing steps and results for the first 20 eigenpairs.

Significance. If the reported accuracy holds, the proposed pipeline is a useful fast surrogate for eigenvalue queries in shape optimization and inverse spectral problems. The paper has genuine strengths: it evaluates on a held-out test set rather than on training data, includes ablations of the preprocessing modules, and reports multiple error metrics. The main limitations are the unaddressed degeneracy of the main-axis alignment for symmetric or nearly symmetric domains, the missing specification of the network output dimensions across eigenvalue indices, and the absence of baselines and runtime comparisons. These gaps affect the reproducibility and the strength of the generality claim, but they are fixable within the scope of the manuscript.

major comments (4)

[Section 2.2, Eqs. (2.6)-(2.8)] The main-axis alignment is not well-defined for domains whose boundary-point covariance matrix has repeated eigenvalues (squares, disks, equilateral triangles, regular polygons). In those cases the eigenvectors in Eq. (2.7) are only determined up to an arbitrary orthogonal transformation of the degenerate eigenspace, and the sign-fixing rule based on left/right and lower/upper standard deviations does not break the tie because those standard deviations are equal by symmetry. As a result, the same geometric domain can be presented to the networks in different orientations depending on the eigensolver and on pixelization noise. Near the degeneracy set the aligned representation is discontinuous: an arbitrarily small perturbation can swap the ordering of the two covariance eigenvalues and rotate the aligned image by 90 degrees. Since the manuscript claims generality to arbitrary 2D domains and does not analyze this degeneracy, this is a load-bearing gap. Please either restrict the claims to domains with distinct covariance eigenvalues, introduce a deterministic and continuous orientation rule (e.g., based on higher-order moments), or add an explicit numerical test on symmetric and near-symmetric shapes showing that the predictions are stable.
[Section 3.1 and Section 4.1] The manuscript does not specify whether one CNN predicts all eigenvalues at once (e.g., an output dimension of 20) or a separate CNN is trained for each eigenvalue index. The same ambiguity applies to the FNO: it is unclear whether one network outputs a single eigenfunction and is retrained for each index, or whether several eigenfunctions are output together. This information is necessary to reproduce the results and to interpret Table 2, Figure 8, and Figure 12, which report per-index metrics. Please state the output dimensionality and whether parameters are shared across eigenvalue indices for both the CNN and the FNO.
[Section 4.1, Table 1] The statement that 'detailed pixelization and main axis alignment significantly improve the model accuracy' is only partially supported by the reported numbers. For the 32x32 setting, MAPE is 1.67% without any preprocessing and 1.75% with main-axis alignment alone, so alignment alone does not improve accuracy; the improvement to 1.15% comes from the combination dp+ma. Please revise the wording to describe the joint effect accurately, or discuss why alignment alone is not beneficial at 32x32.
[Sections 3 and 4] No comparison is made with alternative operator-learning methods (such as DeepONet, a standard FNO used directly for eigenvalues, or a CNN-only baseline) and no runtime measurements are reported. The introduction and conclusion claim that classical methods are 'computationally intensive' and that the proposed method is efficient, but no timing comparison is given. Please add at least wall-clock times for the finite-element data generation versus online CNN/FNO prediction, and, ideally, a baseline comparison with a generic neural operator to calibrate the reported accuracy.

minor comments (6)

[Eq. (2.7)] The eigendecomposition is written as 'M = DM·vM', which is not a valid matrix factorization and is dimensionally inconsistent; it should be M = V D V^T, where V is the matrix of eigenvectors. Please correct this equation.
[Eq. (3.18)] The loss function in Eq. (3.18) is the mean absolute error, but the surrounding text says 'the MSE penalizes large deviations' and no mean squared error is defined. Please make the formula and the description consistent.
[Section 4, statistical robustness] The experiments are run without reporting variance across random seeds or across different train/test splits. Since the dataset is randomly generated, a single split may not be representative; please report means and standard deviations over at least three independent training runs.
[Section 2.1, Eq. (2.3)] The index in the weighted average θ*_{i,i+1} = w·θ_{i-1,i} + (1-w)·θ_{i,i+1} uses θ_{0,1} when i=1, which is not defined unless the vertex index is cyclic; please clarify the indexing convention for the vertex list.
[Reproducibility] No code or data availability statement is included, and the finite-element solver used to generate the dataset is only referenced by a book; details such as the number of mesh elements and the treatment of high-order eigenfunctions would help readers reproduce the experiments.
[Throughout] There are several typographical errors, including 'dateset' in Section 2.2, 'folowing' in Eq. (3.12), and the broken title 'OPERA TOR INFERENCE'; a careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the eigenvalue/eigenfunction predictions are out-of-sample supervised surrogates, not restatements of training inputs.

full rationale

This paper trains CNN and FNO surrogates that map pixelated domain images to FEM-computed eigenpairs. The central predictions are evaluated on a held-out split (20%, 4,000 pairs), so the reported errors are genuine out-of-sample results rather than fitted values. The preprocessing steps (scaling, main-axis alignment, detailed pixelization) use only geometric information about the domain and do not incorporate the target eigenvalues or eigenfunctions. The loss functions are standard supervised losses, and no analytical eigenvalue formula is embedded in the network that would make the output equal to an input by construction. The self-citations to [5] and [19] are not load-bearing in a circular sense: [19] is a standard finite-element reference used to generate labels, and [5] is cited only for a divide-and-conquer design choice. No step in the derivation chain reduces to its own inputs.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

No new physical or mathematical entities are introduced. The method's inputs are images of domains, and outputs are eigenpairs; the central dependencies are the fitted network weights and the FEM-generated labels.

free parameters (5)

Neural network weights (CNN and FNO) = trained on 16,000 domain-eigenpair samples
All predicted eigenvalues and eigenfunctions are direct outputs of these fitted weights; there is no closed-form or first-principles component.
Fourier modes M_max = 16
Hand-chosen number of frequency modes in the FNO; affects representation capacity and accuracy.
Number of Fourier blocks = 4
Hand-chosen depth in the FNO.
Hidden channels = d (image size)
FNO hidden channel count is set equal to the image resolution d; a design choice.
Training hyperparameters (learning rate, batch size, epochs) = lr 0.001 with decay, batch 32/128, 100/50 epochs
Hand-chosen and not ablated; the paper does not report sensitivity to these.

assumptions (4)

domain assumption Finite element solutions at mesh size h ≈ 0.01 are accurate enough that label noise is negligible (Section 2.2)
The training and test labels are FEM-computed eigenpairs; if the FEM discretization error is not small, the reported prediction errors are misleading.
standard math Dirichlet eigenvalues are rotationally and translationally invariant and scale as k^{-2} under uniform scaling (Section 2)
Used to justify scaling and main-axis alignment preprocessing.
domain assumption Principal-axis alignment yields a unique orientation for every domain (Section 2.2, eq. 2.6-2.8)
For domains with degenerate covariance eigenvalues (squares, disks), eigenvectors are not unique; the paper does not analyze stability for these cases, which is load-bearing for the deterministic mapping.
domain assumption Average-pooling-based detailed pixelization captures boundary geometry sufficiently for accurate eigenvalue and eigenfunction prediction (Section 2.2)
The method relies on image representation; the paper shows empirical improvement but no error analysis for the pixelization error.

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Cite this review

Pith. "Pith review of Operator Inference for Elliptic Eigenvalue Problems." pith.science (2026). https://pith.science/paper/OSPRQHFO

@misc{pith2026250415733,
  author       = {Pith},
  title        = {Pith review of: Operator Inference for Elliptic Eigenvalue Problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OSPRQHFO}},
  note         = {Machine review of arXiv:2504.15733}
}

read the original abstract

Eigenvalue problems for elliptic operators play an important role in science and engineering applications, where efficient and accurate numerical computation is essential. In this work, we propose a novel operator inference approach for elliptic eigenvalue problems based on neural network approximations that directly maps computational domains to their associated eigenvalues and eigenfunctions. Motivated by existing neural network architectures and the mathematical characteristics of eigenvalue problems, we represent computational domains as pixelated images and decompose the task into two subtasks: eigenvalue prediction and eigenfunction prediction. For the eigenvalue prediction, we design a convolutional neural network (CNN), while for the eigenfunction prediction, we employ a Fourier Neural Operator (FNO). Additionally, we introduce a critical preprocessing module that integrates domain scaling, detailed boundary pixelization, and main-axis alignment. This preprocessing step not only simplifies the learning task but also enhances the performance of the neural networks. Finally, we present numerical results to demonstrate the effectiveness of the proposed method.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

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

[1] [1]

Ben-Shaul, L

I. Ben-Shaul, L. Bar, D. Fishelov, and N. Sochen, Deep learning solution of the eigenvalue problem for differential operators, Neural Computation, 35 (2023), pp. 1100–1134

work page 2023

[2] [2]

Bhattacharya, B

K. Bhattacharya, B. Hosseini, N. B. Kovachki, and A. M. Stuart , Model reduction and neural networks for parametric PDEs , The SMAI journal of computational mathematics, 7 (2021), pp. 121–157

work page 2021

[3] [3]

Bubuˇska and J

I. Bubuˇska and J. Osborn , Eigenvalue problems. Handbook of numerical analysis, Vol. II , North-Holland, Amsterdam, 1991

work page 1991

[4] [4]

X. Dai, Y. Fan, and Z. Sheng , Subspace method based on neural networks for eigenvalue problems, arXiv preprint arXiv:2410.13358, (2024)

arXiv 2024

[5] [5]

H. Du, Z. Li, J. Liu, Y. Liu, and J. Sun , Divide-and-conquer DNN approach for the inverse point source problem using a few single frequency measurements , Inverse Problems, 39 (2023), pp. Paper No. 115006, 19 pp

work page 2023

[6] [6]

M. Feit, J. Fleck Jr, and A. Steiger , Solution of the Schr¨ odinger equation by a spectral method, Journal of Computational Physics, 47 (1982), pp. 412–433

work page 1982

[7] [7]

J. Han, J. Lu, and M. Zhou , Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach , Journal of Computational Physics, 423 (2020), p. 109792

work page 2020

[8] [8]

J. He, L. Li, and J. Xu , Approximation properties of deep ReLU CNNs , Research in the mathematical sciences, 9 (2022), p. 38

work page 2022