Accelerating Eigenvalue Dataset Generation via Chebyshev Subspace Filter
Authors: Hong Wang
Jie Wang
Jian Luo
huanshuo dong
Yeqiu Chen
Runmin Jiang
Zhen huang
Venue: ICLR 2026 Poster (2026)
Keywords: Data Generation
Eigenvalue Problem
AI4PDE
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)

Abstract: Eigenvalue problems are among the most important topics in many scientific disciplines. With the recent surge and development of machine learning, neural eigenvalue methods have attracted significant attention as a forward pass of inference requires only a tiny fraction of the computation time compared to traditional solvers. 
However, a key limitation is the requirement for large amounts of labeled data in training, including operators and their eigenvalues.
To tackle this limitation, we propose a novel method, named **S**orting **C**hebyshev **S**ubspace **F**ilter (**SCSF**), which significantly accelerates eigenvalue data generation by leveraging similarities between operators---a factor overlooked by existing methods. 
Specifically, SCSF employs truncated fast Fourier transform (FFT) sorting to group operators with similar eigenvalue distributions and constructs a Chebyshev subspace filter that leverages eigenpairs from previously solved problems to assist in solving subsequent ones, reducing redundant computations.
To the best of our knowledge, SCSF is the first method to accelerate eigenvalue data generation. 
Experimental results show that SCSF achieves up to a $3.5\times$ speedup compared to various numerical solvers.

Published as a conference paper at ICLR 2026
ACCELERATING EIGENVALUE DATASET GENERATION VIA CHEBYSHEV SUBSPACE FILTER
Hong Wang 12,33 Jie Wang Jian Luo
Huanshuo Dong! Yeqiu Chen 1,2
Runmin Jiang' Zhen Huang'
University of Science and Technology of China
CAS Key Laboratory of Technology GIPAS, University of Science and Technology of China 3 MoE Key Laboratory of Brain inspired Intelligent Perception and Cognition, University of Sci- ence and Technology of China
wanghongl 700@ma1 1 ustc.edu.co.
ABSTRACT
Eigenvalue problems among the most important topics in many scientific dis- ciplines. With the recent surge and development of machine learning, neural eigenvalue methods have attracted significant attention as forward pass of infer- ence requires only a tiny fraction of the computation time compared to traditional solvers. However, key limitation is the requirement for large amounts of labeled data in training, including operators and their cigenvalues. To tackle this limitation, we propose novel method, named Sorting Chebyshev Subspace Filter (SCSF), which significantly accelerates eigenvalue data generation by leveraging similarities between operators- factor overlooked by existing methods. Specifically, SCSF employs truncated fast Fourier transform sorting group operators with similar eigenvalue distributions and constructs a Chebyshev subspace filter that leverages eigenpairs from previously solved problems to assist in solving subsequent ones, reducing redundant computations. To the best of our knowledge, SCSF is the first method to accelerate eigenvalue data generation. Experimental results show that SCSF achieves up to a 3.5x speedup compared to various numerical solvers.
INTRODUCTION
Solving eigenvalue problems is important challenge in fields such as quantum physics (Pfau et al.
2023), fluid dynamics (Schmid) 2010), and structural mechanics (Wen al. (022). Traditional numerical solvers, such as the Krylov-Schur algorithm (Stewart 2002), often suffer from prohibitively high computational costs when tackling complex problems. To overcome these computational
challenges, recent advancements in deep learning (Schütt al. 2017 Li et al.) 2020 Luo et al.) have demonstrated remarkable success as one forward pass only necessitates a tiny fraction of the computation time compared to numerical solvers, often in milliseconds.
Despite their success, data-driven approaches face fundamental limitation: the reliance on labeled datasets. Training neural networks requires large-scale labeled data, which is often generated using computationally expensive traditional methods. It usually takes dozens hours or even days. For example, the QM9 dataset (Ramakrishnan al. 2014) contains 1.34 10S molecular data points, each produced by solving Hamiltonian operator eigenvalue problems. These calculations typically employ traditional algorithms, whose computational costs can escalate dramatically with increasing problem complexity, like finer grid resolutions or higher accuracy requirements. This scalability issue represents a sigmificant bottleneck for generating the labeled data needed train deep learning models. Furthermore, the diversity of scientific problems leads to the need for unique dataset for each scenario, which further intensifies this challenge of computational intractability. As a result, the high computational expense of generating eigenvalue data severely limits the application of
data-driven approaches (Ren et al. 2026 Zhang et al. 20233 Huang et al. 2025 [iu et al. 2024
2025c Wu et al.) 2025 Luo et al. 2024 Liu et al. 2025b Zhang et al. 2024 Wang et al. Hou
et al.
2026 Zhang et al. 2026).
*Corresponding author.

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Published as a conference paper at ICLR 2026
Payn SepcLooncO
c. p. (o))) a()))y(o) Aay(s)
107
(1P5)
A1
Solver
(AP
operator
Time
eigenvalue
-persler/mn.aa...
A
Solver
(A
DATA SET
problem
generator
3
4]:
6
spendeeoon.aa
1000
Solver
Number of Etgemalluss
Figure 1: Left. Generation process of the eigenvalue dataset: 1. Generate set of random problem parameters. 2. Derive the corresponding operators based on these parameters. 3. Convert the operators into matrices using discretization methods. 4. Independently solve for the matrix eigenvalues using numerical solvers. 5. Obtain the matrix eigenpairs, converting them into the operator eigenpairs. Assemble the dataset. Right. Results of average computation times actoss various algorithms based on the number of eigenvalues solved on the Helmholtz operator dataset.
In particular, the dataset generation process typically involves six key steps, as illustrated in Figure | (left). Among these steps, solving the eigenvalue problem is the most computationally demanding (step 4), accounting for 95% of the total processing cost (Hughes) 2012). Existing data generation methods typically compute the eigenvalues of each matrix in the dataset independently. However, operators in the dataset often share similarities, as they describe related physical phenomena, which process. Existing approaches, however, fail to leverage these similarities, leading to significant computational redundancy. Previous works (Wang et al., 2024; Dong et al. [2024] Liu et al. 2025a) have demonstrated the potential of lever- aging similarity to significantly reduce generation time of linear system datasets. However, how to effectively exploit matrix similarity to accelerate eigenvalue datasets generating remains an unknown problem.
can largely simplify and accelerate the eigenvalue-solvie
To address this problem, we introduce a novel data generation approach, named Sorting Chebyshev Subspace Filter (SCSF). SCSF designed to the eigenpairs of similar problems to reduce redundant computations in the eigenvalue solving process, thereby accelerating eigenvalue dataset generation. Specifically, at the beginning. SCSF employs sorting algorithm based on truncated Fast Fourier transform (FFT), which arranges these problems efficiently, enhancing the adjacent correlation between problems in the queue and laying the groundwork for sequential solving. Then, SCSF accelerates the convergence of iterations and significantly reduces computation times by constructing a Chebyshev subspace filter, which solves the problem aided by the eigenpairs from previous problem solving. The core design of SCSF is to identify and exploit the close spectral distributions and invariant subspaces within these eigenvalue problems. SCSF coordinates the sequential resolution of these systems rather than treating them discrete entities. This improved approach not only alleviates the computational demands of the eigenvalue algorithm but also significantly speeds up the generation of training data for data-driven algorithms. We summarize our contributions as follows:
To the best of our knowledge, SCSF is the first method to accelerate operator eigenvalue dataset generation by exploiting intrinsic operator similarity through a novel sorting and filtering framework.
By using truncated FFT sorting and the Chebyshev filtered subspace iteration, we introduce novel approach that transforms dataset generation into sequence eigenvalue problems.
Comprehensive experiments demonstrate that SCSF substantially reduces the computational cost of eigenvalue dataset generation. As demonstrated in Figure [ (right), our method achieves up to a 3.5x speedup compared to state-of-theeart solvers.
2 PRELIMINARIES
2.1 DISCRETIZATION OF EIGENVALUE PROBLEM
This section outlines the discretization of operator eigenvalue problems into matrix form. Our method focuses on solving these matrix problems, which is the most time-consuming step in generating
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the required eigenvalue datasets. As shown in Figure I (left), these problems typically solved by numerical discretization methods such as FDM Strikwerda 2004; LeVeque 2002). These discretization techniques embed the fiitt-ddimensional space, thereby transforming operator eigenvalue problems into matrix eigenvalue problems. We provide simple example to clarify the discussed processes. A detailed process Specifically, we discuss the case that uses FDM to solve the eigenvalue problem of the two-dimensional Poisson operator, transforming it into a matrix eigenvalue problem:
Hilbert space of operators into appropriate
be found in Appendix C
k(z, V ul y) = Au( v)-
(1)
We map the problem onto a 2 grid (i.e., N₂ N₂ 2 and Ax = Ay), where both the variable Way and the coefficients kt,j follow row-major order. This setup facilitates the derivation of the matrix eigenvalue equation:
1 1
0
4
1
0
11,1
11,1
112
0
1
1
f1,2
U1,2
0
22,1
1
1
4
1
W2,1
12,1
0
0
k2,2
0
4
[2,2]
12,22
k12
By employing various methods to generate the parameter matrices P =
Such as
K2
122
utilizing Gaussian random fields (GRF) or truncated polynomials, we can derive Poisson operators characterized by distinct parameters.
Typically, training a neural network requires number of data from 10 10 (Lu et al. 2019). Such a multitude of eigenvalue systems, derived from the same distribution of operators, naturally exhibit a high similarity (Soodhalter et al.) (222).. It is precisely this similarity that key to the effective acceleration of SCSF. We can conceptualize this as the task of solving sequential series of matrix eigenvalue problems:
j= L; i= 1,2, ...., .N
where L the number eigenvalues to be solved, is the number cigenvalue problems, the matrix
A E C"**, the eigenvector uy E (and the eigenvalue (i) E C vary depending on the operator.
We define the eigenpairs as (A()) V(9), with ( diag(>]) ..... (1)). V(O [u[1 ... .0.11..1)))
(8) and
[A[*] (2 1.............
1-.. < 11
2.2 THE CHEBYSHEV POLYNOMIALS AND CHEBYSHEV FILTER
This section details the Chebyshev filter, which is constructed using Chebyshev polynomials. Dis- tinguished by their optimal uniform approximation properties and efficient three-term recurrence relations, these are among the most widely utilized orthogonal polynomials (Mason Handscombi 2002 Rivlin 2020). This filtering technique is particularly effective for our method because it can efficienlly construct a polynomial that isolates target spectral interval, allowing to use the solution of one eigenvalue problem to significantly accelerate the next.
The Chebyshev polynomials C.n (t) of degree m are defined on [-1, 1] and are expressed as
C. (t) cos(m cos 1 (t)),
(4)
Algorithm 1: Chebyshev Filter (Berljafa et al. 2015)
Input: Matrix A E C"xa, vectors Yn E Ca degree m E N, and parameters X, c, e E IR. Output: Filtered vectors Y., = Cm (Yo), where each vector Ym.j is filtered with Chebyshev polynomial of degree m.
A (A (ccll,/ee,
01 e/(A c);
2 Y = AY;
for i 1,...,
do 0+11 = 1/ - -
Y+1111m11 Y1,1:-11 Y4 -0 AY 004109YY.m.kii
3

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a
Operaaoor generator
Dataset
Subspace iteration
Solution
Chebyshev Filter
eigenvector
Subspace teeration
Solution
((z)
e
Subspace iteration
Solution
random subspace
eigenailues entimation
Truncated FFT Sort by p!))
spectral transformation
Hd1 Subspace iteration
Solution
f
eggenv ector
SCSF
Chebyshev Filter
(() Cale
#2 Subspace iteration
Solution
selspare
Chebyshev Filter
eigeovalues estimation
H3 Subapace iteration
Solution
Figure 2: Algorithm Flow Diagram: Generation of operators to be solved. b. Discretization of operators into matrices. e. Apply SCSF algorithm to sort matrices, obtaining a sequence with strong correlations. d. Other algorithms independently solve eigenvalue problems. d1, d2, d3. SCSF algorithm utilizes Chebyshev subspace iterations to sequentially solve the eigenvalue problems. e. Assembly of eigenvalue pairs into a dataset. f. Amplification of the interval of interest through spectral transformation. g. Replacement of initial subspaces with previously solved invariant subspaces.
(t) commonly referred to the Chebyshev polynomial of the first kind, satisfies the following recurrence relation:
(t) 21Cm (t) i1)-
For a Hermitian matrix E Ca and vectors Yo E Cn we use the three-term recurrence relation
that defines Chebyshev polynomials in vector form:
Cm+1 (Y)) = 2AC,.. (Y)) - Cmo)(XY), CN, (m)Y)).
(6)
The computation of Cm (Yo) and the Chebyshev filter is described in Algorithm Let A' denote
the previously solved related matrix, with (X/x vf) in ascending order, and (X2 X, E [a,, B]. In
Algorithm 1 the parameter À is typically approximated by X1 while c = a+9 and e B-a represent the center and half-width of the interval [a, 3). providing estimates for the spectral distribution of A.
METHOD
In this section, we introduce novel method, named the sorting Chebyshev subspace filter (SCSF),
a fast data generation approach that efficiently solves eigenvalue problems by leveraging intrinsic spectral correlations among operators. SCSF incorporates two key components: (1) truncated fast Fourier transform (FFFT) approach for efficiently sorting operator eigenvalue problems and (2) the Chebyshev filtered subspace iteration (ChFST) employed for sequential solving. By integrating these components, SCSF can use spectral information from the previous eigenvalue problem solving aid the next eigenvalue problem solving, thus accelerating the eigenvalue data generation.
We first introduce the sorting algorithm that leverages the spectral similarities and provides the time complexity analysis in Section 3.1 Then we give an introduction the Chebyshev filtered subspace iteration in Section 3.2 Figure shows the overview of our SCSF. Generally, the truncated FFT sorting algorithm ensures that successive matrices in the sequence exhibit close relations. Then ordered sequence enables ChFSI to effectively utilize prior information, thereby accelerating the solution process (Berljafa al. 2015).
3.1 THE SORTING ALGORITHM
To benefit the successive solving sequence of the eigenvalue problem, we need sorting algorithm that pulls matrices with similar spectral properties, like invariant subspaces, close enough in the solving sequence, so that solving the current matrix in sequence be easily boosted by the previous
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Algorithm 2: The Truncated FFT Sorting Algorithm
Input: Sequence of eigenvalue problems to be solved A E CaX", corresponding parameter
matrix pit) E C 1,2, .......,.... .N, Po is the truncation threshold for low frequencies, and low E CPeXPo
Output: Sequence for eigenvalue problems mat"
Initialize the list with sequence seqo 21.2. 2, ,N,, seqmat is an empty list;
2 Set ig = 1 as the starting point. Remove 1 from sequ and append 1 to seg
3 for i = 1, 1,... .N do
Let Plow = Truncpo (FFT(P())). Perform truncated FFT on matrix p(i) to extract low-frequency information;
5 for i == 1, . and dis = 1000 do
for each j in sedo do
dis, = the Frobenius norm of the difference between pflo and p()) loww
if dis; < dis then
dis = disj and jmin j;
10
Remove Jmii from sego. append Jmin to seqmat and set ig = Jmini Get the sequence for eigenvalue problems segmati
solving. Recalling Section 2.1 the matrix (6) is the operator matrix derived through numerical discretization (e.g., Finite Difference Method) from the parameter matrix p(t) (e.g., the coefficient ((x, 3) in the Poisson equation) (Lu et al. 2022 Li et al. 2020). For instance, in the Poisson equation,
the values jj.j from p(() become part of the discretized sparse matrix A as shown in Eq. 2 A naive strategy
use the Frobenius distance of the parameter matrices p(() to perform a greedy sort (Wang et al. 2024). And by repeatedly fetching without reservation from the remaining matrix in the dataset, we can reorganize the solving sequence so that the successive solving can benefit from the re-ordered sequence.
However, the main computational cost of such a naive sorting algorithm arises from repeatedly calculating the distances between different matrices P, which is directly related to the matrix dimension- that is, the resolution of operators. Existing works (Holmes 2012] Li al,, 2020) have shown that the key variables that affect operators stem from the low-frequency components of the parameter matrices P, while high-frequency components often represent noise or irrelevant data. Based on this insight, to reduce computational overhead during sorting, we first perform a truncated FFT on the parameter matrices to extract the low- wow-frequency information before sorting. We then sort by comparing the Frobenius distances between these low-frequency components.
As shown in Algorithm 2. suppose we have N eigenvalue problems, the parameter matrices p(() CPxp, and the low-frequency truncated matrices Tow E CPoXpo, The computational complexity of directly using greedy algorithm is O(N2p²) Our sorting algorithm's complexity consists of two main parts: 1. FFT Computation: The complexity of FFT is O logp) per matrix. For N matrices, this totals O(N'² log p). 2. Greedy Sorting: The subsequent greedy sorting algorithm has a complexity of O(N2p)). Overall, the total complexity is ON2 + -N² log p). Since Po & and p & N, our sorting algorithm effectively reduces computational cost. In addition, we conducted qualitative theoretical analysis of the truncated FFT to demonstrate its rationality. Details can be found in Appendix
3.2 CHEBYSHEV FILTERED SUBSPACE ITERATION
After the sorting algorithm, obtain sequence of eigenvalue problems that exhibit strong correla- tions between consecutive problems. We employ the Chebyshev filtered subspace iteration (Manteuf)
fel 1977 Saad 2011 Winkelmann al. 2019 Berljafa et al. 2015) that leverages the eigenpairs (A
of the previous problem 10-1) accelerate the iterative convergence of the subse-
quent problem A(), thereby significantly enhancing computational performance. We focus the most common scenario in eigenvalue problems where the operator self-adjoint; in this case, the corresponding matrix A is Hermitian.
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Algorithm 3: Chebyshev Filtered Subspace Iteration
Input: Eigenvalue problem A eigenpairs (A(i-1), V(i-1)) of the previous eigenvalue
problem ((-1)) where A A(--1) = diag(> (2(--1) ;....: ))
V(-=) 11 I... |0 (6-1)] and filter degree m.
Output:
Wanted eigenpairs (A (8) v(0).
Initialize empty arrays/matrices (A, V), and set Ag = A(i-1), Vo V(--1);
2 repeat
Apply Chebyshev filter: Vo Cm (V));
Perform QR orthonormalization on QR [V];
Compute Rayleigh quotient G QU (
Solve the reduced problem GW = W Ag. and update Vo = V.W;
Lock converged eigenpairs into (A, V):
until the number of converged eigenpairs 2 L;
Return eigenpairs (A v(9) (Á, V):
Algorithm 3 outlines the process of ChFSI for solving the i-th eigenvalue problem A (1 < i < N) where L eigenvalues need to be solved. The initial approximate invariant subspace (i- and spectral distribution A (f-11) are derived from the eigenvectors and eigenvalues of the previous problem ( in the sequence. This initialization strategy, illustrated in Figure 2 (g), acts "warm start", where high-quality initial subspace significantly reduces the computational cost of subsequent iterations. The parameter m denotes the polynomial degree in the filter function, e.g., in Figure (D), the Chebyshev filter modifies the operator's 's spectrum to amplify the target region, effectively transforming the problem into equivalent one that is easier solve. For the first eigenvalue problem (1)) in the sequence, the initial iterative subspace Vo and initial spectrum Åg randomly generated.
20. As depicted
In line 3, the Chebyshev filter is applied using the vector form of Chebyshev polynomials; details be found in the preliminaries Section 2.2 After the Chebyshev filtering step, the vector block Vo spanning the invariant subspace may become linearly dependent. To prevent this, orthonormalization performed (line 4) using QR decomposition based on Householder reflectors. Line com- putes the Rayleigh quotient of A using the orthonormalized Vo, projecting the eigenvalue problem onto a subspace that approximates the desired eigenspace. In line 6, the reduced eigenvalue problem is diagonalized, and the computed eigenvectors are projected back the original problem. At the end of the Rayleigh-Rite step, relative residuals of the computed eigenvectors are calculated; converged eigenpairs are locked, and non-converged vectors are set to be filtered again (1ine 7).
Assuming m is the degree of the polynomial, n is the dimension of the matrix A, and L is the number of eigenvalues be solved, the computational complexity per iteration comprises: Chebyshev filter: O(m2² L) 2. QR factorization: O(nL²) 3. Rayleigh-Rizz procedure: O(n L + nL² + L
Residuals check: O() Since m >> 1 and n > L, the Chebyshev filtering step is the most computationally intensive.
The acceleration of the Chebyshev filtered subspace iteration heavily depends on selecting approxi- mate invariant subspaces and eigenvalues that promote rapid convergence in subsequent iterations. Proper sorting amplifies their impact, reducing the number of iterations required. This underscores the critical importance of the sorting algorithm in our method.
4
EXPERIMENT
4.1 EXPERIMENTAL SETTINGS
To comprehensively assess the performance of our approach SCSF against other algorithms, we conducted extensive experiments, each simulating the generation of operator eigenvalue dataset. We primarily compared the average computation times across different numbers of eigenvalues solved and various matrix sizes. These tests encompassed four distinct datasets and five mainstream

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Published as a conference paper at ICLR 2026
Table 1: Comparison of average computation times (in seconds) for eigenvalue problems using various algorithms. The first row lists different algorithms, the first column details the datasets, including matrix dimensions and solution precisions (relative residual), and the second column shows the number of eigenvalues computed for each matrix. The best algorithm is in bold. The symbol '.' denotes the result of method that fails to converge under the given setting.
Dataset
Eigsh
LOBPCG
KS
JD
ChFSI
SCSF (ours)
Poisson
14.20
73.03
23.76
270.2
24.00
12.85
2500
300
26.27
151.5
45.95
920.8
38.03
25.61
lc-12
400
36.86
265.3
72.32
2691
57.41
33,91
Ellipse
41.82
139.2
61.77
414.3
43,90
24.08
4900
300
62.47
264.1
110.5
1446
60.69
29.88
le-10
400
87.19
459.7
188.7
3386
67.13
34,60
Helmholtz
200
151.7
129,9
98.34
489.6
107.1
31.31
6400
400
253.5
460.4
283.0
3829
121.5
40.52
le-8
600
398.8
1031
329.6
146.2
51.32
Vibration
200
397.9
333.7
272.0
1230
300.8
85.70
10000
400
635.6
1170
768.8
310.5
107.2
le-8
600
1037
2716
857.8
382.3
131.4
eigenvalue solving algorithms, with SCSF consistently delivering commendable results. The detailed
data provided in Appendix E.I and the related work is discussed in Appendix B
Baseline. Our focus solves the eigenvalue problem of matrices derived from self-adjoint differential
operators, typically consisting of large Hermitian matrices. We benchmarked against the following
mainstream algorithms implemented in libraries widely used: 1. Eigsh from SciPy (implicitly restarted Lanczos method) (Virtanen et 2020), 2. Locally optimal block preconditioned conjugate gradient
(LOBPCG) algorithm from SLEPc (nnazevv 2001] Hernandez et al.) 2009). 3. Krylov-Schur (KS)
algorithm from SL.Pe (Stewart 2002), 4. Daviid-n (JD) algorithm from SI.EPc (Sleijpen
Van der Vorst 2000), 5. Chebyshev filtered subspace iteration (ChFSI) (Berljafa et al. 2015) with
random initialization. For detailed information, please refer to Appendix D.I
Datasets. To explore the adaptability of the algorithm across different matrix types, we investigate four distinct operator eigenvalue problems: 1. Generalized Poisson operator; Second-order elliptic partial differential operator; 3. Helmholtz operator; 4. Fourth-order vibration equation. For thorough description of the datasets and their generation, please refer to Appendix [.2
All experiments focus computing the smallest L cigenvalues in absolute value and their corre- sponding eigenvectors. For the runtime environment, experimental parameters, and parallelism setup, please see the Appendixes D.3 D.4, D.6 The hyperparameter analysis experiments, runtimes for various components of SCSF, the reliability of data generated by traditional algorithms, can be found in Appendixex E.4
and E.5
We note that all experiments use relative residual as the metric for solution precision, with its definition provided in Appendix D.5 SCSF is numerical algebra algorithm that allows for adjustable solution precision as needed. It is purely an acceleration technique and does not alter the solution results at the specified precision. The solution precision for all experiments significantly higher than the typical relative eтror range of neural networks (1e-1 to le-5), making it effectively ground truth. Therefore, the datasets generated by different numerical algorithms will not affect the training performance of neural networks.
least 1c-8, which is
4.2 MAIN EXPERIMENT
Table showcases selected experimental data. From this table, we can infer several conclusions: First, across all settings, our SCSF consistently has the lowest computation cost. The most significant improvements appeared in the Helmholtz dataset, where SCSF demonstrated speedups of 8x, 20x, 6x, x5x, and 3.5x compared to Eigsh, LOBPCG, KS, JD, and ChFSI algorithms, respectively. These results confirm that SCSF effectively reduces inherent redundancies in sequential eigenvalue problems, substantially accelerating operator eigenvalue dataset generation.
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Moreover, as the number of eigenvalues L solved per matrix increases, the speed advantage SCSF other algorithms becomes more pronounced. For instance, on the second-order elliptic operator dataset, when solving for 200 eigenvalues, SCSF is 2.5 times faster than the Krylov-Schur method and 5.5 times faster at 400 eigenvalues. This ef- ficiency stems from SCSF inheriting approximate invariant subspaces from previous solutions, effec- tively leveraging available information to expand the initial search space. Consequently, SCSF re- quires minimal additional iterations as L. increases, resulting in modest computation time growth.
Time
300
Stpee
Scay Eigh
CHFS)
5C5P
200
302. .0000
50.00
5000
7000
0020
6050
10000
Marrix Dimenssion
Figure Plot of average computation time ver- sus matrix dimension for solving 400 eigenval- ues with precision Te- 12 on the generalized Poisson operator dataset.
Besides, the performance disparity across different datasets significant. For example, on the general- ized Poisson operator dataset, SCSF is only about 10% faster than Eigsh, yet it leads by 4-7 times on the Helmholtz dataset. This difference can be attributed to the numerical properties of different operators and the matrix assembly formats, which directly influence algorithmic performance.
We also conducted additional experiments to show that the impact of the matrix dimension is also significant. Results are shown in Figure 3 SCSF performs noticeably better as matrix dimensions increase. Below the matrix dimension of 3600, SCSF and Eigsh show comparable efficiency. However, beyond 5000, SCSF significantly outperforms Eigsh and other algorithms. For more details about matrix dimension influence, we refer to the results in Appendix E.2
This phenomenon can be explained through operator matrix approximation. A fixed operator has invariant eigenvalues and eigenfunctions. Varying matrix dimensions correspond embedding the operator in different finite- miite-dimensional linear spaces. For a fixed number of eigenvalues L, larger matrices yield more accurate approximations of the true eigenvalues. That is, larger matrices reduce computational noise and enhance operator similarity visibility, enabling SCSF to utilize these similarities more effectively for superior performance. For comparisons with neural networks, similarity impact, and edge-case performance, see Appendices
E.6 E.6 E.7 and E.8
4.3 EFFICACY OF CHEBYSHEV SUBSPACE FILTER
Table 2: Impact of initial subspace modifications on average computation time (in seconds) for different algorithms. denotes the modified version. The first row lists algorithms, and the first column shows the number cigenvalues L computed. The best algorithm bold, and s.r indicates the result of method that fails to converge under the given setting.
L
Eigsh
Eigsh*
LOBPCG
LOBPCG*
KS
KS*
JD
JD
SCSF (ours)
200
151.7
150.2
129.9
95.9
98.34
100.6
489.6
760.1
31.31
300
208.8
206.3
270.1
199.8
179.9
185.2
1803
3101
38.67
400
253.5
249.1
460.4
362.1
283.0
292.2
3829
6374
40.52
500
324.6
315.3
717.3
573.7
314.2
317.4
46.70
600
398.8
394.7
1031
866.0
329.6
335.7
51.32
To analyze the efficacy the Chebyshev subspace filter, we conducted the following experiments. After sorting, the initial vector subspace for the cxisting algorithms was set to the eigenvectors from the previous problem (the modified version **'). We compared the computational time across different methods. All experiments were conducted the Helmholtz operator dataset, with matrix dimension of 6400 and a tolerance of 1e-8. The results are shown in Table
First, the computation time for SCSF in all experiments efficacy of the Chebyshev subspace filter. This also highlights that the Chebyshev subspace filter is the optimal choice for leveraging problem similarity reduce redundancy.
minimal, clearly demonstrating the
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Second, the impact of initial setup modifications varied across algorithms: 1. LOBPCG accelerated significantly due to its subspace-base basue logic, similar to SCSF, where initialization strongly influences convergence. 2. Eigsh and KS remained largely unaffected they rely initial vectors and Krylov iteration, making problem similarity less impactful. 3. JD showed a performance decline. This is because its performance is sensitive to the size of the initial subspace. Our modification altered the default dimension of the initial subspace.
Table 3: Performance comparison of SCSF with and without sorting. The first column lists the number of eigenvalues computed, while subsequent columns display average computation times, average iteration counts, total Flop counts, and filter Flop counts. Experiments used the matrix dimension of 2500 and precision
on the generalized Poisson operator dataset.
Time (8)
Iteration
Flops
Filter Flops
w/o sort
w/o sort
w/o sort
w/o sort
sort
20
8.248
2.971
19.70
9,880
519.7
298.4
485.8
280.8
100
14.18
9.891
18.77
15.38
1984
1332
1798
970.1
200
18.45
12.85
36.30
33.67
4459
3944
3654
3192
300
34.59
25.61
47.50
39.18
8967
7544
6985
5702
400
42.60
33.91
47.43
45.18
12022
11182
9087
8338
4.4
EFFICACY OF SORTING ALGORITHMS
We analyze the performance of the sorting algorithm module from two perspectives: 1. Comparing the performance of SCSF algorithm with and without 'sorting' shown Table 2. Evaluating the effectiveness of different sorting algorithms as detailed in Tables and We note that if the setting is "w/o sort', SCSF approximately equivalent directly using the Chebyshev subspace filter. Unlike the ChFSI used in the main experiments, the initialization of each solve in the "w/o sort' SCSF is set based on the information obtained from solving previous problem (following the default unsorted sequence).
Firstly, Table Table indicates that incorporating sorting improve SCSF speed to 1.3 to 2.8 times, duce the number of iterations by 5% to 50%, and decrease total Flops by 7% to 43%. The effect of sorting is more pronounced with smaller numbers of solutions L. This is because when L is large, the inherited subspace already contains most of the necessary correlation information, diminish- ing the impact of sorting. Moreover, the Flops in the Filter component constitute over 70% of SCSF's computational load. A detailed time anal- ysis of different aspects of SCSF can found Appendix E.3] Additionally, the "w/o sort' SCSF achieves computational speedup
Table 4: Comparison of average computation times (in seconds) for different sorting algo- rithms, with the first column indicating dataset size. Experiments used the matrix dimension of 6400 on the Helmholtz dataset.
Greedy Total
Truncated FFT Sort (ours)
Size
FFT
Greedy
Total
10a
0.114
0.0016
0.0147
0.0163
10
7.328
0.0164
1.421
1.438
10*
592.7
0.1658
150.9
151.1
1.2 to 1.5
times compared to the ChFSI used in the main experiments. The primary difference lies in their initialization strategies: ChFSI uses random initialization for each solve, whereas the "w/o sort' SCSF leverages information from the previous problem for initialization. This indicates that, even without sorting, there is a certain level of similarity between problems the dataset. Such similarity can effectively accelerate the solving process.
Secondly, as shown Table our designed truncated FFT sorting algorithm incurs significantly lower time cost compared to the complete greedy sorting in SKR (Wang et al. (2024), with its benefits becoming more pronounced as the dataset size increases. In the truncated FFT sorting algorithm, the FFT contributes minimally computational overhead but significantly reduces the time required for subsequent greedy sorting.

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Table 5 validates the accuracy of efficient sorting against the expensive greedy approach. To ensure fair- ness, averaged the results over 100 independent runs to eliminate fluctuations. The results for 'Greedy' and "Ours' identical because the sequences produced by both algorithms are over 98% identical. Consequently, the downstream solver incurs exactly the same operation cost, demonstrating that our low-cost sorting achieves the same acceleration performance as the high-cost greedy approach.
Table Comparison average com- putation times and iteration counts for different sorting algorithms using SCSF. Experiments used the matrix dimension 6400 on the Helmholtz dataset, preci- sion le-8, and targeting 400 eigenvalues.
w/o sort
Greedy
Ours
Time (s) Iteration
66.66
40.52
40.52
10.4
5.5
5.5
Furthermore, our experiments demonstrate that the choice of the truncation threshold, Po. is robust. We found that excellent performance is achieved with relatively small Po (e.g., Po 20), value much smaller than the parameter matrix dimension This is because performance stabilizes once a sufficient number of low-frequency components are captured, making unnecessary to retain large number of components. For related experiments, please refer to Appendix 4.3
5
CONCLUSIONS AND FUTURE WORK
Conclusions In this paper, we addressed the critical bottleneck of generating large-scale eigenvalue datasets for training neural operators. We introduced SCSF, the first method to accelerate eigenvalue dataset generation by exploiting operator similarity. By integrating a truncated FFT sorting algorithm with Chebyshev subspace filter, SCSF transforms the generation task into an efficient sequential solving problem. Our method achieves up to a 3.5x speedup over traditional solvers, significantly reducing computational redundancy. By lowering key barrier entry, SCSF provides valuable tool for advancing research in the AI for Science community.
Limitations and Future Work Despite SCSF's demonstrated efficiency, we have identified three key directions for future development. First, we aim extend the current framework from Hermitian linear operators to non- mon-eemiitian and non-linear operators. Second, the correlation between operators currently measured using the Frobenius norm of their parameter matrices. We plan to explore more effective, problem-specine metrics rather than relying on fixed one. Finally, SCSF's effectiveness hinges on continuous mapping between parameters and operators, rendering ineffective when this mapping is discontinuous. Future work will involve measuring correlation directly from the operators or their matrix representations bypass the challenges of discontinuous parameterization.
ACKNOWLEDGEMENTS
The authors would like 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 (No. 10222D0119801)) the National Natural Science Foundation of China (Nos. U23A20388, 62021001), and the Smart-Gria National Science and Technology Major Project (No. 20252D0805500),
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A USAGE OF LLMS
Throughout the preparation of this manuscript, Large Language Models (LLMs) were utilized as a writing and editing tool. Specifically, we employed LLMs to improve the clarity and readability of the text, refine sentence structures, and correct grammatical errors. All final content, including the core scientific claims, experimental design, and conclusions, was conceived and written by us, and we take full responsibility for the final version of this paper.
B RELATED WORK
B. EIGENVALUE DATASETS AND NEURAL EIGENVALUE METHODS
Eigenvalue datasets are widely utilized in neural eigenvalue methods. In molecular chemistry research, eigenvalue algorithms are commonly employed to determine critical molecular properties, such as orbital energy levels (Kittel McEuen 2018). These properties form the foundation of datasets and obtained by solving the cigenvalue problem of the Schrödinger equation and the Hamiltonian operator (Helgaker et al. 2013). Prominent datasets in this domain include QM7 (Blum & Reymond 2009), QM9 (Ramakrishnan et al., 2014), ANI-1 (Smith et al., 2017), and MD17 (Chmiela et al. 2017). In materials science, eigenvalue algorithms are often applied to solve for electronic band structures and density of states in materials. Representative datasets in this field include the materials project (Jain et al., (2000) and OQMD (Kirklin et al.) 2015). These datasets have been extensively used train and validate neural eigenvalue methods (Schütt al. 2017; Bartók al., 2017 Rupp it al. 2012). driving ad advancements in molecular property prediction and materials design. In Hludd dynamics and structural mechanics, eigenvalue algorithms are frequently utilized for modal analysis. Recently, many data-driven modal analysis algorithms have emerged, requiring eigenvalue datasets corresponding to differential operators for training (Murata et al. 2020 Iwata & Kawahara 2023) Alford-Lago et al. 2022 Brunton et al. 2020: Akkad & He 2023 Wang et b). Additionally, some studies leverage operator eigenvalue datasets to optimize algorithms. For instance, Luo et al.
accelerates the solution of linear systems by predicting the eigenfunctions of operators.
B.2 EIGENVALUE DATA GENERATION ALGORITHMS
Training data-driven algorithms require a large amount of labeled eigenvalue data. Typically, the generation of these high-precision data is obtained by traditional algorithms. In the field of computa- tional mathematics, solving operator eigenvalue problems often involves utilizing various discretiza- tion methods such as finite difference methods (FDM) (Strikwerda] (004), finite element methods (FEM) (Hughes 2012, Johnson 2012). These discretization methods transform operator eigenvalue problems into matrix cigenvalue problems, which are then solved using the corresponding matrix algorithms. For larger matrices, the Krylov-Schur algorithm (Stewart, 22022, Jacobi-Davidsse Davidson (Slei jpen & Van der Vorst 2000), and locally optimal block preconditioned conjugate gradient (LOBPCG) (Knyazev 2001) among the most frequently employed algorithms (Golub & Van Loan 2013).
Nonetheless, traditional methods were not designed for dataset generation, resulting in high com- putational costs, which have become significant barrier to the advancement of data-driven ap- proaches (Zhang et al., 2023 Hao et al. 2022 Chen et al.] 2025). Recent data augmentation research (Brandstetter et al.) 2022 Liu al.] 2023) has led to the development of methods that preserve symmetries and conservation laws, enhancing model generalization and data efficiency. Wang al. (2024); Dong et al. (2024) report acceleration in the process of solving linear equations, thereby speeding up the generation of PDE datasets.
However, these improvements largely focus on neural networks or the rapid solution of linear sastem-baee PDEs, without discussing optimizations in the generation of eigenvalue datasets.
B.3 CHEBYSHEV FILTER TECHNIQUE
The Chebyshev filter technique originates from polynomial approximation theory, where the core concept involves using Chebyshev polynomials to accelerate the convergence of eigenvalues (Zhoul & Saad 2007). This technique constructs a polynomial filter that selectively amplifies spectral components in specified interval, thereby speeding up the solution of specific eigenvalues. This
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technique is particularly effective in dealing with sequence eigenvalue problems (Sand 2011 Zhou et al. 2006a) and has been applied in various contexts, such as stability analysis in electronic structure (Preper al., 2016 Banerjee et al. (201)) and quantum chemical computations (Mohr et al. Zhou et al. 2014 2006b).
2017
Due to the chaotic and disordered nature of eigenvalue problems in the dataset, directly applying the Chebyshev filter technique fails to accelerate dataset generation. To further adapt this technique to the generation of operator eigenvalue datasets, have developed a specialized sorting algorithm that transforms dataset generation into sequence eigenvalue problems. Throughout the solving process, eigenpairs obtained from previous solutions are used construct Chebyshev filters, accelerating subsequent solutions.
C FROM DIFFERENTIAL OPERATOR TO MATRIX EIGENVALUE PROBLEM: AN EXAMPLE
C.I OVERVIEW
The general methodology for solving the eigenvalue problems of differential operators numerically, employing techniques such as Finite Difference Method (FDM), Finite Element Method (FEM), and Spectral Method, can delineated through the following pivotal steps (Strikwerda) 2004 Hughes
2012
Johnson
2012
Le Veque 2002
Wei et al. 2024):
Mesh Generation: This step involves dividing the domain, over which the differential operator defined, into discrete grid. The grid could be composed of various shapes, including squares, triangles, or more complex forms, depending on the problem's is geometry.
2. Operator Discretization: The differential operator is transformed into its discrete counterpart. Essentially, this maps the operator from an representation.
Hilbert space to a diiee-dimessional
Matrix Assembly: In this phase, the discretized operator represented in a matrix form. For lincar differential operators, this involves creating system operators, iterative methods akin to Newton's 's iteration are employed, transforming the problem into a sequence of matrix eigenvalue problems.
of matrix eigenvalue problems. For nonlinear
4. Applying Boundary Conditions:
This involves discretizing and applying boundary conditions specific the differential operator in question, which are then incorporated into the matrix system.
5. Solving the Matrix Eigenvalue Problem: entails solving the matrix for its cigenvalues and cigenvectors, which correspond the eigenvalues and eigenfunctions of the original differential operator.
This stage, often the most computationally intensive,
Obtaining the Numerical Solution: The final step involves mapping the obtained numerical solutions back onto the original domain, analyzing them for accuracy and stability, and interpreting them in the context of the initial problem.
C.2 EXAMPLE
To illustrate how the FDM can transform the wave equation into a system of matrix eigenvalue problems, let's consider concrete and straightforward example. Assume we aim to solve one- dimensional wave equation's 's operator eigenvalue problem, expressed as
duu
Xu,
dx
the interval L]. The boundary conditions u(0) u(L) 0, signifying fixed-end conditions. In this context, u(x) denotes the eigenfunction, and À represents the eigenvalue.
1. Mesh Generation:
Using the central difference quotient, divide the interval [0, L] into N + 1 evenly spaced points, including the endpoints. The distance between adjacent points denoted as Ar
Operator Discretization: This step involves formulating the difference equation. At each inte- rior node, which excludes the endpoints and totals N points, apply central difference
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approximation for the second derivative, represented
du dx²
R u(x(+1) 2u(x)) + u (x)-1)
(Az)²
3. Matrix Assembly: In this phase, the discretized operator is represented in matrix form. Following
the approximation, we construct the matrix A, an N tridiagonal matrix, crucial for the computations. The matrix A is constructed
0
1
2
1
0
2
A
(Ar)²
1 ", 0
0
2
Applying Boundary Conditions: For the wave equation with boundary conditions u(0) u(L) O, these fixed-end conditions are integrated into the matrix equation. In the FDM framework, the values at the endpoints (ug and uN) are zero, directly reflecting the boundary conditions. The impact of these conditions is encapsulated in the matrix A. affecting the entries related to ú1 and uN-1 (the grid points adjacent to the boundarics). The tridiagonal matrix incorporates these boundary conditions, ensuring that the computed eigenfunctions satisfy u(0) u(L) 0.
The final computational step involves solving the matrix eigenvalue problem, expressed Au Au. This includes determining the eigenvalues À and corresponding eigenvectors u, which are discrete approximations of the eigenfunctions of the original differential equation.
Solving the Matrix Eigenvalue Problem:
Obtaining the Numerical Solution: By solving the eigenvalue problem, we obtain numerical solutions that approximate the behavior of the original differential equation. These solutions reveal the eigenvalues and eigenvectors and provide insights into the physical phenomena modeled by the equation.
D DETAILS OF EXPERIMENTAL SETUP
D.1 BASELINE
The baseline algorithms were implemented using the following numerical computing libraries:
Eigsh: A SciPy (v1.14.1) implementation wrapping ARPACK's SSEUPD and DSEUPD functions, which compute eigenvalues and eigenvectors using the Implicitly Restarted Lanezos Method. The default parameters were used.
Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG): Implemented in SLEPc (v3.21.1) with default parameters.
Krylov-Schur (KS): Implemented SLEPe (v3.21.1) with default parameters.
Jacobi-Davidson (JD): Implemented in SLEPe (v3.21.1). The implementation uses 'begsl" as the linear equation solver, 'bjacobi' as the preconditioner, and sets the linear equation solving precision le-5.
ChFSI: Implemented in ChASE (v1.6) with default parameters.
D.2 DATASET
All operators in this paper use Dirichlet boundary conditions.
1. Generalized Poisson Operator
We consider two-dimensional generalized Poisson operators, which can be described by the following
equation (Li et al. 2020 Rahman al.] 2022 Kovachki et al. 2021: Lu et al. 2022 Gao et al. 2025):
(K(,, v) Vh( X)(r,v),
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In our experiment, K (2, y) is derived using the Gaussian Random Field (GRF) method. We convert these operators into matrices using the central difference scheme of FDM. The parameters inherent to the GRF serve as the foundation for sort scheme.
Second-Ordes Elliptic Partial Differrntial Operator
We consider general wooddimensional second-order elliptic partial differential operators, which are frequently described by the following generic form (Evans) 2022; Bers et al. 1964):
Lu = Urz Mrz + 012llay + 022"ly + 01 02tly + agu = Xu,
where do, d1,, (22 01 d12, 022 are constants, and f represents the source term, depending on æ, y- The variables My are the dependent variables and their partial derivatives. The equation is classified as elliptic 4011022 > a/2-
In our experiments, a11 022, d1, 02 do are uniformly sampled within the range (-1, 1), while the coupling term 412 is sampled within (-0.01,
We then select equations that satisfy the elliptic condition to form our dataset. We convert these operators into matrices using the central difference scheme of FDM. The coefficients 0o, d1, 02 011, 412, 022 serve as the foundation for our sort scheme.
3. Helmholtz Operator
We consider two-dimensional Helmholtz operators, which can be described by the following equa- tion (Zhang al. 2022):
+ (2, Au(x,y),
Physical Contexts in which the Helmholtz operator appears: 1. Acoustics; 2. Electromagnetism; 3. Quantum Mechanics.
In Helmholtz operators, is the wavenumber, related to the frequency of the wave and the properties of the medium in which the wave is propagating. our experiment, p(æ; y) and k(2, y) are derived using the GRF method. The parameters inherent to the GRF serve as the foundation for our sort scheme.
4. Vibration Equation
We consider the vibration equation for thin plates, which can be described by the following eigenvalue problem (uee et al. 2018):
= Ap(a, y)u(,, y).
Physical contexts in which the vibration equation appears: Structural dynamics of thin plates; Modal analysis in mechanical engineering, 3. Vibrational behavior of elastic materials.
In this equation, (xx y) represents the flexural rigidity of the plate, p(z, y) is the density distribution, and X corresponds to the eigenvalue, which is related to the natural frequencies of the system. The eigenfunction u(x, y) describes the mode shapes of vibration.
In our experiment, (xr, y) and p(æ, y) are derived using the GRF method. The parameters inherent to the GRF serve as the foundation for our sorting scheme.
D.3 ENVIRONMENT
To ensure consistency in evaluations, all comparative experiments conducted under uniform computing environments. Specifically, the environments used are detailed as follows:
Platform:
Docker version 4.33.1 (windows 11)
Operating System: Ubuntu 22.04.3 LTS
Processor: CPU AMD Ryzen 8945HS w, clocked 4.00 GHz
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D.4 EXPERIMENTAL PARAMETER CONFIGURATION
All baseline methods were implemented using their default parameters from respective libraries. For SCSF, the following configurations were adopted:
The size of the inherited subspace varies according to the number of eigenvalues to be computed. Specifically, when calculating 20, 100, 200, 300, and 400 eigenvalues, the corresponding subspace sizes are set to 4, 20, 40, 60, and 80, respectively.
The filter degree parameter m is consistently set to 20 across all experiments.
Truncation threshold for low frequencies is consistently set to 20 across all experiments.
Each experiment generates a dataset consisting of 1,000 samples. In this paper, the Experi- mental tables report the average solving time for each eigenvalue problem.
D.5
ERROR METRICS
Relative Residual Error:
To estimate the bias of the eigenpair (ū, X) predictions, we employ relative residual error as follows:
||4 Aöllz
Relative Residual Error
||
Here, ő represents the eigenfunction predicted by the model, and denotes the eigenvalue predicted by the model. When À is the true eigenvalue and is the true eigenfunction, the Relative Residual Error equals O.
D.6 RELATIONSHIP WITH PARALLELIZATION AND EXPERIMENTAL SETTING
The SCSF framework is designed to be complementary parallel computing architectures; the relationship is both orthogonal and synergistic. Fundamentally, SCSF accelerates the serial processing of sequence of related eigenvalue problems. In practical, large-scale setting, total dataset of N problems can be partitioned into M independent chunks. Subsequently. M instances of the SCSF algorithm can be executed in parallel across M computing units, with each computing unit responsible for solving one chunk.
To ensure fair and direct comparison of algorithmic efficiency under practical, parallelized condi- tions, all experiments reported in this paper were conducted using the Message Passing Interface (MPI) with cores in parallel.
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E EXPERIMENTAL DATA AND SUPPLEMENTARY EXPERIMENTS
E. MAIN EXPERIMENTAL DATA
As shown in Tables
SCSF showed the best performance among all tested configurations
Table 6: Comparison of average computation times (in seconds) for eigenvalue problems using various algorithms on the generalized Poisson operator dataset. The first row lists different algorithms, and the first column shows the number of eigenvalues L computed for each matrix. Matrix dimension 2500, precision le-12
L
Eigsh
LOBPCG
KS
JD
ChFSI
SCSF (ours)
150
9.15
46.8
14.9
138
17.3
7.95
200
14.2
73.0
23.8
270
24.0
12.9
250
19.8
109
34.3
553
30.2
19.0
300
26.3
152
45.6
921
38.0
25.7
350
31.5
203
58.4
1732
45.8
29.8
400
36.9
265
72.3
2691
57.4
33.9
450
42.8
342
87.3
3708
74.2
38.3
Table 7: Comparison of average computation times (in seconds) for eigenvalue problems using various algorithms on the second-order elliptic operator dataset. The first row lists different algorithms, and the first column shows the number of eigenvalues L computed for each matrix. Matrix dimension 4900, precision = le-10.
L
Eigsh
LOBPCG
KS
JD
ChFSI
SCSF (ours)
150
31.35
91.80
40.65
214.80
38.37
19.62
200
41.82
139.20
61.77
414.30
43.90
24.08
250
52.17
197.04
84.65
861.44
53.42
28.00
300
62.47
264.10
110.50
1446.00
60.69
29.88
350
74.59
355.18
147.01
2324.88
64.94
31.52
400
87.19
459.70
188.70
3386.00
67.13
34.60
450
100.28
577.67
235.56
4629.38
76.32
40.05
Table 8: Comparison average computation times (in seconds) for eigenvalue problems using various algorithms on the Helmholtz operator dataset. The first row lists different algorithms, and the first column shows the number of eigenvalues L computed for each matrix. Matrix dimension = 6400, precision 1e-8. The symbol 5.' denotes data not recorded due to excessive computation times.
Eigsh
LOBPCG
KS
JD
ChFSI
SCSF (ours)
200
151.70
129.90
98.34
489.60
107.12
31.31
300
190.84
273.08
192.88
1601.08
113.73
37.78
400
253.50
460.40
283.00
3829.00
121.53
40.52
500
344.60
720.33
310.21
135.73
47.41
600
398.80
1031.00
329.60
146.24
51.32
E.2 ANALYSIS OF THE INFLUENCE OF MATRIX DIMENSION
As demonstrated in Table 10 the impact of matrix dimension algorithm performance reveals several key insights. For matrices below dimension 3600, SCSF and Eigsh show comparable efficiency. However, SCSF's advantages become increasingly pronounced matrix dimensions grow larger.
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Table 9: Comparison of average computation times (in seconds) for eigenvalue problems using various algorithms on the Vibration operator dataset. The first row lists different algorithms, and the first column shows the number of eigenvalues computed for each matrix. Matrix dimension 10000, precision 1e-8. The symbol denotes data not recorded due to excessive computation times.
L
Eigsh
LOBPCG
KS
JD
ChFSI
SCSF (ours)
200
397.9
333.7
272.0
1230
300.8
85.70
300
516.8
750.0
520.0
3600
305.0
96.50
400
635.6
1170
768.8
310.5
107.2
500
820.0
1950
810.0
350.0
120.0
600
1037
2716
857.8
382.3
131.4
Table 10: Comparison of different algorithm computation time (in seconds) for varying matrix dimensions using the generalized Poisson operator dataset. Results show average computation times for solving 400 eigenvalues with a precision of 1e-12.
Matrix Dimension
Eigsh
LOBPCG
KS
JD
ChFSI
SCSF (ours)
2500
36.86
265.30
72.32
2691.00
57.41
33.91
3600
66.41
387.20
116.50
2990.00
102.4
65.41
4225
89.13
467.74
151.36
3548.13
126.2
70.79
4900
121.90
546.20
187.80
3886.00
153.5
74.23
5625
186.21
691.83
251.19
216.8
85.11
6400
282.80
860.00
337.70
282.2
93.86
8100
707.95
1412.54
707.95
435.1
114.82
10000
3162.28
2511.89
1995.26
590.3
158.49
At dimension 10000, SCSF achieves remarkable speedups: 20x faster than Eigsh, 16x faster than LOBPCG, 13x faster than KS, and 3.7x faster than ChFSI. This phenomenon can be attributed to how larger matrix dimensions result in fewer errors and noise in the computed eigenvalues, allowing SCSF better exploit similarities between problems. Additionally, the JD algorithm becomes computationally intractable at and above dimension 5625, while SCSF maintains stable performance even at high dimensions.
E.3 ANALYSIS OF COMPUTATIONAL. TIMES FOR SCSF COMPONENTS
Table 11: Analysis of Computational Times (in seconds) for SCSF Components.
All
Filter (line 3)
QR (line 4)
RR (line 5)
Resid (line 6)
Sort
9.89c+0
7.4le+0
3.12e-1
9.76e-1
7.95e-1
1.5le-2
We conducted a statistical analysis of the average time consumption for each component of the SCSF algorithm on the generalized Poisson operator dataset, with matrix dimension of 2500 and the number of eigenvalues to be solved set to 100. The results presented in Table [I The notation 'line x' within parentheses corresponds to line x in Algorithm 3 'ALL' denotes the total time consumption, and 'sort' represents the average time required by the sorting algorithm. It is evident that the filtering process accounts for over 70% of the total time consumption, which aligns with our theoretical analysis in Section 3.2
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E.4 ANALYSIS OF HYPERPARAMETERS
E.4.1
DEGREE PARAMETER
Table 12: Average Computational Times (in seconds) of SCSF under Different Degree Parameters m.
Deg 12
16
20
24
28
32
36
40
Time (s)
43.92
39.79
40.52
40.64
40.85
41.13
41.19
43.50
We investigated the impact of different degree parameters the performance of SCSF, As shown in Table 12 the experiments were conducted on the Helmholtz operator dataset with a matrix dimension of 6400, solution accuracy of le-8, 400 eigenvalues be solved, and an inherited subspace size of 80. The degree parameter m, as described in A Algorithm 3 primarily controls the order of the Chebyshev polynomial. The results indicate that varying within the range of 12 to 40 has minimal effect on the computation time of SCSF. Therefore, as long as m is chosen within a reasonable range, its specific value does not significantly influence the performance. In the main experiments of this paper, m is fixed at 20.
E.4.2 SUBSPACE DIMENSION
Table 13: Average Computational Times (in seconds) of SCSF under Different Subspace Dimension.
Dim
50
60
70
80
90
100
110
120
Time (s)
43.28
44.35
42.43
40.52
39.65
37.43
38.28
38.58
We examine the influence of different inherited subspace sizes on the performance of SCSF, As presented Table [3] the experiments are conducted on the Helmholtz operator dataset with matrix dimension of 6400, solution accuracy of 1e-8, 400 eigenvalues to be computed, and a degree parameter set
The results demonstrate that as the inherited subspace size increases, the computation time of SCSF initially decreases and then rises, reaching its minimum around a size of 100. The reduction in computation time at the front end is attributed to the enriched initial subspace with more available information as the inherited subspace grows. Conversely, the increase in computation time at the back end is due to the significantly higher overhead of performing Chebyshev filtering with larger inherited subspace.
Overall, as long the inherited subspace size is set within reasonable range, its impact SCSF remains minimal. our experiments, we consistently set the inherited subspace size to 20% of the number of eigenvalues to be computed.
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E.4.3 TRUNCATION THRESHOLD FOR Low FREQUENCIES
Table 14: Average computational times (in seconds) of SCSF under different truncation thresholds.
No sort
Po =
10
Po
20
Po = 30
Pp = 40
Greedy
One-sided distance
0.95
0.89
0.85
0.85
0.85
0.85
Sort time (s) Average solve time (s)
0
110
151
193
246
593
66.7
52.2
40.5
40.5
40.5
40.5
We measure the similarity between matrices by computing the cosine of the principal angles between their 10-dimensional invariant subspaces (spanned by the smallest 10 eigenvectors in modulus) (one- sided distance), Smaller values indicate higher similarity. As presented in Table [4] the experiments are conducted on the Helmholtz operator dataset with a matrix, a solution accuracy of le-8, 400 eigenvalues to be computed, and degree parameter m set 20, 10k data problems, parameter matrix P with dimension 80, and varying truncation frequencies Po
The results demonstrate that sorting significantly increases inter-problem correlation in the dataset (explaining the performance gain). The truncation parameter affects sorting time, sorting quality, and solver time. For Po 2 20, solver time becomes stable, showing diminishing returns. This reflects the interplay between sorting and Chebyshev iteration. In the main experiments of this paper, Po is fixed at 20.
E.5 RELIABILITY OF GENERATED DATA AS GROUND TRUTH
A key concern was whether the data generated by our method, which relies numerical solvers, is reliable 'ground truth' for training neural networks. To address this, we trained a NeurKItt (Luo et al. 2024) model generalized Poisson datasets generated by various solvers (including our SCSF) different matrix dimensions. The precision for all solvers was set to a high tolerance of 10-12.
Table 15: Impact of data generation method on the performance of a trained NeurKItt model. The consistent final loss indicates that data from all tested solvers serves as a reliable ground truth.
Generation Method
Matrix Dimension
Generation Time
NeurKItt Principal Angle Loss
Figsh
2500 / 6400 / 10000
10h / 80h / 800h
0.06 / 0.06 / 0.06
LOBPCG
2500 6400 / 10000
70h 240h 700h
0,06 0.06 0.06
ChFSI
2500 / 6400 / 10000
16h / 44h / 160h
0.06 / 0.06 0.06
SCSF (ours)
2500 6400 / 10000
9h / 26h 45h
0,06 / 0.06 0.06
E.6 COMPARISON WITH SUPERVISED AND UNSUPERVISED NEURAL NETWORK METHODS
To clarify the significance of accelerating dataset generation for the dominant supervised learning paradigm, we conducted an experiment comparing the performance and resource trade-off's of different categories of eigensolvers. We evaluated our method (SCSF), a traditional solver (Eigsh), supervised neural network INt Luo et al. (2024)), and two state-of-the-ae unsupervised neural networks (NeuralEF Deng al. (2022) and NeuraiSVDi Ryu et al. (2024)) on a 2D Helmholtz problem (solving for the smallest 100 eigenvalues, matrix dimension 6400).
The results, presented in Table T6 highlight the distinct characteristics of each approach. Unsu- pervised methods obviate the need for pre-generated datasets but require substantial 'solving time' for each new operator, as they essentially perform an optimization from scratch. In contrast, super- vised methods offer near-instantaneous inference but demand significant upfront investment in both data generation and model training. Our method, SCSF, dramatically reduces the data generation bottleneck for these powerful supervised models.
As shown in Table 15 the final performance of the trained NeurKItt model (measured by Principal Angle Loss) identical regardless of which high-precision solver used for data generation
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Table 16: Comparison of different eigensolver paradigms on 2D Helmholtz problem. 'Solving Time' for unsupervised methods refers to the entire optimization process required to find the solution for single operator instance.
Category
Algorithm
Solving Time
Training Time
Dataset Gen. Time
Relative Residual
Method
SCSF (random init)
10-3
Traditional
Eigsh
min
10-6
Supervised NN
NturKItt
0.1s
3h
20h
10-2
Unsupervised NN
NeuralEF
2h
10
Unsupervised NN
NeuralSVI
3h
10-2
the specific matrix dimension (for dimensions 2 2500). This demonstrates that the discretization and solver errors are orders of magnitude smaller than the neural network's 's approximation error, confirming that the generated data serves as highly reliable ground truth for training purposes.
E.7 PERFORMANCE BOUNDS AND THE IMPACT OF DATASET SIMILARITY
To provide theoretical insight into SCSF's performance bounds, we conducted an experiment to quantify the relationship between dataset similarity and acceleration. We generated sequence of Helmholtz operator problems where each subsequent problem is a slight perturbation of the previous one. The magnitude of this perturbation reflects the dataset's internal similarity. A smaller perturbation size indicates higher similarity. The experiment was run on the Helmholtz dataset (dimension 6400, L 200 eigenvalues).
Table 17: Average solution time (seconds) function of dataset similarity (perturbation size). Lower perturbation implies higher similarity. The experiment was run on the Helmholtz dataset (dim 6400, 200).
Perturbation Size
Eigsh
LOBPCG
ChFSI
SCSF (w/o sort)
SCSF
50%
151
130
107
76
27
10%
150
129
107
48
23
1%
152
130
107
14
6
0% (Identical)
151
130
107
2
2
Standard Generation
152
130
107
82
31
Independent Problems
152
130
107
107
107
The results in Table show that SCSF's performance strongly correlated with dataset similarity. As problems become more similar (perturbation size decreases), the speedup increases dramatically. The experiment also highlights the effectiveness of our sorting algorithm; SCSF consistently outperforms SCSF without sorting ('SCSF w/o sort') across various similarity levels. In the theoretical limit of identical problems (0% perturbation). the solution found in just a few iterations. Conversely, for completely independent problems, SCSF's performance gracefully degrades that of ChFSI, as expected.
E.8 ANALYSIS OF FAILURE CASES: DISCONTINUOUS DATASETS
The core assumption of SCSF that the dataset is generated from process with underlying continuity, allowing our sorting algorithm to group similar problems effectively. To investigate the behavior of SCSF when this assumption (Helmholtz and Poisson). We varied the proportion of Helmholtz operators from 100% (a fully continuous dataset) down to 0% (a different fully continuous dataset), with mixed ratios between representing varying degrees of discontinuity. (dimension 6400, L 200 eigenvalues) and solved them sequentially.
violated, We simulated gradual mixing of the two datasets
Table 18 presents the results. As expected, the performance gain of SCSF is reduced in this discontin- uous scenario because the inter problem correlation is disrupted, diminishing the effectiveness of the sorting module. However, in this challenging case, SCSF still provides notable speedup
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Table 18:
Performance on discontinuous dataset created by mixing Helmholtz and Poisson problems. All times are in seconds.
Helmholtz %
Eigsh
LOBPCG
ChFSI
SCSF (w/o sort)
SCSF
100% (Helmholtz)
152
130
107
46
31
75%
154
203
116
94
78
50% (1:1 mix)
154
280
132
118
98
25%
152
359
141
108
80
0% (Poisson)
149
454
153
81
52
baseline solvers, demonstrating a degree of robustness. The performance of 'SCSF' approaches that of "SCSF (w/o sort)", confirming that the sorting component's 's benefit is tied to dataset continuity.
E.9 COST-BENEFIT ANALYSIS OF THE SORTING ALGORITHM
To address the trade-off between the cost of sorting and its benefits, we analyzed its computational overhead. Our analysis shows that the cost of the Truncated FFT Sort is negligible in the context of large-scale dataset generation. For example, as shown in Tf the main paper, sorting a dataset of 10a samples takes approximately 151 seconds. In contrast, solving single eigenvalue problem from the Helmholtz dataset can take over 250 seconds with a standard solver like Eigsh. For a full dataset of this size, the sorting overhead constitutes less than 0.1% of the total generation time.
The benefit, however, is substantial. As shown in Table 3 sorting reduces the number of solver iterations by 5-50% and total floating-point operations (Flops) by 7-43%. Furthermore, our Truncated FFT Sort is significantly more cost-effective than naive greedy sort, achieving nearly identical final solver performance at a fraction of the computational cost (see Tables 4 and (5). Given this highly favorable cost-benefit ratio, the sorting step is a crucial and efficient component of the SCSF framework.
E. SENSITIVITY TO PARAMETERIZATION
To assess the sensitivity of the proposed parameter sorting different PDE parameterizations, conducted an additional experiment comparing two distinct discretization methods for the Helmholtz problem: the Finite Difference Method (FDM) (utilized in the main text) and the Finite Element Method (FEM/Galerkin).
As shown in Table 19| SCSF demonstrates consistent speedups across both discretization schemes. For the FEM dataset, we utilized a dimension of 10,000 with tolerance of le-8.
Table 19: Comparison of SCSF performance under different parameterizations (Helmholtz). Times are in seconds. '_' indicates non-convergence or excessive time cost.
Dataset
Eigsh
LOBPCG
KS
JD
ChFSI
SCSF (ours)
200
151.7
129.9
98.34
489.6
107.1
31.31
FDM (Central Diff)
400
253.5
460.4
283.0
3829
121.5
40.52
Dim: 6400, Tol: le-8
600
398.8
1031
329.6
146.2
51.32
200
623.1
515.7
365.6
1885
450.9
116.7
FEM (Galerkin)
400
938.5
1746
1166
481.3
142.9
Dim: 10000, Tol: 1e-8
600
1499
1299
599.9
196.3
These results confirm that SCSF remains highly effective regardless of the discretization scheme, provided the relationship between parameters and operators remains continuous.
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E.11
GENERALIZATION OF TRUNCATION LEVEL (po)
To justify this universal constant, we calculated the ratio of the Frobenius norm of high-frequency components (frequency > 20) the total norm for each dataset. The results are presented in Table
20
Table 20: Ratio of high seghfrreuency component energy (frequency > different PDE datasets.
Po 20) to total energy across
Dataset
Poisson
Ellipse
Helmholtz
Vibration
High-frequency Ratio (> Po) 3,4%
3.7%
4.4%
4.8%
As shown, high-frequency components account for less than 5% of the total energy across all diverse PDE families. This confirms that Po 20 consistently captures over 95% the structural information inherent in the parameter matrices.
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F THEORETICAL ANALYSIS OF TRUNCATED FFT SORTING
Our sorting algorithm approximates the distance between operators to optimize the solving sequence. To analyze the theoretical validity of this approximation, we consider the isometric property of the Discrete Fourier Transform (DFT). Let p() and p() denote the discretized parameter matrices (e.g., diffusion coefficients) for two distinct problems. By Parseval's identity, the DFT is unitary transformation (up scaling factor), implying that the Frobenius distance in the spatial domain is strictly equivalent to that in the frequency domain:
| p() || |FFT(P()) - )
where FFT(-) denotes the normalized DFT. The SCSF algorithm computes a proxy distance using truncated spectrum, retaining only frequencies within low-frequency bandwidth po- Consequently, the exact distance can be decomposed into the sorting metric and residual truncation error:
|| pb) ||
[runnc (AP)II
Trunc pa (AP))|
SCSF Metric
Truncation Error Eij
where AP = FFT(P() - p()) and Trunc is the truncation operator.
The magnitude of the error term Ety is governed by the spectral decay properties of the underlying physical operators. The parameter matrices in our dataset, generated via Gaussian Random Fields (GRF) or representing physical media, typically exhibit smoothness or piecewise smoothness corre- sponding to Sobolev regularity H² with s >> 0. Harmonic analysis dictates that the spectral energy of such functions decays polynomially or exponentially with frequency magnitude |k]. Specifically, the Fourier coefficients satisfy |Px| S [k - Therefore, the truncation error is bounded by the tail energy of the spectrum:
|AAP Opp 2otd),
where the spatial dimension. This decay implies that the high-frequency components contribute negligibly to the global topological distance between operators. This is also the fundamental reason why truncated FFT sorting is effective.
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Reviews:
[{"review_id": "tNU6qONTqr", "reviewer": "Reviewer_k8SQ", "summary": "The paper tackles the challenge of accelerating eigenvalue dataset generation for parametric operator learning frameworks (e.g., DeepONet). These frameworks typically require labeled data consisting of operator-eigenpair pairs, which are expensive to generate using traditional solvers. The authors propose a method, **Sorting Chebyshev Subspace Filter (SCSF)**, to reduce this computational cost by exploiting structural similarities among related operators.\n\nThe core idea is that by transforming a collection of eigenvalue problems into a correlated sequence, one can recycle subspace information and reduce redundant computation.\nTheir approach combines two main components:  \n- **Parameter sorting via a truncated FFT**, which orders problems such that consecutive operators are spectrally similar.  \n- **Chebyshev filtered subspace iteration**, which reuses eigenpair information from previously solved problems to accelerate convergence for subsequent ones.\n\nThey evaluate the proposed method with several problem settings to demonstrate the efficacy.", "questions": "1. How sensitive is the proposed parameter sorting to the different parameterization of the PDEs?\n2. How sensitive is the sorting algorithm to the choice of truncation level \\(p_0\\)? Does it generalize across different PDE families? (I found a discussion in Appendix, but I think it's better to mention this explicitly in the main text. Is $p_0=20$ universally acceptable constant over all possible PDE families? Should it scale for a harder problem?)\n3. Can the authors have any theoretical insights or potentially feasible direction for an analysis of the FFT-based ordering? \n4. In Table 5, why are the numbers in the column Greedy and column Ours exactly the same?\n\n#### **Suggestions**\n- Consider adding a paragraph in Section 2, before Section 2.1, to overview what will come in the section. This would resolve the logical gap in the current version. \n- Table captions are instructed to come above tables, not below.\n\n---\n\nOverall, I believe that the paper addresses a timely and practical problem of community's interest. While the proposed idea is not entirely new, the combination of FFT-based sorting and the classical Chebyshev filter seems to provide a nice, practical method in the modern ML context. This can be a useful technique for neural-net-based, scalable numerical analysis.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- The paper addresses a practical bottleneck in the data generation pipeline for neural operator training, which is both relevant and timely.  \n- The paper proposes a systematic integration of classical numerical linear algebra tools (Chebyshev filtering) with a new idea (FFT-based ordering) in a modern ML context.  \n- The experiments demonstrate consistent empirical speedups across diverse PDE operators.\n- The paper is generally well written and structured.", "weaknesses": "- In terms of novelty, the proposed FFT-based parameter sorting seems to be the only new component on top of the Chebyshev filter.\n- The paper does not analyze any theoretical properties of the proposed sorting algorithm. \n- While the writing is generally clear, the introduction of the Chebyshev filter in Section 2.2 looks unmotivated.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "WQJTB4MDsa", "reviewer": "Reviewer_k3N6", "summary": "The paper suggests the use of Chebyshev polynomials to as a filter to accelerate the creation of eigenvalue data sets for training neural nets. The suggested SCSF algorithm uses FFT to transform operators into a representation that allows efficient sorting of operators in terms of their distance. The runs subspace iteration using a Chebyshev filter to accelerate the convergence os the eigen-problem, by initializing from the previously solved-for closest operator. \n\nResults are presented for several different eigen problems of interest. SCSF is compared with old and fairly basic eigenvalue solvers. The comparison include one baseline that uses Chebyshev subspace iteration as a filter.", "questions": "please see the above", "limitations": "", "rating": 2, "confidence": 4, "soundness": 3, "presentation": 1, "contribution": 2, "strengths": "Results include a set different experiments in terms of capturing both accuracy, running time, and efficacy of the sorting process.\nThe stated problem is of interest to the community of numerical analysts.", "weaknesses": "1.\tThe paper doesn’t cover or compare with the two works of Wang et al. 2024,  and Dong et al 2024. For example, Wang et al. 2024 is using the same similarity search, but uses a Krylov iteration instead Chebyshev filtering. Why is that not compared to in the text and in experiments? What are the advantages of using the Chebyshev polynomials of Wang et al’s Krylov iteration.  I would expect experimental benchmarking to go beyond standard packages of eigensolvers. \n\n2.\tThere should be a dedicated related work section to compare and contrast this work and prior relevant art.\n\n3.\tI fill there is disconnect between the process described in f and g in fig 2, and what is explained in section 3.2\n\n4.\tThe paper is missing motivation as to why is it useful to use Chebyshev polynomials for this problem.\n\n5.\tThere is no explanation as how A is derived from P, is it A=P^T*P? is is A= (P+P^T)/2?\n\n6.\tIt’s not clear if FFT is run on P or on A? in Fig 2 it is applied on A, in algorithm 2 it is applied on P.\n\n7.\tThe authors claim in their statement of contribution in the introduction that ‘To the best of our knowledge SCSF is the first method to accelerate the operator eigenvalue problem”, however Wang et al 2024 and Dong et al 2024 have shown acceleration before for a similar type of problems. \n\n8.\tSame problem with the statement about the sorting algorithm novelty – it was already state in section E.2.2 of Wang et al.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "n15tJhRc5W", "reviewer": "Reviewer_uA7R", "summary": "Training neural operators for eigenvalue problems requires a lot of data and the main bottleneck for generating data for\neigenvalue problems is doing that linear operation. In this paper, the authors propose SCSF which identifies and solves sequentially problems that have close spectral distributions.", "questions": "* Are there other types of filtering that could be used? Is there any particular reason as to why Chebyshev polynomials\n  are a good choice overall? Maybe something related to the expected decay rate of the linear operators in certain\n  problems? I could've missed this discussion in the paper.\n* I understand that the goal of the paper is to get rid of a critical bottleneck in generating large-scale eigenvalue\n  datasets but there is no training of neural operators. One fear is that indeed SCSF might reduce runtime in the\n  data preparation but that the eigenvalue estimates are worse hence worsening the performance of neural operators\n  trained on this data. Could you please comment on this concern?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "* The paper is well-written and well motivated. It is indeed interesting to see work trying to improve eigenvalue\n  computations as they are indeed a core component in several important applications.\n* The objective of the paper seems unique and novel.\n* The method explanation is thoroughly developed with great attention to detail.\n* The baseline comparisons are quite strong.", "weaknesses": "* No code is shared through a anonymized link? Actually the sorting and Chebyshev Filtering does not seem trivial to implement. How are the authors planning to promote adoption and reproducibility?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "LnQUnGLy0P", "reviewer": "Reviewer_drTD", "summary": "The authors present SCSF, a sorting Chebyshev subspace filter, which reduces the time to solve eigenvalue problems numerically. The authors present an approach which uses a truncated FFT in to identify low order modes of operators, and sort them by similarity. The problems are solved sequentially, where the solution of the next problem is sped up by incorporating a bias of the eigenvectors and eigenvalues of the previous solution. The authors present a set of experiments on several families of PDEs as well as with a sufficient comparison with alternative solvers.", "questions": "- What modifications are necessary to apply this approach to operators which it is not intended for? Or how might it perform on more general operators?\n- Could the first highlighted contribution bullet in the introduction be modified to be more specific? \n- How sensitive is the approach to the choice of sorting metric (the truncated FFT distance)? Would other similarity metrics—e.g., based on operator spectra or principal angles—change performance?\n- How does the method behave when the dataset contains operators from distinct distributions (i.e., heterogeneous or discontinuous parameter spaces)? The appendix touches on this, but more quantitative analysis might help clarify failure modes.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "- The paper was very well written and easy to follow. Although I do not actively do research in this field, I felt I was able to understand the problems, methods, and contributions quite well. \n- The presented approach has strong and consistent gains in time to solution. \n- The authors present many experiments which demonstrate the effectiveness of their approach in many scenarios. As I read, there were 3 occurrences where I read a claim and then made a note to ask for an ablation, only to find the authors had already performed such an experiment and included results either in the main text, or in the appendix.", "weaknesses": "- The scope of the paper is relatively limited. It mainly benefits Hermitian linear operators. \n- There is an assumption of dataset similarity built in to this approach. I wonder what failure modes might arise in situations where this does not hold true. Perhaps it would be an interesting study to gradually mix two different operator distributions and observe how the approach compares to a baseline, in the average and worst-case situations. Some theoretical understanding could also be beneficial on this point as an alternative to an empirical study.\n- The work shares many similarities to other approaches (Chase, ChFSI), which hinder the absolute novelty of their approach. This also makes statements such as \"the first method to accelerate the operator eigenvalue data generation\" a bit too broad. However, the authors have made sure to provide experiments which demonstrate their empirical advantages over these approaches. Perhaps a short statement which makes the differences in SCSF explicit would be beneficial. \n- I always recommend authors to add a short statement on the limitations of the approach. \n\n- (very minor typo) line 122 on page 3, should state \"exhibit a high similarity\" instead of \"exhibit a highly similarity\".", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)