Title: STNet: Spectral Transformation Network for Solving Operator Eigenvalue Problem
Venue: NeurIPS 2025
Accepted: yes
Keywords: AI for Science

Operator Eigenvalue Problem

Scientific Computing
Abstract: Operator eigenvalue problems play a critical role in various scientific fields and engineering applications, yet numerical methods are hindered by the curse of dimensionality. Recent deep learning methods provide an efficient approach to address this challenge by iteratively updating neural networks. 
These methods' performance relies heavily on the spectral distribution of the given operator: larger gaps between the operator's eigenvalues will improve precision, thus tailored spectral transformations that leverage the spectral distribution can enhance their performance. Based on this observation, we propose the **S**pectral **T**ransformation **Net**work (**STNet**).
During each iteration, STNet uses approximate eigenvalues and eigenfunctions to perform spectral transformations on the original operator, turning it into an equivalent but easier problem.
Specifically, we employ deflation projection to exclude the subspace corresponding to already solved eigenfunctions, thereby reducing the search space and avoiding converging to existing eigenfunctions. 
Additionally, our filter transform magnifies eigenvalues in the desired region and suppresses those outside, further improving performance.
Extensive experiments demonstrate that STNet consistently outperforms existing learning-based methods, achieving state-of-the-art performance in accuracy.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes STNet (Spectral Transformation Network), a deep learning framework designed to solve operator eigenvalue problems efficiently, particularly in high-dimensional settings. STNet enhances convergence by applying spectral transformations to the operator using approximate eigenvalues and eigenfunctions obtained during training. Specifically, it employs two key methods: a deflation projection that eliminates previously learned eigencomponents to reduce redundancy and a filter transform that amplifies target spectral regions to accelerate convergence. By integrating these modules into an iterative neural network training process, STNet reformulates the original eigenvalue problem into a series of more tractable ones. Experiments across harmonic, Schrödinger, and Fokker-Planck operators demonstrate that STNet outperforms existing learning-based and traditional methods in accuracy.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- This paper is overall well written and easy to follow. The method and results are clearly presented.\n- It combines the deflation and filter transforms into a neural-network-based solver for the operator eigenvalue problem, and this method is described clearly. \n- The work addresses a challenging and important problem: solving high-dimensional eigenvalue problems, which arise in many scientific and engineering domains. By demonstrating scalability and improved performance over other learning-based baselines, STNet has potential for meaningful impact in both scientific computing and machine learning communities.\n\nWeaknesses:\n- The method is primarily evaluated on synthetic benchmark problems. While these are standard in the literature, applying STNet to real-world or more complex physical systems would better demonstrate its broader applicability. \n- Lack of comparison with traditional eigenvalue methods, like the convergence and computational cost. \n- There is no explicit study on the convergence for different matrix sizes, and how it scales.\n\nQUESTIONS:\n1. Can you compare the convergence with the traditional power method under different matrix sizes? \n2. Are there any advantages in computational cost and memory usage for large-scale matrices? \n3. Can STNet be applied to real-world problems where the operator or domain is not analytically defined or is noisy?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-01"}

comments: SUMMARY:
The authors  propose STNet based on the power method for estimating eigenvalues of linear operators. From the experimental results, it can be observed that there is a significant improvement in accuracy compared to previous neural network methods and traditional methods. The experimental analysis also roughly leads to the conclusion that STNet's two innovations, Deflation Projection and Filter Transform, can play their respective roles in different scenarios.

STRENGTHS AND WEAKNESSES:
### Strengths:
1. Deflation projection and filter transform are simple and intuitively effective.
2. The paper is well-written with clear and fluent presentation that makes the methodology easy to understand.

### Weaknesses:
Several issues need to be addressed or supplemented:
1. Convergence and stability analysis with graphical illustrations is needed.
2. It is observed that the absolute error of λ in the w/o F and D case is still much lower than other methods when converted to relative error. Does this mean that the performance improvement does not come from F and D? Additionally, consider showing the results of PMNN (only showing principal eigenvalue is OK).

QUESTIONS:
see weaknesses

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-02

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces STNet, a neural network-based approach to solving linear operator eigenvalue problems. Traditional numerical methods have the curse of dimensionality problem, so deep learning-based methods have recently been suggested. The authors were inspired by classical numerical techniques, specifically deflation projection and filter transform. By excluding eigenvalues and eigenfunctions which are already been computed and amplifying the spectral region of interest, the authors improve the deep learning-based method.\n\nSTRENGTHS AND WEAKNESSES:\n* Strengths:\n1) A good attempt to improve existing deep learning-based methods by incorporating classical numerical ideas.\n\n* Weakness:\n1) Ambiguity in the Abstract\n\n Although deflation and filter transform are classical numerical techniques, the abstract presents them as if they are the authors’ original ideas. There is no mention in the abstract that these techniques originate from traditional numerical methods.\n Precisely, since they are described only as tools for resolving spectral distribution issues in deep learning, the abstract creates the impression that these are problems unique to neural network methods. This could easily mislead readers into thinking that the techniques themselves are newly developed by the authors. A clear statement is needed so readers can immediately recognize the origin of these methods.\n\n2) Lack of theoretical justification\n\n While it is a valid idea to incorporate classical methods, there is not enough theoretical analysis to back up this approach. In particular, there is no guarantee that deflation projection and filter transform, when applied in a deep learning context, will preserve the spectral properties of the original operator.\n\n3) Unclear implementation of the differential operator\n\n The paper only stated that the differential operator 𝐿 is implemented via automatic differentiation, without further details. It is unclear how L is implemented in code, or how deflation and filtering (lines 7–8 in Algorithm 2) are carried out. The description is too vague.\n\n4) Lack of details in the experimental setup\n\n The experiments do not state the loss functions used in each case explicitly. It makes it difficult to understand or reproduce the experiments.\n\n5) Main contents and Appendix\n\nIt seems that deflation projection and filter transform are key ideas of this paper. Therefore, the content currently in Sections 2.3 and 2.4 should be moved from the appendix into the main text and explained in more detail. Additionally, the differences between the baselines in Section 4 are also mainly described in the appendix; these should be elaborated more clearly in the main body to improve the reader’s understanding.\n\nQUESTIONS:\n1) Could you clarify or justify the points raised above?\n2) The method is described as being specific to linear operators in the limitations section. Is this a structural limitation of the approach, or just an implementation choice?\n3) Are there no comparisons with other more accurate than FDM or specialized traditional methods?\n4) Did you compare with FDM combined with deflation or filtering?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a network based eigen-solver, following several works in the field. The main idea is solving for several eigenfunctions in parallel, while using the deflation technique to decouple the different eigenvalue problems. This way each STNet module is focused on solving the problem within a pre-specified range of eigenvalues. For the algorithm to work properly, one needs a reasonable estimation of the expected eigenvalue range.  They use a filter transform to amplify the eigenvalues in a certain desired range and improve convergence. The results appear to be good, compared to 3 other NN-based methods, achieving better accuracy on several standard eigenvalue problems. Experiments with various variations and an ablation study is carried out. Code of the algorithm is supplied.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1. The method appears sensible and may be able to compute well middle-range eigenvalues, which are often harder to compute using iterative methods.\n\n2. Experiments show high accuracy, compared to other similar NN-based algorithms.\n\n3. There is transparency, where code is supplied and could be readily used upon publication.\n\n\nWeaknesses and questions:\n\n1. I did not see details on architecture of the network. Code is of course not a valid way to communicate the architecture. There seem not to be ablation on the architecture.\n\n2. The method may highly depend on having a good approximation of the eigenvalues expected. Not clear what  happens when the range is erronous.\n\n3. What is the complexity of the process? There are some matrix inversions for instance, which are computationally demanding.\n\n4. Running time, compared to other methods. How many iterations are needed on average? \n\n5. Rate of convergence? Although a full theoretical analysis is very hard and beyond the scope. It would good to show at least a plot of convergence: error vs. iterations (or actual time).\n\n6. Grid resolution. Seems to be only one resolution tested.\n\n7. Are there no more recent works on the subject from 2024 and 2025? Please update the related work section.\n\nQUESTIONS:\nsee above\n\nLIMITATIONS:\nsee above", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-03"}