STNet: Spectral Transformation Network for Solving Operator Eigenvalue Problem
Authors: Hong Wang
Jiang Yixuan
Jie Wang
Xinyi Li
Jian Luo
huanshuo dong
Venue: NeurIPS 2025 poster ()
Keywords: AI for Science
Operator Eigenvalue Problem
Scientific Computing
Primary area: machine_learning_for_sciences

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.

STNet: Spectral Transformation Network for Solving Operator Eigenvalue Problem
Hong Wangl22: Yixuan Jiangl Jie Wang Xinyi Li', Jian Luo Huanshuo Dong! 12.
University of Science and Technology of China
CAS Key Laboratory Technology in GIPAS, University of Science and Technology of China MoE Key Laboratory of Brain-inspired Intelligent Perception and Cognition, University of Sci- ence and Technology of China
Tsinghua University
wanghong177000msi..uuass edu...edd.ccom en, jiangyix25tmails wtw...a...t.aa...ee. en, jiewangxbustc.edu..
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 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 this observation, we propose the Spectral Transformation Network (STNet). During each iteration, STNet uses approximate eigenvalues and eigenfunctions 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 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-base based methods, achieving state-of-the-an performance accuracy
Introduction
The operator eigenvalue problem is a prominent focus in many scientific fields 10
351 and
engineering applications 8 13. However, traditional numerical methods are constrained by the curse of dimensionality, as the computational complexity increases quadratically or even cubically with the mesh size 43
These ap- proaches reduce the number of parameters by replacing the matrix representation with a parametric nonlinear representation via neural networks. By designing appropriate loss functions, it updates parameters to approximate the desired operator cigenfunctions. These methods only require sampling specific regions without designing discretization mesh, significantly redueing the algorithm design cost and unnecessary approximation errors [9] Moreover, neural networks generally exhibit stronger
promising alternative is using neural networks approximate eigenfunctions 36
"Equal contribution.
1 Corresponding author.
OOr code is available at http://gitub.ccmmjyy1//x
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Ground Truth
Prrdet
STNet
Figure I: Left. Absolute error results of zero eigenvalues for the Fokker-Plance operator computed using various algorithms, the x axis represents the operator dimension. Right. Comparison of the eigenfunctions of the 2D Harmonic operator computed by STNet and the ground truth.
expressiveness than linear matrix representations, requiring far fewer sampling points for the same problem compared traditional methods [34] 14].
Despite these advantages, the performance such methods strongly depends on the operator's 's spectral distribution: if the target cigenvalues differs greatly to each other, the algorithm converges much more faster; otherwise, it may suffer from inefficient iterations. To improve convergence, spectral transformations can be designed based on the spectral distribution, reformulating the original problem into an equivalent but more tractable one. However, since the real spectrum of the operator is initially unknown, existing approaches do not optimize spectral properties through such transformations.
To address this limitation, we propose the Spectral Transformation Network (STNet). By exploiting approximate eigenvalues and eigenvectors learned during the iterative process, STNet applies spectral transformations the original operator, modifying its spectral distribution and thereby converting into an equivalent problem that converges more easily. Concretely, we employ deflation projection to remove the subspace corresponding already computed eigenfunctions. This not only narrows the search space but also prevents subsequent cigenfunctions from collapsing into the same sub- space. Meanwhile, our filter transform amplifies eigenvalues within the target region and suppresses those outside it, promoting rapid convergence to the desired eigenvalues. Extensive experiments demonstrate that STNeet significantly surpasses existing methods based on deep learning, achieving state-of of-thh-art performance in accuracy. Figure Tillustrates selection of the experimental results.
2 Preliminaries
2.1 Operator Eigenvalue Problem
We primarily focus the eigenvalue problems of differential operators, such à Mathematically, an operator L: Hi H2 is mapping between Hilbert spaces. Considering a self-adjoint operator L. defined on domain f C RD, the operator eigenvalue problem can be expressed in the following form
dr ay A, etc.
Le Au in Ω
(1)
where RDP serves as the domain; the eigenfunction and À the eigenvalue. Typically, often necessary solve for multiple cigenvalues, A, I,...,L.
2.2 Power Method
The power method is classical algorithm designed approximate the eigenvalue of operator C in the vicinity of given shift a. By applying the shift (often chosen as an approximation to the target eigenvalue), the original eigenvalue problem effectively transformed into an equivalent problem for the new operator (L a In each iteration, the current approximate solution is multiplied by this new operator, thereby amplifying the component associated with the eigenvalue closest to This iterative procedure converges to the desired eigenvalue, shown in Algorithm [1].

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Algorithm Power Method for the Operator L
1: Input: Operator L., shift o, initial guess v0, maximum iterations Aimax convergence threshold
2: Output: Eigenvalue À near a.
3: e0 / /||e01
4: for
to mmax do
5: ok p² /Ipp
and solve (C (C-ol) =
6: if 11 2
then
f Co
7: À
(ut cky
and break.
8: end if
9: end for
In each iteration, solving the linear system (I) -k-1 equivalent applying the operator (C-o7) -1) to ett11, Afterward, normalizing v² helps maintain numerical stability. Convergence is typically assessed by evaluating the error | - - II. ensuring that the final solution meets the desired accuracy. The fundamental reason for the convergence of the power method lies in the repeated application of (C af)- which progressively magnifies the component of of in the direction of the eigenfunction with eigenvalue closest to a. For a more detailed introduction to the power method, please refer to the Appendix B.1 B.1
2.3 Deflation Projection
The deflation technique plays critical role in solving eigenvalue problems, particularly when multiple distinct eigenvalues need to be computed. Deflation projection is an effective deflation strategy that utilizes known eigenvalues and corresponding eigenfunctions modify the structure of the operator, thereby simplifying the computation of remaining eigenvalues [39].
The core idea of deflation projection is construct an operator P, often defined P(u) (u, en Y where en is a known eigenfunction. This operator is then used to modify the original operator L into a new operator:
B = L A1P.
(2)
In B, the eigenvalue Å₁ associated with es is effectively removed from the spectrum of C. Additional details on deflation projection can be found in Appendix B.2
2.4 Filter Transform
The filter transform is widely used in numerical linear algebra to enhance the accuracy of eigenvalue computations [39]. By constructing a suitable filter function F(C), the operator undergoes spectral transformation that amplifies the target eigenvalues and suppresses the irrelevant ones. The filter transform can effectively highlight the desired spectral region without altering the corresponding
eigenfunctions
43 Further details the filter transform can be found in Appendix B.3
3 Method
3.1 Problem Formulation
We consider the operator eigenvalue problem for a differential operator L defined on domain C "R Our goal is to approximate the eigenvalues A; near a given shift a and their corresponding eigenfunctions v, satisfying
Lu; = X; V/s i = 1,,,.... L.
(3)
To achieve this, employ L neural networks parameterized by 0, Each neural network NNN((;; maps the domain into R, providing an approximation of the eigenfunction Uy!
NNc(=; Ω + R, i= 1,2,
In order to represent both the functions and the operators numerically, we discretize Ω by uniformly randomly sampling N points:
S = (x, (.).....)))
11) 2x
Ω j= 1.2....,
(5)
3

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Filter
Deflation
STNet
differential operator C
Spectral Transformation Network
NNv((01)
update
sampling points { f-1
eigenvalues (
achicved socurne
updant
6,
NN2((22)
filter transform
ON
oulput
deflation projection
(a)
4,
(N.
updarie
(b)
Figure 2: Overview of the STNet. (a) Introduction to the inputs and outputs. (b) STNet comprises multiple neural networks, each tasked with predicting distinct eigenvalues. the accuracy of the solution reaches the expectation, then STNet will output the result.
Correspondingly, each neural network N (N; (=;((;00) output vector Y; IR-Y, which approximate the values of the eigenfunction ū;(-) = NN2(=;0 0.) at these sampled points:
i (m)) Yi)), i == 1..,..... I,,
== 111,2.... N.
The approximate eigenvalues À are then obtained by applying C to the computed eigenfunctions ūs:
x (0 Coe)
i 1.2....,
(0), U.)
Here, the differential operator acts on the functions via automatic differentiation. We iteratively update the neural network parameters 0; using gradient descent, aiming minimize the overall residual. Specifically, we formulate the following optimization problem:
1 ΣΣ Σ (,,))) us (xx))
min 0,E0
where Q denotes the parameter space of the neural networks. This approach does not require any training data, as it relies solely on satisfying the differential operator eigenvalue equations over the domain Q. Finally, this procedure provides approximations X; of the true eigenvalues A, i 1,...,; L.
3.2 Spectral Transformation Network
Inspired by the power method and the power method neural network (45, we propose STNet to solve eigenvalue problems, as shown in Figure 2 In STNet, replace the function o from the k-th iteration of the power method with 7 (x) NNr;; 0ft), where each neural network is implemented via multilayer perceptron (MLP). Since neural networks cannot directly perform the inverse operator (L (L-o11 ol) we enforce (f (C-ol)ō o xs f through suitable loss function. The updated parameters 8h + g then yield g = NN: 1 )). Algorithm shows the detailed procedure of STNet.
Classical power method convergence is strongly influenced by the spectral distribution of the op- erator, which is unknown initially and thus difficult optimize against directly. However, as the iterative process starts, we can get additional information-succ and eigenfunctions. Using these results the spectral transformation of the original operator can greatly improve subsequent power- metho iterations. In Algorithm 2 we introduce two modules to improve the performance. Their impact on the operator spectrum is shown in Figure 3
already computed eigenvalues
Deflation projection uses already computed eigenvalues and eigenfunctions construct projection that excludes the previously resolved subspace, preventing convergence to known eigenfunctions and reducing the search space.
Filter transform employs approximate eigenvalues to construct spectral transformation (filter function) that enlarges the target eigenvalue region and suppresses others, boosting the efficiency of STNet.

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Algorithm 2 Spectral Transformation Network
1: Input: Operator L. over domain Ω C RD, shift o, number of sampling points N, number of cigenvalues L, learning rate convergence threshold maximum iterations Kmax.
2: Output: Eigenvalues Åo = i
3: Uniformly randomly sample N points {æ,} in Z to form dataset S.
4: Randomly initialize the network parameters 6P. as well as the normalized ös and set b = a.
== l..... L.
5: for = to knux do
6:
( 3 E.S. i (xy) N NN (z
7:
C = =D;(C).
1...., L
// Deflation Projection // Filter Transform
8:
C =F((C',
i= 1, I...,,L
9:
ū (1,) (|
i= l.......
10:
Loss 2j=1 10
(a.)) (x,)),
I,...,I. L.
11:
gf+1 = 0
Ve, Loss R
i= I.....L
// Parameter Update
for to L do
13:
if Loss < then
14:
6= Loss, k XΙ
10 Cot) (0) 5it)
15:
end if
16:
end for
if' e;
e for all then Convergence achieved; break.
18:
else
20:
Update deflation projection and filter function: Di, F,, i= 1, 1,....,,44 L.
end if
22:
end for
3.2.1 Deflation Projection
Suppose have already approximated the eigenvalues 'Ι .e,..., and their corresponding cigenfunctions U, . U; To compute the i-th eigenfunction, we focus on the residual subspace orthogonal to the subspace spanned by these previously computed eigenfunctions. The deflated projection is then defined as
D.(C) L Qi-1 Ex-1 P.
(9)
Here Q,-1 maps each vector (a1, ......... 04--1 Ri-1 the function Σ k= Or ūk thus reconstructing functions from the span of (Ø1...., 0 } 27_ is the transpose of Q And E, is diagonal operator that scales each üg by its corresponding eigenvalue λ&-
By employing the deflation projection, the gradient descent search space of the neural network is constrained to orthogonal the subspace spanned by [Ø1 . 1- This projection prevents the neural network output NNc (0,) from converging to the invariant subspace formed by known eigenfunctions, thereby enhancing the orthogonality among the outputs of different neural networks N NN)...)..... ... NN NNc( (0,...)) 1). On the one hand, this reduction in the search space accelerates the convergence toward the eigenfunctions U;; On the other hand, it improves the orthogonality among the neural network outputs, which reduces the error in predicting the eigenfunction ë;
In practice, we use the approximate eigenvalues and eigenfunctions with the smallest error in iterations to construct the deflation projection. This allows us update adaptively, ensuring that the method remains effective when calculating more eigenfunctions.
3.2.2 Filter Transform
During the iterative process, we can obtain approximate eigenvalues X, and assume the corresponding true eigenvalues lie within [N, E, E, E). where & tunable parameter, typically & = 0.1 or f 1. We employ rational dunction-based filter transform the original operator simultancously

---

(a)
(b)
Deflation Projection
D.
Filter
R
Transform
Figure Illustration of the modules' impact on the operator spectrum: (a) Deflation projection sets the solved eigenvalues to zero, (b) Filter transform enlarges the target eigenvalue region and suppresses others.
amplify the eigenvalues in these intervals and thus improve convergence performance. Specifically, we transform
C -
[C- - (C) - (7., + +)))
(10)
=0=0
By contrast, the basic power method shift-invert strategy, & (C ot) can be viewed special case of this more general construction. In STNet, we simulate the inverse operator via suitably designed loss function. Therefore, the corresponding pseudocode filter function removes the inverse, namely:
i-1
F((C) = [[ [c (7.. - (C (2i + {) 30 =
(11)
When X lies within [A; E, X3 + E), the poles X; + E make [[;()))) sufficiently large for the corresponding eigenvector vt- This repeated amplification causes that direction to dominate in the subsequent iterations, while cigenvalues outside those intervals are gradually suppressed. Conse- quently, the method converges more efficiently the desired eigenvalues.
4 Experiments
We conducted comprehensive experiments to evaluate STNet, focusing on:
Solving multiple eigenvalues in the Harmonic eigenvalue problem.
Solving the principal eigenvalue in the Schrödinger oscillator equation.
Solving zero eigenvalues in the Fokker-Planck equation.
Comparative experiment with traditional algorithms.
The ablation experiments.
Baselines: For these experiments, selected three learning-based methods for computing operator eigenvalues as our baselines: 1. PMNN [45]; 2. NeuralEF [7] 3. NeuralSVD (38). NeuralSVD and
NeuralEF were implemented using the publicly available code provided by the authors of NeuralS VD PMNN was implemented using the code provided by the authors of PMNN An introduction to related works can be found in Appendix A In the comparative experiments with traditional algorithms, chose the finite difference method (FDM) 24).
Experiment Settings: To ensure consistency, all experiments were conducted under the same com- putational conditions. For further details on the experimental environment and algorithm parameters, please refer to Appendices C.I and
4.1 Harmonic Eigenvalue Problem
Harmonic eigenvalue problems are common in fields such as structural dynamics and acoustics, and can be mathematically expressed as follows [5 33]:
Ao Au, in Ω
(12)
on 00.
Here A denotes the Laplacian operator. We consider the domain Ω = [0, 1]P where D represents the dimension of the operator, and the boundary conditions Dirichlet. In this setting, the eigenvalue

---

Table 1: Relative comparison for eigenvalues of Harmonic operators. The first row lists the methods, the second row lists eigenvalue indices, and the first column lists the operator dimensions. The most accurate method is in bold.
NeunalEF
NeumiSVD
STNet
Method A1
Ag
Ag
A
A1
Aa
Aa
Aa
A
Aa
As
Ag
Dim =1
1.98c=0
2.53e40
8.89-1
7.1e-1
1.72e-2
1.83-22
1.86-2
9.386-1
6,38c-11
861e-3
2.13e-2
4.05e-1
Dim 2
1.27e40
1.66041
1.61e42
1.3tes1
7.25e-3
7,50e-3
7826-3
7.640-3
5.07e-7
1.01e-3
2.30e-3
2.53e-3
Dim =
3.98-11
7.24c40
1.16+41
1.1e+1
2.8e-2
3.56--2
3.36-2
3.310-2
4.66e-6
1.60e-6
1.05e-6
2.20e-5
problem has analytical solutions, with cigenvalues and corresponding cigenfunctions given by:
7 n ni EN,
D
App
II sin(ng (13)
These experiments aim to calculate the smallest four eigenvalues of the Harmonic operator in 1, and dimensions. Since the PMNN model only computes the principal eigenvalue and cannot compute multiple eigenvalues simultaneously, it is not considered for comparison.
Firstly, demonstrated in Table the accuracy of STNet in most tasks is sigmificantly better than that of existing methods. This enhance- ment primarily stems from the deflation pro- jection. It effectively excludes solved invariant subspaces during the multi-eigenvalue solution process, thereby preserving the accuracy of mul- tiple eigenvalues. This strongly validates the efficacy of our algorithm.
Table 2: Residual comparison for eigenpairs for solving 5-dimensional Harmonic eigenvalue prob- lems. The first row indicates the eigenpair index.
Index
(v1, Ax)
(e2, Ax)
(ts, (s))
(Va, Aa)
NeuralSVD
1.90+00
2.60+00
2.70c+0
3,02c+0
NeuralEFF
3,45e+
2.10++1
1.83e11
4.864e-4
3.060e-3
5.980e-3
4.447e-3
Secondly, in 5-dimension, STNet consistently maintains a precision improvement at least three orders of magnitude. As shown in Table 2 this is largely due to the STNet computed eigenpairs having smaller residuals (defined as I| Aöll2, see Appendix C.3 for details), indicating that STNet can effectively solve for accurate eigenvalues and eigenfunctions simultaneously.
4.2 Schrödinger Oscillator Equation
The Schrödinger oscillator equation is a com- problem in quantum mechanics, and its time-independent form is expressed as follows:
Table 3: Relative eиrоr comparison for the principal eigenvalues of oscillator operators. The first row lists the methods, and the first column lists the operator dimensions. The most accurate method is in bold.
1 2 Av + Vib- = Evp, in Ω [0, (14)
where v is the wave function, A represents the Laplacian operator indicating the kinetic energy term, V is the potential energy within Q and E denotes the energy eigenvalue (38) [6]. This equation is formulated in natural units, simpli- fying the constants involved. Typically, the po-
Method
PMNN
NeuralEFF
NeuralSVD
STNet
4.58e+0
1.25e.3
7.24e-7
3.74e+0
5.95e
2.35e-6
Dim 55
1.570-1
1.78+40
3,32+-
1.29c-1
tential V (Z .D) Eke z characterizes multidimensional quadratic potential. The principal eigenvalue Eg and corresponding eigenfunction vo are given by:
:
D E. - 2' v'
II
(15)
This experiment focuses on calculating the ground states of the Schrödinger equation one, two, and five dimensions, i.e., the smallest principal eigenvalues.
Firstly, as shown Table the STNet achieves significantly higher precision than existing algorithms in computing the principal cigenvalues of the oscillator operator.
7

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Furthermore, the accuracy of STNet surpasses that of PMNN. Both are designed based on the concept of the power method. When solving for the principal eigenvalue, the deflation projection loss may be considered inactive. This outcome suggests that the filter transform significantly enhances the accuracy.
4.3 Fokker-Planck Equation
The Fokker-Planck equation is central to statistical mechanics and is extensively applied across diverse fields such as thermodynamics, particle physics, and financial mathematics [4] 21 12]. can be mathematically formulated follows:
D AVe Xu, in - [0, V(x) = sin Σ cs cos(s:)
(16)
Here (zx) is a potential function with each coefficient varying within [0.1, 1J. A the cigenvalue, and the eigenfunction. When the boundary conditions are periodic, there are multiple zero eigenvalues.
The eigenvalue at zero significantly impacts the numerical stability of the algorithm during iterative processes. This experiment investigates the computation of two zero eigenvalues for the Fokker- Planck equations with different parameters in 1, 2, and 5 dimensions. Due to the inherent limitation of the PMNN method, which can only compute single eigenvalue, restrict our analysis to calculating one eigenvalue when employing this approach.
Table 4: Absolute error comparison for the zero eigenvalues of Fokker-Planck operators across algorithms. The first row lists methods, the second row lists the eigenvalue index, the first column lists the Fokker-Planck parameter, and the second column lists the operator dimensions. The most accurate method bold.
Method
PMNN
NeuralEF
NeuralSVD
STNet
Dim
A1
A1
X2
A1
A2
A1
A2
1.16+00
4.98e-2
1.05e+0
7.19e-1
1.02e+0
1.17e-3
8,75e-3
0.5
2
1.1
6.71e.2
1.57e+0
3.33e-1
1.03e+0
5.26e-6
5.14e-2
1.17e+0
2.11e+0
9.17e+0
2.11es0
4.82e+0
3.90e-3
1.29c-1
8.60e-1
5.21e-1
5.95e-1
2.73e-1
3.19e-1
3.86e-2
2,33e-1
1.0
2
8.30c-1
6.58c-1
8.45c-1
2.75c-1
3.94c-1
1.99e-2
3,91e-2
5
7,58c-1
7.71e-1
1.02c40
2.01e-1
3,08c-1
5.64e-2
2.67e-2
As indicated in Table the STNet algorithm significantly outperforms existing methods in computing the zero eigenvalues of the Fokker-Plance operator, effectively solving cases where the eigenvalue is zero. mainly due to the filter function, which performs a spectral transformation on the operator, converting the zero eigenvalue into other cigenvalues that easier to calculate without changing the cigenvector.
4.4 Comparative Experiment with Traditional Algorithms
This experiment compares the accuracy of STNet and the traditional finite difference method (FDM) with central difference scheme under varying grid densities 24). Both methods aim to compute the four smallest eigenvalues (A₃ A4) of the 5D harmonic operator.
As shown in Table STNet consistently achieves higher accuracy than FDM across all eigenvalues. While FDM's precision improves with increasing grid density, this comes at the cost of exponentially higher memory consumption. For instance, as the grid points increases from 45 to 455, the relative error for X decreases from 3.20 x 10-1 to 3.82 x 10 -3 but memory usage grows from 0.0001 GB to 22.9 GB. This demonstrates the inefficiency of FDM -igh-dimensional problems, where maintaining accuracy requires impractical number of grid points.
In contrast, STNet employs uniform random sampling instead of fixed grids, enabling achieve superior accuracy with fewer parameters and lower memory requirements. For example, with grid

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Table 5: Relative comparison for eigenvalues of 5D Harmonic operators. The first column lists the methods, the second column lists grid points, and the third column lists memory consumption (in GB). Columns X1 to 2 list the eigenvalue relative errors.
Method
Grid Points
Memory (GB)
A
A2
A
A
46
1.0le-4
3,20e-1
1.04e+0
1.04++0
1.04++0
7
1,86c-3
1.26c-1
4.15c-1
4.15e-1
4.15c-1
g
6.74e-3
8.10e-2
2.68e-1
2.68e-1
2.68e-1
FDM
155
9.05e-2
3.17e-2
1.05e-1
1.05e-1
1.05e-1
255
1.19e40
1.20e-2
4.00e-2
4.00e-2
4.00e-2
356
6.48e+0
6.26e-3
2.09e-2
2.09e-2
2.09e-2
455
2.29e+1
3.82e-3
1.27e-2
1,27e-2
1.28e-2
ST
99
1.14c+0
4.62e-5
1.59e-5
1.04e-5
2.32e-4
density of 9", STNet achieves relative errors of 2.32 x 10 for X4. outperforming FDM by orders of magnitude. Its memory usage remains relatively low 1.14 GB, highlighting its scalability and efficiency in high-dimensional eigenvalue problems. By leveraging the expressive power of neural networks, STNet effectively approximates eigenfunctions without relying dense grids. This makes it promising alternative to traditional numerical methods for solving high-dimensional operators. A more detailed comparison with traditional algorithms is provided in Appendix D.4
Traditional algorithms and neural network- -ased methods each excel in different scenarios. In low-dimensional problems, traditional algorithms faster and can improve accuracy by increasing grid density. However, in high-dimensional problems, the number of required grid points grows exponentially with dimensionality, making grid-based methods impractical. For example, while a 2D problem requires 1002 grid points, a 5D problem requires 100 grid points. Neural network- based algorithms, such as STNet, offer an effective solution to these challenges, providing high accuracy without the need for dense grids.
4.5 Ablation Experiments
We conducted ablation experiments to validate the ef- fectiveness the deflation projection and filter trans- form modules. As shown in Table the results for "w/o F" indicate that removing the filter transform significantly reduces solution accuracy. In the cases of "w/o D" and "w/o F and D," " while the residuals remain small, the absolute errors for d2 and Åg notably larger compared to A₁- This suggests that without the deflation projection module, the network converges exclusively to the first cigenfunction i1 cor- responding Å₁ failing capture subsequent eigen- functions. These findings underscore the critical roles of both modules: the filter transform enhances accu- racy through spectral transformation. The deflation projection removes the subspace of already solved eigenfunctions from the search space, enabling the computation of multiple eigenvalues.
Table A comparison of different settings of STNet for the 2-dimensional Harmonic eigen- value problem. "w/o" denotes the absence of a specific module, "F" represents the filter transform module, and "D" indicates the de- flation projection module.
Index
A Absolute Error
Residual
(#1: A1)
1.02e-5
4.12c-3
STNet
(V2, Xa)
3.04e-2
1.24ct
(es, Xs)
6.76e
1.43et]
(ei, A1)
6.73e-5
1.35e-2
F
(#2; Aa)
5.10e-2
4.72cs1
(Vs, As)
1.06e-1
1.70442
(#1, A1)
1.42e-5
4.12e-3
(#2; Az)
7.09e-
(va, Aa)
2.97c+1
1.09c-2
Additionally, experiments detailing the performance of STNet function of model depth, model width, and the number of sampling points are provided in Appendix D.I] Runtimes and convergence processes for selected experiments are presented in Appen- dices D.2 and D33
(01: A1)
6.73e-5
1.35e
(e2, Xa)
2,96++1
1.45e-2
(vg. Aa)

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Limitations and Conclusions
In this paper, propose STNet, learning- approach for solving operator eigenvalue problems. By leveraging approximate eigenvalues and eigenvectors from iterative processes, STNet uses spectral transformations to reformulate the operator, enhancing convergence properties. Experiments show that STNet outperforms existing deep learning methods, achieving seee-of-theearr accuracy.
While STNet shows strong performance solving operator eigenvalue problems, several limitations and avenues for future exploration remain: Although STNet utilizes spectral transformations, the potential benefits of broader matrix preconditioning techniques have not been investigated. Its current scope is limited to linear operators, with future work needed to address nonlinear eigenvalue problems.
Acknowledgements
The authors would like to thank all the anonymous reviewers for their insightful comments and valuable suggestions. This work was supported by the National Key R&D Program of China under contract 2022ZD01 19801, and the National Nature Science Foundations of China grants U23A20388, 62021001.
10

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References
[1] Amartya S Banerjee, Lin Lin, Wei Hu, Chao Yang, and John E Pask. Chebyshev polynomial filtered subspace iteration in the discontinuous galerkin method for large-scale electronic structure calculations. The Journal of chemical physics, 145(15), 2016.
[2] Ido Ben-Shaul, Leah Bar, Dalia Fishelov, and Nir Sochen. Deep learning solution of the eigenvalue problem for differential operators. Neural Computation, 35(6): 100-1134, 2023.
[3] Ido Ben-Shaul, Leah Bar, and Nir Sochen. Solving the functional cigen-problem using neural networks. arXiv preprint wwxiv:2007.10266 2020.
[4] AG Buchan, CC Pain, F Fang, and IM Navon. A pod reduced-order model for eigenvalue problems with application to reactor physics. International Journal for Numerical Methods in Engineering, 95(12):1011-000)
2013.
[5] Q Chen and YW Chan. Integral finite element method for dynamical analysis of elastic- viscoelastic composite structures. Computers & Structures, 74(1):51-6)) 2000.
[6] Alice Cuzzocrea, Anthony Scemama, Wim Bricls. Saverio Moroni, and Claudia Filippi. Variational principles in quantum monte carlo: The troubled story of variance minimization. Journal of chemical theory and computation, 16(7):4203-427) 2020.
[7] Zhijie Deng. Jiaxin Shi, and Jun Zhu. Neuralef: Deconstructing kernels by deep neural networks. In International Conference on Machine Learning, pages 4976-4992. PMLR, 2022.
[8] Huaian Diao, Hongjie Li, Hongyu Liu, and Jiexin Tang. Spectral properties of an acoustic- elastic transmission eigenvalue problem with applications. Journal of Differential Equations, 371:629-633 659, 2023.
[9] Huanshuo Dong, Hong Wang, Haoyang Liu, Jian Luo, and Jie Wang. Accelerating pdc data generation via differential operator action in solution space. arXiv preprint arXiv:240..05957, 2024.
[10] Mohamed H Elhareef and Zeyun Wu. Physics-informed neural network method and application to nuclear reactor calculations: A pilot study. Nuclear Science and Engineering, 197(4):601-622 2023.
[11] Lawrence Evans. Partial differential equations, volume 19. American Mathematical Society, 2022.
[12] Till Daniel Frank. Nonlinear Fokker-Planck equations: fundamenials and applications. Springer Science Business Media, 2005.
[13] Wenhan Gao and Chunmei Wang. Active learning based sampling for high- high-dimensional nonlinear partial differential equations. J. Comput. Phys., 475:111848, February 2023.
[14] Wenhan Gao, Ruichen Xu, Yuefan Deng, and Yi Liu. Discretization-inaaaanne?? on the discretization mismatch errors in neural operators. The Thirteenth International Conference on Learning Representations, 2025.
[15] Gene H Golub and Charles F Van Loan. Matrix computations. JHU press, 2013.
[16] David J Griffiths and Darrell Schroeter. Introduction to quantum mechanics. Cambridge university press, 2018
[17] Jiequn Han, Jianfeng Lu, and Mo Zhou. Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion monte carlo like approach. Journal of Computational Physics, 423:109792, 2020.
[18] Liu Haoyang, Wang Jie Cai Yuyang. Han Xiongwei, Kuang Yufei, and HAO Jianye. Optitree: Hierarchical thoughts generation with tree search for LLM optimization modeling. In The ninhh Annual Conference on Neural Information Processing Systems, 2025.
11

---

[19] Cuiyu He, Xiaozhe Hu, and Lin Mu. A mesh-free method using piecewise deep neural network for elliptic interface problems. Journal of Computational and Applied Mathematics, 412:1 14358, 2022.
[20] Henry Jin, Marios Mattheakis, and Pavlos Protopapas. Physics-informed neural networks for quantum eigenvalue problems. 2022 International Joint Conference on Neural Networks (IJCNN), pages 1-8. IEEE, 2022.
[21] Richard Jordan, David Kinderlehrer, and Felix Otto. The variational formulation of the fokker- planck equation. SIAM journal on mathematical analysis, 29(1):1-17, 1998.
[22] Walter Kohn. Nobel lecture: Electronic structure of matter-wave functions and density functionals. Reviews of Modern Physics, 71(5):1253, 1999.
[23] Yufei Kuang, Jie Wang, Haoyang Liu, Fangzhou Zhu, Xijun Li, Jia Zeng, HAO Jianye, Bin Li, and Feng Wu. Rethinking branching on exact combinatorial optimization solver: The first deep symbolic discovery framework. In The Twelfth International Conference on Learning Representations, 2024.
[24] Randall J Vequee Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems. SIAM, 2007.
[25] Xijun Li, Fangzhou Zhu, Hui-Ling Zhen, Weilin Luo, Meng Lu, Yimin Huang, Zhenan Fan, Zirui Zhou, Yufei Kuang, Zhihai Wang, et al. Machine learning insides optverse ai solver: Design principles and applications. arXiv preprint arXiv:2401. 005600, 2024.
[26] Haoyang Liu, Yufei Kuang, Jie Wang, Xijun Li, Yongdong Zhang, and Feng Wu, Promoting generalization for exact solvers via adversarial instance augmentation, 2023.
[27] Haoyang Liu, Jie Wang. Zijie Geng, Xijun Li, Yuxuan Zong, Fangzhou Zhu, Jianye HAO, and Feng Wu. Apollo-MILP: An alternating ceddiction-correcttio neural solving framework for mixed-integer linear programming. In The Thirteenth International Conference on Learning Representations, 2025.
[28] Haoyang Liu, Jie Wang, Wanbo Zhang. Zijie Geng, Yufci Kuang, Xijun Li, Bin Li, Yongdong Zhang, and Feng Wu. MILP-studio: MILP instance generation via block structure decompo- sition. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024.
[29] Hongyu Liu, Haoyang Liu, Yufei Kuang. Jie Wang, and Bin Li. Deep symbolic optimization for combinatorial optimization: Accelerating node selection by discovering potential heuristics. In Proceedings of the Genetic and Evolutionary Computation Conference Companion, pages 2067-2075, 2024.
[30] Jian Luo, Jie Wang, Hong Wang, Zijie Geng, Hanzhu Chen, Yufei Kuang, et al. Neural krylov iteration for accelerating linear system solving. Advances in Neural Information Processing Systems, 37:128636-128667 2024.
[31] Qitan Lv, Tianyu Liu, and Hong Wang. Exploiting edited large language models general scientific optimizers. arXiv preprint arXiv:2503.09622 2025.
[32] Cun-Qiang Miao and Wen-Ting Wu. On relaxed filtered krylov subspace method for non-symmetric eigenvalue problems. Journal of Computational and Applied Mathematics, 398: 11 13698, 2021.
[33] Ronald B Morgan and Min Zeng. Harmonic projection methods for large non-symmetric eigenvalue problems. Numerical linear algebra with applications, 5(1)333-55 1998.
[34] Thao Nguyen, Maithra Raghu, and Simon Kornblith. Do wide and deep networks learn the same things? uncovering how neural network representations vary with width and depth. arXiv preprint arXiv:2010.1 15327, 2020.
[35] David Pfau, Simon Axelrod, Halvard Sutterud, Ingrid von Glehn, and James Spencer. Natural quantum monte carlo computation of excited states. arXiv preprint arXiv:2308. 16848, 2023.
12

---

[36] David Pfau, Stig Petersen, Ashish Agarwal, David GT Barrett, and Kimberly L Stachen- feld. Spectral inference networks: Unifying deep and spectral learning. arXiv preprint arXiv: 1806.022155 2018.
[37] Tianle Pu, Zijie Geng, Haoyang Liu, Shixuan Liu, Jie Wang. Li Zeng, Chao Chen, and Changjun Fan. RoME: Domain-robust mixture-of-expere for MILP solution prediction across domains.
In The Thirty-ninth Annual Conference on Neural Information Processing Systems, 2025.
[38] J Jon Ryu, Xiangxiang Xu, HS Erol, Yuheng Bu, Lizhong Zheng, and Gregory W Wor- nell. Operator svd with neural networks via nested low-rank approximation. arXiv preprint arXiv:2402.03655, 2024.
[39] Yousef Saad. Numerical methods for large eigenvalue problems: revised edition. SIAM, 2011.
[40] Pablo Salas, Luc Giraud, Yousef Saad, and Stéphane Moreau. Spectral recycling strategies for the solution of nonlinear eigenproblems in thermoacoustics. Numerical Linear Algebra with Applications, 22(6):1039-1033 2015.
[41] Roel Van Becumen. Rational krylov methods for nonlinear eigenvalue problems, 2015.
[42] Hong Wang. Zhongkai Hao, Jie Wang. Zijie Geng, Zhen Wang, Bin Li, and Feng Wu. Ac- celerating data generation for neural operators via krylov subspace recycling. arXiv preprint arXiv:2401.09516 2024.
[43] David S Watkins. The matrix eigenvalue problem: GR and Krylov subspace methods. SIAM, 2007.
[44] Jan Winkelmann, Paul Springer, and Edoardo Di Napoli. Chase: Chebyshev accelerated sub- space iteration eigensolver for sequences of hermitian eigenvalue problems. ACM Transactions on Mathematical Software (TOMS), 45(2):1-34, 2019.
[45] Qihong Yang. Yangtao Deng. Yu Yang, Qiaolin Hc, and Shiquan Zhang. Neural networks based power method and inverse power method for solving linear eigenvalue problems. Computers & Mathematics with Applications, 147:14-2 24, 2023.
[46] Zherui Yang, Zhehao Li, Kangbo Lyu, Yixuan Li, Tao Du, and Ligang Liu. Learning sparse approximate inverse preconditioners for conjugate gradient solvers gpus. In Advances in Neural Information Processing Systems, 2025.
[47] Bing Yu et al. The deep ritz method: deep dearning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics, 6(1):1-12, 2018.
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A Related work
Recent advancements in applying neural networks to eigenvalue problems have shown promising
31 Innovations such spectral inference networks which model eigenvalue problems as kernel problem optimizations solved via neural networks. Neural eigenfunctions (NeuralEF) 77. which significantly reduce computational costs by optimizing the costly orthogonalization steps, are noteworthy. Neural singular value decomposition (NeuralSVD) employs truncated singular value decomposition for low-rank approximation to enhance the orthogonality required in learning functions (38].
results 42
30
26
27
28
37
23
29
25
18
(SpIN)
[3
Another class of algorithms originates from optimizing the Rayleigh quotient. The deep Ritz method (DRM) utilizes the Rayleigh quotient for computing the smallest cigenvalues, demonstrating significant potential [47]. Several studies have employed the Rayleigh quotient to construct variation- free functions, achieved through physics-informed neural networks (PINNs) 3 Extensions of this approach include enhanced loss functions with regularization terms to improve the learning accuracy of the smallest eigenvalues [20]. Additionally,
[17] reformulates the eigenvalue problem as fixed-point problem of the semigroup flow induced by the operator, solving it using the diffusion Monte Carlo method. The power method neural network (PMNN) integrates the power method with PINNs, using an iterative process to approximate the exact eigenvalues [45] closely. While PMNN has proven effective in solving for a single eigenvalue [45], it has yet to be developed for computing multiple distinct eigenvalues simultancously.
B Background Knowledge and Relevant Analysis
B.1 Convergence Analysis of the Power Method
Suppose A _xx= and V-1 AV diag(A1..... An) with V [en] [A A2l 2 ... Anl. The pscudocode for the power method is shown below [5]
Un). Assume that
Algorithm 1: Power method for finding the largest principal eigenvalue of the matrix
Given A E I an X n matrix, an arbitrary unit vector (0) E IR",, the maximum number of iterations kiax, and the stopping criterion e.
2 for k = 1, 2,.... kimox do Compute y (k) = (x-1-
Normalize (A) (4)
Compute the difference = (|x (x
5 < then
Record the largest principal eigenvalue using the Rayleigh quotient.
( (R), (a)(
(x))), x (A))
The stopping criterion is met, and the iteration be stopped.
Let us examine the convergence properties of the power iteration. If
x (0)
aili + a2 + + an Un
and 01 * then
A anti
Vi
a1
A
2=2
Since (k) E spaan{ ( we conclude that
((22) A
dist (span(x (*) swwnspanu span(u1)) =
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It is also easy to verify that
1 - 100 A2
Since A1 is larger than all the other eigenvalues in modulus, it is referred to as the largest principal eigenvalue. Thus, the power method converges if Y is the largest principal and (0 has component in the direction of the corresponding dominant eigenvector ø1.
In practice, the effectiveness of the power method largely depends on the ratio / as this ratio
determines the convergence rate. Therefore, applying specific spectral transformations to the matrix increase this ratio can significantly accelerate the convergence of the power method.
B.2 Deflation Projection Details
Consider the scenario where we have determined the largest modulus eigenvalue, A1 and its cor- responding eigenvector, vi, utilizing an algorithm such the power method. These algorithms consistently identify the eigenvalue of the largest modulus from the given matrix along with an associated eigenvector. We ensure that the vector V1 is normalized such that ||v/||2 1. The task then becomes computing the subsequent eigenvalue, Xg. of the matrix A. A traditional approach to address this is through what is commonly known as a deflation procedure. This technique involves rank-one modification to the original matrix, aimed at shifting the eigenvalue A while preserving all other eigenvalues intact. The modification is designed in such a way that X2 emerges as the eigenvalue with the largest modulus in the adjusted matrix. Consequently, the power method can be reapplied this updated matrix extract the eggevvalueeeigeveetto pair X2, U22
When the invariant subspace requiring deflation is one-dimensional, consider the following Proposi- tion [.I] The propositions and proofs below are derived from [39] P90.
Proposition B.1. Let V1 be an eigenvector of A of norm I, associated with the eigenvalue \ and let A = - -o off. Then the eigenvalues of A₁ are À X - a and X3 - J = 2, 3, ........ Moreover, the Schur vectors associated with X = 1, 2, ,n. are identical with those of A.
Proof. Let AV VR be the Schur factorization of A, where R is upper triangular and V is orthonormal. Then we have
A Here, e the first standard basis vector. The result follows immediately.
o V = VR duieo = - -ee-
According to Proposition B.I once the eigenvalue λ and eigenvector tn are known, we can define the deflation projection matrix P. I T-A10101 to compute the remaining eigenvalues and eigenvectors.
When deflating with multiple vectors, let qi ......... Qi be a set of Schur vectors associated with .......... qj- Thus, the eigenvalues A2, A2...., ......,,,, >j. We denote by Q, the matrix of column vectors Qi, 92....,
Q2 = (1, 92. 4,] is an orthonormal matrix whose columns form basis of the eigenspace associated with the eigenvalues λ1, .2,..., An immediate generalization of Proposition B.l is the following [39] P94.
Proposition B.2. Let Ej be the x j diagonal matrix Σ, diag(a o2..... Gg). and Q2 n x orthogonal matrix consisting of the Schur vectors of associated with A1...., eigenvalues of the matrix
...... Ag. Then the
A, = A Q
are X; = = for j and S > for i > j. Moreover, its associated Schur vectors are identical with those of A.
Proof. Let AU = UR be the Schur factorization of A. We have
A₁U = [A Q, UR - E,
where E, = [e1, E2,
e,]. Hence
AU = U E,
and the result follows.
22

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According to Proposition B.2. if A is normal matrix and the eigenvalues A1..... along with their corresponding eigenvectors vi.... ..... Vj are known, we can construct the deflation projection matrix P V, V to compute the remaining eigenvalues and eigenvectors. Here, Σ; diag( 01 .2....,.7....................... oT) and V = U1, 02 U2..... Uy,,,,,
B.3 Filtering Technique
The primary objective of filtering techniques to manipulate the eigenvalue distribution of matrix through spectral transformations [39] This enhances specific eigenvalues of interest, facilitating their recognition and computation by iterative solvers. Filter transformation functions, F(x). typically fall into two categories:
Polynomial Filters, expressed as P(x), such as the Chebyshev filter [32
2. Rational Function Filters, often denoted as P(x))/(zz), such as the shift-inver method 41 43) Below, we describe this strategy in detail.
Shift-Invert Strategy The shift-invert strategy applies the transformation (A - oI) to the matrix A, where a is scalar approximating a target eigenvalue, termed as shift. This operation transforms each eigenvalue Ả of A into x-o amplifying those eigenvalues close in the transformed matrix, making them larger and more distinguishable [4).
For instance, consider the power method, where the convergence rate primarily governed by the ratio of the matrix's 's largest modulus eigenvalue to its second largest. Suppose matrix A has three principal eigenvalues: Å 10, X2 and λ Our objective is compute X1, the largest eigenvalue. In the original matrix A, the convergence rate of the power method hinges on the spectral gap ratio, defined as:
Spectral Gap Ratio
= 3.33
A2
Applying the
transformation with 9.5 strategically selected close to Å1, the new
eigenvalues f are recalculated as:
1
X
This results in transformed eigenvalues:
P1 P2 0.133, Mg 00.225
Under this transformation, μl emerges as the dominant eigenvalue in the new matrix, with the other eigenvalues significantly smaller. Consequently. the new spectral gap ratio escalates to:
New Spectral Gap Ratio 0.133
15.04
This enhanced spectral gap notably accelerates the convergence of the power method in the new matrix configuration.
Filtering techniques are often synergized with techniques like the implicit restarts of Krylov al- gorithms [43] 15] (66), employing matrix operation optimizations to minimize the computational demands of evaluating matrix functions. These methods enable more precise localization and compu- tation of multiple eigenvalues spread across the spectral range, particularly vital in physical [40 and materials science [22] simulations where these eigenvalues frequently correlate with the system's fundamental properties
C Details of Experimental Setup
C.1 Experimental Environment
To ensure consistency in our evaluations, all comparative experiments were conducted under uniform computing environments. Specifically, the environments used are detailed follows:
CPU: 72 vCPU AMD EPYC 9754 128-Core Processor
GPU: NVIDIA GeForce RTX 4090D (24GB)
23

---

C.2 Experimental Parameters
Ve and NeuralEF were implemented using the publicly available code provided by the au- thors of NeuralSVD (https://githu) com/ jongharyu/noural-svea PMNN was implemented using the code provided by the authors of PMNN (attps://github cwt.coom///aaa /ut.meeeeooovaiii///// PMNN IPMNN). No modifications were made to their original code for the experiments, except for changes the target operator to be solved. All experiments are fully reproducible. The detailed baseline experimental parameter files and code are provided in the supplementary materials.
NeuralSVD and NeuralEF: (Using the original paper settings)
Optimizer: RMSProp with learning rate scheduler.
Learning rate: le-4, batch size: 128
Neural Network Architecture: layers = [128, 128, 128]
Laplacian regularization set to 0.01, with evaluation frequency every 10000 iterations.
Fourier feature mapping enabled with size of 256.
Neural network structure: hidden layers of 128,128, using softplus the activation function.
PMNN and STNet
Optimizer: Adam
Learning rate: le-4
Neural Network Architecture: Assuming d is the dimension of the operator. For d = or 2, layers [d, 20, 20, 20, 20, 1]. For d=5, layers [d, 40, 40, 40, 40, 1]. For the else cases, layers = [d, 40, 40, 40, 40, 1].
For the I-dimensional problem, the number of points is 20, 000, with 400, 000 iterations. For the 2-dimensional problem, the number of points is 40, 000 = 200 200, also with 400, 000 iterations. For the 5-dimensional problem, the number of points is 59, 0149 95, with 500, 000 iterations.
C.3 Error Metrics
Absolute Error:
We employ absolute error to estimate the bias of the output eigenvalues of the model:
Absolute Error = | A/-
Here À represents the eigenvalue predicted by the model, while À denotes the true eigenvalue.
Relative Error:
We employ relative error to estimate the bias of the output eigenvalues of the model:
[ - A
Relative Error
|A]
Here Ā represents the eigenvalue predicted by the model, while denotes the true eigenvalue.
Residual Error:
To further analyze the error in eigenpair (7, Å) predictions, we use the following metric:
Residual Error | - -xol22-
Here, represents the eigenfunction predicted by the model. When À is the true eigenvalue and is the true eigenfunction, the Residual Error equals 0.
24

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D Supplementary Experiments
D.1 Analysis of Hyperparameters
Model Depth:
Table 7: Consider the 2-dimensional Harmonic problem, with the fixed layer width of 20, and compare the performance of STNet at different model layers. Other experimental details are the same Appendix C.2
The error metric is absolute error.
Eigenvalue
Layer
Layer
Layer
Layer 6
À
1.02e-5
1.42e-5
4.36e-6
1.06e-5
A2
3,04c-2
2.96e-1
8.63e-1
8.21e-1
A
6.76e-2
4.17e-1
1.98+40
1.17e+0
A
1.00e-1
2.00e+1
8.94c+1
3,81e+1
Model Width:
Table 8: Consider the 2-dimensional Harmonic problem, with the fixed layer depth of 3, and compare the performance of STNet at different model widths. Other experimental details are the same as Appendix C.2.
The error metric is absolute error.
Eigenvalue
Width 10
Width 20
Width 30
Width 40
A
1
1.42e-5
3.26e-5
1.57e-5
A2
3.82e-1
2.96e-1
1,50e+0
2.67e+0
A
7.54e-1
4.17e-1
1.59e+0
7.93e+1
1.71e-1
2.00c+1
3.52c+2
1.50e+2
The Number of Points:
Table Consider the 2-dimensional Harmonic problem and compare the performance of STNet at different numbers of points. Other experimental details the same Appendix C.2 metric absolute error.
The error
Eigenvalue 20000 Points
30000 Points
40000 Points
50000 Points
A
1.1le-5
4.40e-5
1.42e-5
4.94e-6
Aa
1.25e+0
3.58e-1
2.96e-1
2.53e-1
A
1.61e40
1.70e-1
4.17e-1
3.73e-1
The influence of model depth, model width, and the number of points on STNet is illustrated in Tables 7 and 9 respectively. Experimental results indicate that STNet is relatively unaffected by changes in model depth and model width. However, it is significantly influenced by the number of points, with performance improving as more points are used.
25

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D.2 Computational Complexity and Runtimes
While neural network-basse based methods may exhibit larger computational overhead than traditional methods in now-dimessional problems, their strength lies in high-dimensional settings where they offer significant precision and computational cost advantages. To illustrate this, we provide runtime comparison for the 5D Harmonic problem in Table TO
Table 10: Runtime comparison for the 5D Harmonic problem. "STNet (partial)" refers the time taken to achieve error of 1.00 x 10-3
Method
Computation Time (s)
Principal Eigenvalue Absolute Error
FDM (459 points)
1.9e+4
3.8e-3
STNet (partial)
1.3e+3
I 1.0e-3
STNet (converged)
3.1e+4
4.6e-5
NeuralEF (converged)
1.2c+4
3,9e-1
NeuralSVD (converged)
1.2c+4
2.8c-2
The results in Table 10 highlight two key points:
1. STNet achieves lower error more rapidly than other learningbbased methods and converges to significantly more accurate solution than NeuralEF and NeuralSVD.
2. The total time to full convergence for STNet is longer. This is an inherent trade-off of the Filter Transform, which iteratively refines the problem unlock further optimization, thus requiring more iterations to reach its high-precision potential.
D.3 Partial Convergence Process
we provide the convergence process for the Harmonic problem in Table II The table shows the absolute error for the first two eigenvalues (A1, A2) at different iteration counts across dimensions 1, and 5.
Table 11: Absolute Error vs. Iterations for the Harmonic problem. convergence for that eigenvalue already achieved in prior stage.
Empty cells indicate that
Dim Eigenvalue
Iter: 100
500
1000
40000
80000
130000
180000
A
1.6e-2
3.5e-3
3.2e-4
2.6e-4
2.7e-5
1.2e-3 8.8e-3
d=1
A2
1.0e+4
2.4e+3
2.8e+3
8.0e+2
9.8e-1
A1
6.9e-2
6.9e-3
5.9e-3
5.3e-6
d=2
X2
6.3e01
5.9e+0
4.2e+0
5.1e-2
A1
5.8e-2
6.6e-3
4.8c-4
7.8c-3
3,9c-3
6.2c-3
3,9e-3
d=5
X2
6.3c+3
4.1e+2
1.5e+2
2.9c+0
5.5e+0
3.8c+0
1.3e-1
D.4
Impact of Discretization on Traditional Methods
critical point of comparison is understanding the primary sources of error in different methodolo- gies. For traditional methods like the Finite Difference Method (FDM), the dominant bottleneck in high-dimensional problems not the algebraic eigenvalue solver's accuracy but the fundamental discretization eтror introduced when mapping a continuous operator onto discrete grid. While advanced solver techniques such as deflation or filtering can improve the precision the matrix eigen- value solution, they cannot overcome the inherent error from the initial discretization. Reducing this error necessitates a denser grid, which leads to an exponential increase memory and computational cost, a challenge known the "curse of dimensionality."
To empirically validate this, we conducted a experiment for the 5D Harmonic problem, comparing STNet against FDM enhanced with Richardson extrapolation- technique designed to improve
26

---

accuracy. The results, presented in Table 12 show that even when FDM is augmented with advanced techniques, it cannot match the accuracy of STNet.
Table 12: Comparison with advanced FDM techniques for the 5D Harmonic problem.
Method
Grid/Points
Memory (GB)
A Error
Ag Brror
Às Error
A Erro
FDM (Richardson Extrap.)
25
3.4e+0
8.6e-3
2.9e-2
2.9e-2
2.9e-2
FDM (with deflation & filtering: Richardson Extrap.)
25
3.4e+0
8.6e-3
2.9e-2
2.9e-2
2.9c-2
FDM (Central Difference)
452
2.2c+1
3.8e-3
1.2e22
1.2o-2
1.2c-2
FDM (with deflation & filtering; Central Difference)
45a
2.2c+1
3.8e-3
1.2e-2
1.2e-2
1.2e-2
STNet
g
1.le+0
4.6e-5
1.6e-5
1.0c-5
2.3c-4
This experiment reinforces that neural network-based methods, which rely mesh-free random sampling, are inherently better suited for high-dimensional problems, as they effectively bypass the discretization step that limits traditional grid-based approaches.
While traditional methods are often superior for low-dimensional problems, their practicality dimin- ishes rapidly in high dimensions due to the "curse of dimensionality". To provide more granular analysis, Table [3) details the convergence time versus accuracy for the 5D Harmonic problem. Our traditional baseline uses 'scipy.eigs' which is based on the highly-optimizer ARPACK library. The data clearly shows that FDM's ' accuracy capped by the grid resolution, and STNet can achieve higher precision. The total time for STNet is longer, but it reaches level of accuracy that is infeasible for FDM under reasonable memory constraints.
Table 13: Detailed convergence analysis for the 5D Harmonic problem (A1). The table shows the time required to reach specific absolute error thresholds. A dash (-) indicates that the method could not reach the specified error due to limitations imposed by grid resolution.
Method
Grid'Sample
Final
Ies0
Emor
Time
Time
EDMreips
26 Tes7)
1.2c40
1.2:-2
2.5c+1s
8.lesls
Se+7)
2e18)
9.4e22
6.7en3s
1.9e44
1.9esss
3.4e-1
7.2s
1/Rexls
1.30xts
6e+4)
1.er0
1.9c11
1.5en3s
1.3ex3s
1.5e00
9.5e=2s
This detailed analysis validates our conclusion that for large-scale, high-dimensiona problems, STNet holds significant advantage both achievable accuracy and memory efficiency. Its core strength lies in its mesh-free random sampling strategy, which leverages the expressive power of neural networks to approximate eigenfunctions directly, thereby circumventing the need to construct and store massive matrix.
27


Reviews:
[{"review_id": "Rz9ppgQJKo", "reviewer": "Reviewer_z6YH", "summary": "The 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.", "questions": "see above", "limitations": "see above", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": true, "final_justification": "The authors have answered most of my concerns in the rebuttal including running time, architecture, issues on convergennce etc. I expect to put these new material in the final version. Thus I believe the work can be accepted.", "rebuttal": "Dear Reviewer,\n\nThank you for your thoughtful and constructive feedback on our paper. We are grateful for the time you have taken to review our work and for your valuable suggestions, which have helped us to identify areas for improvement. We have carefully considered all your comments and provide the following clarifications, which we hope will address your concerns. We believe these explanations will further strengthen our paper, and we hope you will consider raising your score.\n\n---\n\n### **On Network Architecture and Ablation Studies (Weaknesses 1)**\n\nAs noted on page 5, line 108 of our paper, the neural network modules employed in our work are all Multi-Layer Perceptrons (MLPs). The core contribution of STNet lies not in a novel network architecture, but in the use of Deflation Projection and Filter Transform to modify the loss function and the optimization process. For this reason, and to ensure a fair comparison with baseline methods, we adopted the standard MLP architecture used by NeuralSVD and PMNN.\n\nTo analyze the impact of the architecture, we conducted ablation studies on the depth and width of the MLP, with the results presented in the appendix on page 16 (Tables 7 and 8). We agree that designing a network architecture more specifically suited to this task is a promising direction for future research.\n\n### **On the Dependency on Initial Eigenvalue Estimates (Weaknesses 2)**\n\nSTNet does not critically depend on a precise initial estimation of the eigenvalues. A key feature of our method is that the Filter Transform module dynamically adjusts its filter function based on the approximate eigenvalues computed by the network at each iteration. This adaptive process progressively refines the search and is the primary reason STNet achieves higher accuracy than other algorithms. In all our experiments, we used broad, non-specific initial ranges such as [-10, 10] or [-100, 100], demonstrating the method's robustness.\n\n### **On Computational Complexity and Runtimes (Weaknesses 3 & 4)**\n\nWe appreciate your concern regarding computational complexity. As detailed on page 6, lines 144-151, our method circumvents the high cost of direct matrix inversion. We simulate the inverse power iteration process by reformulating the loss function, thus avoiding explicit and computationally demanding matrix inversions.\n\nWhile it is true that neural network-based methods can have a larger computational overhead than traditional methods in low-dimensional problems (as mentioned on page 8, line 248), they offer a distinct precision advantage in high-dimensional settings. We provide a runtime comparison for the 5D Harmonic problem below:\n\n| Method                | Computation Time | Principal Eigenvalue Absolute Error |\n| --------------------- | ---------------- | ----------------------------------- |\n| FDM (45^5 points)     | ~1.9e+04 s        | 3.8e-03                            |\n| STNet (partial)       | ~1.3e+03 s        | 1.0e-03                            |\n| STNet (converged)     | ~3.1e+04 s        | 4.6e-05                            |\n| NeuralEF (converged)  | ~1.2e+04 s        | 3.9e-01                            |\n| NeuralSVD (converged) | ~1.2e+04 s        | 2.8e-02                            |\n\nThis comparison highlights two points:\n\n1. STNet achieves a lower error more rapidly and converges to a significantly more accurate solution than NeuralEF and NeuralSVD.\n2. The final convergence time for STNet is longer. This is an inherent trade-off of the Filter Transform, which iteratively refines the problem to unlock further optimization, thus requiring more iterations to reach its high-precision potential.\n\nWe will include detailed runtime records for all experiments in the final version of the paper.\n\n### **On the Rate of** **Convergence** **(Weaknesses 5)**\n\nWe thank you for this suggestion. A full theoretical analysis of the convergence rate is indeed challenging. To provide empirical evidence, we present the convergence process for the Harmonic problem below. The empty cells indicate that convergence for that eigenvalue has already been achieved.\n\nAbsolute Error vs. Iterations for Harmonic Problem\n\n| Dim  | Eigenvalue Absolute Error | Iter:  100 | 500     | 1,000   | 40,000  | 80,000  | 130,000 | 180,000 |\n| ---- | ------------------------- | ---------- | ------- | ------- | ------- | ------- | ------- | ------- |\n| d=1  |  λ1                        | 1.6e-02    | 3.5e-03 | 3.2e-04 | 2.6e-04 | 2.7e-05 | 1.2e-03 |         |\n|      | λ2                        | 1.0e+04    | 2.4e+03 | 2.8e+03 | 8.0e+02 | 9.8e-01 | 8.8e-03 |         |\n| d=2  | λ1                        | 6.9e-02    | 6.9e-03 | 5.9e-03 | 5.3e-06 |         |         |         |\n|      | λ2                        | 6.3e+01    | 5.9e+00 | 4.2e+00 | 5.1e-02 |         |         |         |\n| d=5  | λ1                        | 5.8e-02    | 6.6e-03 | 4.8e-04 | 7.8e-03 | 3.9e-03 | 6.2e-03 | 3.9e-03 |\n|      | λ2                        | 6.3e+03    | 4.1e+02 | 1.5e+02 | 2.9e+00 | 5.5e+00 | 3.8e+00 | 1.3e-01 |\n\nGiven the space constraints of the rebuttal, we will add comprehensive convergence curves (error vs. iterations and time) for all algorithms in the final manuscript.\n\n### **On** **Grid** **Resolution Analysis (Weaknesses 6)**\n\nIn our main experiments, we fixed the number of sampling points to ensure a fair and direct comparison between methods, with the specific parameters detailed in Appendix C.2 on page 15.\n\nTo analyze the effect of this hyperparameter on STNet, we performed a dedicated analysis by varying the number of sampling points, as presented on page 16, line 511 (Table 9). The results in Table 9 show that as long as the number of sampling points is not set too low, STNet maintains its strong performance, demonstrating its robustness to this parameter.\n\n### **On Updating the Related Work Section (Weaknesses 7)**\n\nThank you for this valuable suggestion. We have conducted a further literature search, including a review of recent citations for key papers like PMNN and NeuralEF. Our search identified several recent papers that focus on optimizations for specific eigenvalue problem scenarios. However, we did not find new works proposing general-purpose methods for computing differential operator eigenvalues that would constitute direct baselines. We will gladly incorporate these newly found, relevant papers into the related work section in the final version to provide a more current context.\n\n---\n\n### **Thanks again** \n\nWe sincerely thank you again for your constructive and detailed feedback, which has helped us identify clear pathways to improve our paper. Should you have any further questions or require additional discussion, please don't hesitate to reach out. If we have adequately addressed your concerns, we would be grateful for your consideration in adjusting your evaluation score accordingly."}, {"review_id": "opzAuB2RE2", "reviewer": "Reviewer_pBsz", "summary": "This 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.", "questions": "1) 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?", "limitations": "Yes", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "* 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.", "was_revised": true, "final_justification": "I am raising my score to a 4. However, some important aspects—such as theoretical justification, implementation clarity, and experimental detail—remain somewhat underdeveloped in the current submission.", "rebuttal": "Dear Reviewer,\n\nThank you for your thorough review and insightful comments on our paper. We sincerely appreciate the time and effort you have dedicated to providing this valuable feedback. We have carefully considered each of your points and offer the following clarifications and planned revisions. We hope these responses will fully address your concerns and lead you to reconsider your assessment of our work.\n\n---\n\n### **On the Clarity of the Abstract (Weakness 1)**\n\nWe agree with your observation. Our intention was not to claim the invention of deflation projection and filter transforms. Our core contribution is to be the **first to successfully integrate these powerful, classical spectral transformation techniques into a neural network framework for solving operator eigenvalue problems**. As you noted, these methods are well-established in traditional numerical analysis, a point we detail in Sections 2.3 and 2.4 of our paper. To prevent any misunderstanding, we will revise the abstract in the final version to explicitly state that our work builds upon these classical numerical methods, clarifying our unique contribution.\n\n### **On Theoretical Justification  (Weakness 2)**\n\nThis is an excellent point, and we appreciate the opportunity to clarify the theoretical underpinnings of our method. The key insight of STNet is that the spectral transformations are applied directly to the original operator $\\mathcal{L}$, transforming it into an equivalent but easier-to-solve problem, $\\mathcal{L''}$. This process, detailed in Sections 3.2.1 and 3.2.2, does not modify the neural network architecture or its learning mechanism.\n\nCrucially, the transformation preserves the spectral properties in an exact, predictable way. It creates an **equivalent problem**, not an approximation. Therefore, if the neural network minimizes the loss to zero for the transformed problem $\\mathcal{L''}$, it provides an exact solution to the original operator eigenvalue problem. The transformation only impacts the convergence path and final precision, not the validity of the solution. Our extensive experimental results, which demonstrate state-of-the-art accuracy, provide strong empirical validation for this theoretical consistency.\n\n### **On the Implementation of the Differential Operator (Weakness 3)**\n\nThank you for pointing out the need for more clarity here. The differential operator $\\mathcal{L}$ is constructed and applied using **automatic differentiation (AD)**. AD allows for the exact computation of derivatives of any order by decomposing functions into a sequence of elementary operations and applying the chain rule. For instance, to apply an operator like $\\frac{d^2}{dx^2} + \\frac{d}{dx}$, we use AD to compute the second derivative and the first derivative of the network's output with respect to its input and sum the results.\n\nThe specific computations for deflation and filtering (lines 7–8 in Algorithm 2 ) are detailed by the formulas in the main text, specifically **Equation 9** (page 5, line 127) and **Equation 11** (page 6, line 150).\n\nWe have also provided the complete, runnable code in the supplementary materials. For example, the operator application and deflation are implemented as follows:\n\nPython\n\n```Plain\n# Automatic differentiation to compute Laplaciandef compute_grad(self, u, x):\n    u_x = autograd.grad(u.sum(), x, create_graph=True)[0]\n    return u_x\n\nu_xx = Nonefor i in range(self.d):\n    # ... code to compute second partial derivatives and sum them ...\n    u_xixi = self.compute_grad(self.compute_grad(u, x[i]), x[i])\n    u_xx = u_xx + u_xixi if u_xx is not None else u_xixi\n\n# Application of deflation projection operatordef apply_deflation(u_xx, u, max_vecs, max_vals):\n    Lu = -u_xx\n    # Sequentially subtract projections onto solved eigenfunctionsfor i in range(len(max_vecs)):\n        projection = max_vals[i] * torch.matmul(max_vecs[i], torch.matmul(max_vecs[i].transpose(0, 1), u))\n        Lu = Lu - projection\n    return Lu\n```\n\nWe will incorporate these explanations, along with more detailed pseudocode and illustrative code snippets, into the main paper to make our implementation crystal clear. We also plan to release our final code as a user-friendly library to facilitate broader use.\n\n### **On Details of the Experimental Setup (Weakness 4)**\n\nWe apologize for any lack of clarity. The loss function is explicitly defined in our paper. On page 4, Algorithm 2, line 10, the loss for each network is given as: $Loss_{i}^{k} = \\frac{1}{N}\\sum_{j=1}^{N} [\\tilde{v}_i^{k-1}(x_j)-\\tilde{u}_i^{k}(x_j)]^{2}$.\n\nThis loss formulation remains consistent across all experiments.  The only part that changes is the definition of the operator $\\mathcal{L}$ within the computation of $\\tilde{u}\\_{i}^{k}$. For full transparency, we will add the specific code snippets for implementing the operator $\\mathcal{L}$ for each problem (Harmonic, Schrödinger, Fokker-Planck) to the appendix in the final version. For instance, the implementation of Harmonic operator is:\n\nPython\n\n```Plain\n# Code snippet for Harmonic operator L\nx = self.x\nu = self.forward(x)\nu_xx = Nonefor i in range(self.d):\n    # ... (AD code as shown in previous point)\nLu = -u_xx # Lu represents the action of the operator L on u\n```\n\n### **On the Organization of** **Main** **Content and Appendix (Weakness 5)**\n\nThis is a very helpful suggestion. We agree that the background on deflation projection and filter transform, as well as a more detailed comparison of the baseline methods, are crucial for understanding our paper's context and contribution. In the camera-ready version, we will move these essential details from the appendix into the main body of the paper to improve readability and flow.\n\n### **On the Method's Applicability to Non-linear Operators (Question 2)**\n\nThe paper focuses on linear operators primarily because this is the most fundamental and intensively studied area for neural network-based eigenvalue solvers. However, this focus represents an implementation choice rather than a hard structural limitation. In traditional numerical analysis, many non-linear and quadratic eigenvalue problems are tackled by reformulating them into a sequence of linear problems that are solved iteratively.\n\nFollowing this established paradigm, STNet can, in principle, be extended to solve these more complex classes of problems. We see the current work as a foundational step, and we plan to develop optimized versions of STNet for non-linear, generalized, and other specific eigenvalue problems in our future research.\n\n### **On Comparisons with Advanced Traditional Methods (Question 3 & 4)**\n\nThis is an important question. As we discuss on pages 8-9 (lines 248-254), the primary bottleneck for traditional methods like FDM in high dimensions is the **discretization error**, not the algebraic eigenvalue solver's error. Techniques like deflation or filtering can improve the accuracy of the matrix eigenvalue solve, but they cannot overcome the fundamental error introduced by discretizing a high-dimensional space onto a grid. Reducing this error requires increasing the grid density, which leads to an exponential increase in memory and computational cost, a problem known as the \"curse of dimensionality\".\n\nTo further validate this point, we have run a new comparison for the 5D Harmonic problem against FDM with Richardson extrapolation, a technique that improves accuracy. The results, shown below, confirm that STNet still achieves significantly higher accuracy with comparable memory usage.\n\n| Method                                                 | Grid/Points | Memory (GB) | lambda_1 Error | lambda_2 Error | lambda_3 Error | lambda_4 Error |\n| ------------------------------------------------------ | ----------- | ----------- | -------------- | -------------- | -------------- | -------------- |\n| FDM (Richardson Extrap.)                               | 255         | 3.40E+00    | 8.60E-03       | 2.90E-02       | 2.90E-02       | 2.90E-02       |\n| FDM (with deflation and filtering; Richardson Extrap.) | 255         | 3.40E+00    | 8.60E-03       | 2.90E-02       | 2.90E-02       | 2.90E-02       |\n| FDM (Central Difference)                               | 455         | 2.20E+01    | 3.80E-03       | 1.20E-02       | 1.20E-02       | 1.20E-02       |\n| FDM (with deflation and filtering; Central Difference) | 455         | 2.20E+01    | 3.80E-03       | 1.20E-02       | 1.20E-02       | 1.20E-02       |\n| STNet                                                  | 95          | 1.10E+00    | 4.60E-05       | 1.60E-05       | 1.00E-05       | 2.30E-04       |\n\nThis experiment reinforces that neural network-based methods, which rely on random sampling rather than fixed grids, are inherently better suited for high-dimensional problems. We will add this experiment and a more detailed analysis to the appendix in our final version.\n\n---\n\n#### **Thanks again** \n\nWe sincerely thank you again for your constructive and detailed feedback, which has helped us identify clear pathways to improve our paper. Should you have any further questions or require additional discussion, please don't hesitate to reach out. If we have adequately addressed your concerns, we would be grateful for your consideration in adjusting your evaluation score accordingly."}, {"review_id": "AdJ06Zrs8h", "reviewer": "Reviewer_Saie", "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.", "questions": "see weaknesses", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### Strengths:\n1. Deflation projection and filter transform are simple and intuitively effective.\n2. The paper is well-written with clear and fluent presentation that makes the methodology easy to understand.\n\n### Weaknesses:\nSeveral issues need to be addressed or supplemented:\n1. Convergence and stability analysis with graphical illustrations is needed.\n2. 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).", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer,\n\nThank you for your valuable feedback and for recognizing the strengths of our work. We appreciate your positive assessment of our paper's clarity and the intuitive effectiveness of our proposed modules. We have carefully considered your suggestions and provide the following responses, which we hope will address your questions and further strengthen our submission.\n\n---\n\n### **On** **Convergence** **and Stability Analysis (Weakness 1)**\n\nWe agree that a graphical illustration of convergence is essential for a complete analysis. To provide a preliminary view of the convergence behavior of STNet, we present the absolute error at different iteration stages for the Harmonic problem below. The empty cells in the table indicate that convergence for that particular eigenvalue was already achieved.\n\nAbsolute Error vs. Iterations for Harmonic Problem\n\n| Dim  | Eigenvalue Absolute Error | Iter: 100 | 500     | 1,000   | 40,000  | 80,000  | 130,000 | 180,000 |\n| ---- | ------------------------- | ------------- | ------- | ------- | ------- | ------- | ------- | ------- |\n| d=1  | λ1                        | 1.6e-02       | 3.5e-03 | 3.2e-04 | 2.6e-04 | 2.7e-05 | 1.2e-03 |         |\n|      | λ2                        | 1.0e+04       | 2.4e+03 | 2.8e+03 | 8.0e+02 | 9.8e-01 | 8.8e-03 |         |\n| d=2  | λ1                        | 6.9e-02       | 6.9e-03 | 5.9e-03 | 5.3e-06 |         |         |         |\n|      | λ2                        | 6.3e+01       | 5.9e+00 | 4.2e+00 | 5.1e-02 |         |         |         |\n| d=5  | λ1                        | 5.8e-02       | 6.6e-03 | 4.8e-04 | 7.8e-03 | 3.9e-03 | 6.2e-03 | 3.9e-03 |\n|      | λ2                        | 6.3e+03       | 4.1e+02 | 1.5e+02 | 2.9e+00 | 5.5e+00 | 3.8e+00 | 1.3e-01 |\n\nFollowing your excellent suggestion, we will add comprehensive convergence plots (error vs. iterations) for all algorithms across all experiments in the final camera-ready version of the paper to provide a clearer visual analysis of the convergence dynamics.\n\n### **On the** **Source** **of Performance Improvement and PMNN Comparison (Weakness 2)**\n\nThis is an insightful question, and we thank you for the opportunity to clarify. You are correct in your observation. As stated on page 5, line 105, STNet is designed as an enhanced version of the Power Method Neural Network (PMNN). The PMNN framework itself is capable of achieving very high precision for a single eigenvalue.\n\nTherefore, the **\"w/o F and D\" case in our ablation study (Table 6) is functionally equivalent to the original PMNN**. The high accuracy for the principal eigenvalue ($ \\lambda_1$) in this setting is indeed inherited from this strong baseline.\n\nThe primary limitation of PMNN, however, is its inability to compute multiple eigenvalues simultaneously. Our key contribution with STNet is the introduction of the **deflation projection and** **filter** **transform**, which extend the high-precision capabilities of the base method to solve for multiple eigenvalues concurrently and more accurately.\n\nThe effectiveness of our modules is clearly demonstrated in Table 6. In the \"w/o D\" setting, the network correctly finds the principal eigenpair $(v_1, \\lambda_1)$ but fails to find the subsequent ones, with the errors for $\\lambda_2$ and $\\lambda_3$ being extremely large. This shows that the network repeatedly converges to the same dominant eigenfunction. The deflation projection module is what resolves this issue, enabling the network to discover multiple distinct eigenfunctions.\n\nTo make this clearer, we will explicitly state in the ablation study section of our final paper that the \"w/o F and D\" setting serves as a direct comparison to the PMNN baseline, and we will revise the text to better articulate how our contributions build upon it to overcome its limitations.\n\n---\n\n#### **Thanks again** \n\nWe sincerely thank you again for your constructive and detailed feedback, which has helped us identify clear pathways to improve our paper. Should you have any further questions or require additional discussion, please don't hesitate to reach out. If we have adequately addressed your concerns, we would be grateful for your consideration in adjusting your evaluation score accordingly."}, {"review_id": "MiXHWGZJJF", "reviewer": "Reviewer_btcB", "summary": "The 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.", "questions": "1. 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?", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": true, "final_justification": "The general applicability of this work remains uncertain, but this work shows great improvement in some domains. The additional data provided in the rebuttal makes it more convincing in the given settings. Thus, I'd like to raise my score.", "rebuttal": "Dear Reviewer,\n\nThank you for your detailed and insightful review of our paper. We are very encouraged that you found our work to be well-written, our method to be clearly presented, and recognized its potential for meaningful impact. We sincerely appreciate the time you have taken to provide this valuable feedback. Your questions have highlighted several areas where we can provide further clarification and strengthen our contribution.\n\nBelow, we address the weaknesses and questions you raised.\n\n---\n\n### **On Comparison with Traditional Methods: Scalability, Cost, and** **Convergence** **(Weaknesses 2 & 3, Questions 1 & 2)**\n\nWe thank you for raising these critical points regarding the comparison with traditional methods. This comparison is central to our work, and we have dedicated **Section 4.4 (\"Comparative Experiment with Traditional Algorithms\")** and **Table 5** in our paper to a direct discussion and data presentation on this topic.\n\n**The Core Advantage: Overcoming the \"Curse of Dimensionality\"**\n\nIt is essential to distinguish between the challenges of high-dimensional problems. For traditional methods like the Finite Difference Method (FDM), the primary bottleneck is not the algebraic eigenvalue solver itself (e.g., the Power Method or ARPACK), but rather the initial **discretization** **of the continuous operator into a matrix**. This process suffers from the \"curse of dimensionality\": to maintain accuracy in higher dimensions, the number of grid points must grow exponentially, leading to an intractable explosion in memory and computational cost.\n\nNeural network-based methods and traditional algorithms excel in different domains. In low-dimensional problems, traditional methods are often faster and highly accurate. However, in high-dimensional settings, the cost of discretization makes them impractical, which is precisely where methods like STNet offer a significant advantage.\n\n**Analysis of Computational Cost and Memory Usage**\n\nOur experimental data clearly illustrates this trade-off:\n\n- **The Dilemma of Traditional Methods**: As shown in **Table 5** for the 5D Harmonic problem, for FDM to reduce the relative error of the first eigenvalue (λ1) from $3.20×10^{−1}$ to a modest $3.82×10^{−3}$, the memory consumption skyrockets from 0.0001 GB ($4^5$ grid) to **22.9 GB** ($45^5$ grid). Even with this massive resource usage, the resulting accuracy is orders of magnitude lower than STNet's. **The Efficiency of STNet**: In contrast, STNet achieves a much higher accuracy of $4.62×10^{−5}$ while using only **1.14 GB** of memory.\n\nTo further address your question, we provide a more detailed analysis of the convergence time and absolute error of the principal eigenvalue for a five-dimensional Harmonic problem. It is important to note that our traditional baseline in the paper uses `scipy.eigs`, which is based on the highly-optimized ARPACK library—a much more powerful solver than a basic power method.\n\n| Method | Grid/Sample Pts | Memory (GB) | Final Abs. Error | Time to reach error of 1e+0 | Time to reach error of 1e-1 | Time to reach error of 1e-2 | Time to reach error of 1e-3 | Total Time |\n| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |\n| **FDM+eigs** | 8e+05 (15^5) | 9.1e-02 | 3.2e-02 | 1.3s | 4.0s | 1.1e+01s | - | 1.1e+01s |\n| | 1e+07 (25^5) | 1.2e+00 | 1.2e-02 | 2.5e+01s | 3.2e+01s | 8.1e+01s | - | 1.1e+02s |\n| | 5e+07(35^5) | 6.5e+00 | 6.3e-03 | 1.8e+02s | 2.5e+02s | 4.9e+02s | 1.9e+03s | 1.9e+03s |\n| | 2e+08(45^5) | 2.3e+01 | 3.8e-03 | 9.4e+02s | 3.1e+03s | 6.7e+03s | 1.9e+04s | 1.9e+04s |\n| **STNet** | 5e+04 | 9.5e-01 | 3.4e-04 | 7.2s | 1.4e+01s | 1.7e+02s | 1.3e+03s | 1.0e+05s |\n| | 6e+04 (9^5) | 1.1e+00 | 4.6e-05 | 9.6s | 1.9e+01s | 1.5e+02s | 1.3e+03s | 3.1e+04s |\n| | 7e+04 | 1.3e+00 | 6.5e-05 | 5.7s | 1.1e+01s | 1.4e+02s | 3.0e+02s | 5.4e+04s |\n| | 8e+04 | 1.5e+00 | 2.5e-05 | 8.9s | 1.6e+01s | 1.6e+02s | 9.8e+02s | 2.9e+04s |\n\n\nThis detailed data further validates our conclusion. For large-scale (high-dimensional) problems, **STNet holds a significant advantage in both computational cost and memory usage**. Its core strength lies in its **mesh-free random sampling strategy**, which leverages the expressive power of neural networks to approximate eigenfunctions directly. This approach completely bypasses the step of constructing and storing a massive matrix, thereby effectively circumventing the curse of dimensionality that plagues traditional grid-based methods.\n\nThe convergence process of all experimental solutions and their computational time and memory requirements will be added to the appendix of the subsequent official version.\n\n### **On Applicability to Real-World Problems (Question 3)**\n\nThis is an excellent question regarding the practical scope of our method. The iterative nature of STNet, which is inspired by the power method, requires the ability to perform an \"operator action\"—that is, to compute the result of the operator $\\mathcal{L}$ acting on the current function approximation (the neural network's output).\n\nThe applicability of STNet to a given problem therefore hinges on this requirement:\n\n- If the mathematical form of the operator is known, even if it's highly complex or the domain is irregular, the operator action can be precisely computed using automatic differentiation. In these cases, STNet is directly applicable and its mesh-free nature is a distinct advantage.\n- If the operator is not analytically defined (e.g., it exists as a \"black box\" simulator) or the data is noisy, our framework could still apply **as long as this operator action can be performed or simulated**. However, if there is no way to evaluate the effect of the operator on a given function, then our method in its current form would not be applicable.\n\nIn summary, the ability to compute the operator action is the main prerequisite. Extending the framework to cases where the operator itself must be learned from noisy data (i.e., inverse problems) is a very important direction for future research.\n\n---\n\n#### **Thanks again** \n\nWe sincerely thank you again for your constructive and detailed feedback, which has helped us identify clear pathways to improve our paper. Should you have any further questions or require additional discussion, please don't hesitate to reach out. If we have adequately addressed your concerns, we would be grateful for your consideration in adjusting your evaluation score accordingly."}]

Author rebuttal:

Meta-review: This paper received the ratings of (5, 4, 4, 4) and the ratings converged toward acceptance after rebuttal. The reviewers recognized STNet's novel integration of classical spectral transformation techniques (deflation projection and filter transform) into neural network-based eigenvalue solvers, achieving state-of-the-art accuracy on benchmark problems. The main concerns centered on limited theoretical justification for why these transformations preserve spectral properties in the neural framework and the lack of real-world applications beyond synthetic benchmarks. The authors provided a strong rebuttal with new convergence analyses, additional experiments comparing against FDM with Richardson extrapolation, and clarified implementation details. All reviewers moved toward acceptance, with reviewer z6YH explicitly supporting acceptance and reviewer pBsz raising their score to borderline accept. While theoretical gaps remain, the empirical evidence and methodological contribution of successfully combining classical numerical techniques with deep learning justify acceptance. The authors should incorporate the promised convergence plots, theoretical clarifications, and additional experimental details in the final version.
Decision: Accept (poster)