Pith. sign in

REVIEW 3 major objections 4 minor 37 references

A data-projected discrepancy between Koopman matrices built from observations and from candidate PDEs identifies the correct governing equation, even though all candidates share a degenerate eigenvalue spectrum.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-01 02:28 UTC pith:GD4SG6SV

load-bearing objection A solid incremental paper whose central empirical claim is undercut by self-consistency in the data generation. the 3 major comments →

arxiv 2607.27728 v1 pith:GD4SG6SV submitted 2026-07-30 math.NA cs.NAeess.SP

Numerical Spectrum Linking: Identification of Governing PDE via Koopman-Chebyshev Approximation with Resampling

Phonepaserth Sisaykeo , Shogo Muramatsu This is my paper
classification math.NA cs.NAeess.SP MSC 65M7093B3065M32
keywords Koopman operatorChebyshev spectral methodPDE identificationdata-projected discrepancyresamplingcoefficient recoveryobservation modelnilpotent generator
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper claims that comparing where two Koopman operators send observed data—not their spectra—reveals which candidate PDE generated the data. It builds both operators in a Chebyshev spectral basis: one from recovered coefficients of the observed field, one from each candidate equation's spatial differential operator. A structural nilpotency result shows the equation-driven operator always has the same degenerate unit eigenvalue for every candidate, so eigenvalue comparisons cannot discriminate. Instead, the data-projected discrepancy, a normalized measure of how differently the two operators act on the recovered coefficient snapshots, is shown to be minimized by the true PDE across four canonical linear candidates and three sampling geometries. If correct, this gives a spectral-domain route from raw spatiotemporal measurements to the governing linear constant-coefficient PDE.

Core claim

PDE identification is performed by comparing operators in the Chebyshev coefficient domain rather than by their spectra. An observation-driven Koopman matrix K̂ is fit to recovered Chebyshev coefficients, while each candidate PDE supplies an equation-driven matrix K* = exp(Δt N), with N the coefficient-space generator. The authors show N is strictly upper triangular for the candidate advection/diffusion operators, hence nilpotent, so every K* has the same degenerate unit eigenvalue regardless of dynamics. Spectral comparison is therefore uninformative. Instead, the data-projected discrepancy d(K*,K̂) = ||(K*-K̂)Â0||_F/||K*Â0||_F, with Â0 the recovered-coefficient snapshot matrix, compares ho

What carries the argument

The load-bearing objects are the coefficient-space generator N of a candidate PDE and the observation model that maps arbitrary sampling grids into the Chebyshev basis. Chebyshev differentiation matrices are strictly upper triangular, so any linear constant-coefficient spatial differential operator without a zero-order term becomes a strictly upper triangular, nilpotent matrix N in coefficient space; consequently K* = exp(Δt N) carries no spectral information. The data-projected discrepancy d(K*,K̂) = ||(K*-K̂)Â0||_F/||K*Â0||_F uses the recovered coefficient snapshot matrix Â0 to compare operator action on data, remaining discriminative despite degenerate spectra. The observation matrix H =

Load-bearing premise

The procedure assumes that the candidate PDEs are linear, constant-coefficient spatial differential operators with no zero-order term, and that the coefficient values in each candidate (a_x, a_y, ν) are known in advance rather than estimated from the data.

What would settle it

Run the four-candidate identification on data generated by the advection-diffusion equation with the candidate diffusivity ν perturbed ±10% around the true value. If the minimum data-projected discrepancy no longer selects the true structural form at the true parameter value, or the margin turns negative, then the method's reliance on a priori exact coefficients is exposed.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

Share X Bluesky LinkedIn Reddit HN

If this is right

  • Under direct Chebyshev, uniform, and irregular sampling, a minimum data-projected discrepancy selects the true PDE among the four candidates, with the correct entry several orders of magnitude below competing entries.
  • The identification margin stays positive for both uniform and irregular sampling once the number of observations reaches the spectral state dimension M=64; additional observations beyond this rank ceiling give no further benefit.
  • The nilpotency result means that eigenvalue-based model selection (e.g., matching Koopman eigenvalues) cannot work for this class of candidate PDEs, and any method relying on spectra alone will fail to discriminate.
  • The framework provides a practical observation-density guideline: collect at least as many independent observations as spectral modes to guarantee full column rank of the recovery matrix and thereby reliable identification.
  • The unified observation model reduces to the direct Chebyshev case when H=I, so the method is a strict generalization of earlier ideal-setting formulations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The nilpotency argument generalizes beyond the four tested examples to any linear constant-coefficient PDE without a zero-order term, so the data-projected discrepancy criterion is likely applicable to a wider candidate library of the same structural type.
  • The requirement that candidate coefficients be known a priori is a practical bottleneck; extending the framework to jointly estimate coefficients (e.g., by minimizing the discrepancy over parameter values) is a natural next step that the paper lists as future work.
  • The sharp transition at N=M suggests a design rule for sensor placement: measurements should be positioned to make the Chebyshev evaluation matrix Φ well-conditioned and full rank, since the number of observations alone does not guarantee identifiability.
  • For nonlinear or parameter-varying PDEs, the linear-operator comparison would need a different basis or a nonlinear extension; the data-projected discrepancy itself is basis-dependent and would not transfer unchanged.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes a numerical framework, 'numerical spectrum linking', for identifying the governing PDE of a spatiotemporal process from discrete observations. It constructs an observation-driven Koopman matrix K̂ from Chebyshev spectral coefficients recovered from arbitrary sampling grids via a unified observation model, and an equation-driven Koopman matrix K* = exp(ΔtN) from candidate PDEs. The central theoretical observation is that for the candidate class (linear, constant-coefficient, no zero-order term), the Chebyshev coefficient-space generator N is strictly upper triangular and nilpotent, so K* has a single degenerate unit eigenvalue; the paper therefore introduces a data-projected discrepancy d(K*,K̂) (Eq. 24) as the identification criterion. Experiments on four advection/diffusion PDEs under direct Chebyshev, uniform, and irregular sampling, plus an observation-density study, are reported.

Significance. The mathematical statements in the paper are correct: the Chebyshev differentiation matrices are strictly upper triangular for the chosen ordering, N is nilpotent, and the rank transition of the observation matrix Φ at N = M is a clean, falsifiable prediction. The unified observation model and the data-projected discrepancy are sensible design choices, and the paper makes code available. However, the empirical validation is not yet convincing: the data-generation procedure is underspecified, and if the reference trajectories are produced by integrating the same spectral ODE used to build K*, the identification results are close to tautological. The paper also assumes known coefficients and noiseless data. If the experimental concerns are addressed, the framework could be a useful contribution to spectral Koopman-based PDE identification; at present the central claim is not fully supported.

major comments (3)
  1. [IV-A and III-B] The reference-trajectory generation is unspecified. The text says only that 'a finer internal integration step of 1.0e-5 is employed', but no initial condition, boundary conditions, or numerical solver is stated. If, as the natural reading suggests, the trajectories are generated by integrating the spectral ODE a_dot = N a (Eq. 18) with the same truncated Chebyshev operator N used to build K* = exp(ΔtN) in Eq. (20), then the data exactly satisfy a_{k+1}=K*a_k and K̂ obtained from Eq. (17) coincides with K* by construction. The near-machine-precision diagonal entries in Table I would then be consistency checks, not evidence of PDE identification. Please specify the full data-generation protocol and re-run the experiments with data produced by an independent PDE solver (e.g., finite differences or a pseudospectral method with explicit boundary conditions), so the recovered coefficients are
  2. [IV-A, Eqs. (37)-(40)] The candidate coefficients a_x, a_y, ν are never assigned numerical values, and the paper does not state whether the K* matrices are built using the same coefficient values used in the data-generating dynamics. Since d(K*,K̂) in Eq. (24) depends on these values, the method as described assumes the coefficients are known a priori. This should be stated explicitly, and the authors should clarify that the framework identifies operator structure, not coefficients. If coefficient estimation is intended, the present formulation does not yet address it; the conclusion's mention of 'unknown coefficients' as future work (Section V) confirms that this is a limitation.
  3. [IV and V] The empirical support is limited to noiseless data with known candidate coefficients, and no comparison is made to existing PDE-identification methods such as PDE-FIND. The abstract's claim that the framework 'accurately identifies the governing PDE from observations' is therefore only supported under idealized conditions. A noise-robustness analysis and at least one comparison with a baseline method would be necessary to substantiate the practical relevance argued in the introduction. This is especially important because the identification margin in Table II is small (≈3×10^-3), and its behavior under realistic perturbations is unknown.
minor comments (4)
  1. [III-F, after Eq. (35)] The sentence 'This condition on N is therefore not an independent assumption but a direct consequence of the shape of Φ' is unclear and appears erroneous; please revise or remove.
  2. [Notation throughout] N is used both for the number of snapshots (Section III-A) and the number of observations (Section IV-A), while M denotes both the truncation multi-index (Section II-B) and the state dimension (Section III-C). This overloading is confusing; suggest N_t, N_obs and d or M_tot.
  3. [Eq. (24)] The denominator uses K* A0; a brief justification that this normalization does not bias the criterion toward candidates with small ||K* A0|| would improve readability. (Not a correctness issue in the current experiments.)
  4. [Reproducibility] The Code Ocean capsule is mentioned only in the footnote; for reproducibility, the main text should describe which scripts generate each table/figure and what software versions were used.

Circularity Check

1 steps flagged

Experimental validation is self-consistent by construction: reference data appear to be generated by the same truncated Chebyshev spectral ODE that defines K*, so the reported identification success is baked into the setup, while the comparison criterion itself is not definitionally circular.

specific steps
  1. other [Section IV-A (Experimental Setup) with Eqs. (18)-(20), Section III-B]
    "To generate accurate reference trajectories, a finer internal integration step of 1.0×10−5 is employed during numerical simulation. ... By discretizing in time with step size ∆t, the evolution of the coefficient vector is approximated as a_{k+1}≈K⋆ a_k, where the equation-driven Koopman matrix is given by K⋆ := exp(∆tN)."

    The reference trajectories are simulated under the same truncation M=8, time step ∆t, and (judging from the ~1e-12 diagonal entries in Table I) the same coefficient-space generator N used to construct K*. If the internal integration solves the spectral ODE ȧ=Na (18), then the recovered coefficient snapshots satisfy a_{k+1}=exp(∆tN)a_k to internal accuracy, so the least-squares estimate (17) yields K^≈K* for the true candidate by construction. The diagonal entries of Table I then measure only internal integration error, not the ability to identify a PDE from independent observations. Since the paper specifies no PDE-level boundary conditions or solver for candidates (37)-(40), the experiments reduce to comparing K^ with the same operator that generated the data.

full rationale

The mathematical derivation is not definitionally circular: K*=exp(∆tN) is derived from a candidate PDE via Chebyshev differentiation, K^ is estimated from recovered observation coefficients via least squares, and the data-projected discrepancy (24) compares two independently constructed objects. The nilpotency observation in Remark 1 is a genuine property of Chebyshev differentiation and does not rely on self-citation. However, the numerical validation is compromised: the reference trajectories appear to be generated by the same truncated Chebyshev spectral ODE (18) that defines the candidate operators. Under that setup, the true candidate is guaranteed to minimize (24) to near machine precision, independent of whether the method works on genuine PDE observations. The only self-citation ([30]) is provenance, not load-bearing. The score reflects this partial, experiment-level circularity rather than a collapse of the framework itself.

Axiom & Free-Parameter Ledger

2 free parameters · 7 axioms · 0 invented entities

The central claim rests on standard Chebyshev spectral theory and on the restriction to four linear constant-coefficient PDEs whose coefficients are known a priori. No new physical entities are introduced, and the main assumptions are the noiseless setting, the fixed candidate library, and the consistency of the data-generation with the spectral truncation.

free parameters (2)
  • Candidate PDE coefficients (a_x, a_y, nu) = not fitted; set to the true values used to generate data
    The equation-driven K* for each candidate is built with fixed coefficient values. The identification procedure never estimates these coefficients, so the method's success depends on them being known a priori.
  • Truncation order M (M1=M2=8) = 64 spectral coefficients
    Fixed throughout; the observation-density threshold N=M=64 is relative to this choice.
axioms (7)
  • standard math Chebyshev polynomials form an orthogonal basis and DCT-II gives the orthonormal coefficient transform (Eqs. (8)-(12)).
    Invoked in Section II-B; standard spectral approximation theory.
  • standard math The Chebyshev differentiation matrix D^(d) is strictly upper triangular under the chosen coefficient ordering, so any linear combination or product of such matrices (first/second derivatives) is strictly upper triangular.
    Used in Remark 1 (Section III-C) to conclude nilpotency of N for all candidates.
  • ad hoc to paper Candidate PDEs are restricted to linear constant-coefficient spatial differential operators with no zero-order term (Eqs. (37)-(40)).
    This restriction is what makes N strictly upper triangular; the method as presented does not apply to nonlinear or lower-order terms.
  • ad hoc to paper The candidate coefficients (a_x, a_y, nu) are known a priori and are not estimated from data.
    Section IV-A lists the candidate equations with symbolic coefficients but never states they are known; the equation-driven K* is built with these values.
  • domain assumption Observations are noiseless; robustness to noise is deferred to future work.
    All experiments are noiseless; the future-work paragraph mentions noise robustness.
  • domain assumption The observation matrix Phi attains full column rank at N=M for the tested grids, so the least-squares recovery is unique.
    Section IV-F observes the rank transition; the paper acknowledges N>=M alone does not guarantee full rank, but the guideline relies on it for these grids.
  • ad hoc to paper The numerical reference trajectories are generated consistently with the Chebyshev spectral truncation used to build K*.
    Section IV-A describes time integration but not the spatial discretization; if data came from a different discretization, the equation-driven operator would not match.

pith-pipeline@v1.3.0-daily-deepseek · 16838 in / 16693 out tokens · 172335 ms · 2026-08-01T02:28:34.273256+00:00 · methodology

0 comments
Cite this review

Pith. "Pith review of Numerical Spectrum Linking: Identification of Governing PDE via Koopman-Chebyshev Approximation with Resampling." pith.science (2026). https://pith.science/paper/GD4SG6SV

@misc{pith2026260727728,
  author       = {Pith},
  title        = {Pith review of: Numerical Spectrum Linking: Identification of Governing PDE via Koopman-Chebyshev Approximation with Resampling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GD4SG6SV}},
  note         = {Machine review of arXiv:2607.27728}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

A numerical framework is proposed for identifying governing partial differential equations (PDEs) from observational data by establishing a link between observation-driven and equation-driven Koopman operators in a common Chebyshev spectral domain. In contrast to data-driven approaches such as dynamic mode decomposition (DMD), which approximate Koopman operators without explicitly relating them to differential operators, the proposed framework constructs finite-dimensional Koopman operators using Chebyshev spectral representations, thereby enabling direct comparison between data-derived dynamics and candidate governing PDEs. A unified observation model together with a least-squares coefficient recovery formulation is introduced to recover Chebyshev spectral coefficients from observations obtained on arbitrary sampling grids. This provides a numerically consistent interface between practical observations and Chebyshev-based Koopman analysis. Numerical experiments under direct Chebyshev, uniform, and irregular sampling configurations demonstrate that the proposed framework accurately identifies the governing PDE from observations. An observation-density study shows that reliable PDE identification is consistently achieved once sufficient independent observations are available for stable coefficient recovery, providing a practical guideline.

Figures

Figures reproduced from arXiv: 2607.27728 by Phonepaserth Sisaykeo, Shogo Muramatsu.

Figure 1
Figure 1. Figure 1: Conceptual motivation for linking different representations of dynam [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Relationship between Chebyshev spectral representation and Koopman operator theory. The system state is mapped to Chebyshev coefficients via [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Overview of the proposed numerical spectrum linking framework. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: illustrates the nonzero pattern of N for a representa￾tive candidate (Advection-Diffusion, the only candidate with nonzero advection and diffusion coefficients simultaneously), for visual intuition only, and the exact nilpotency NM = 0 is already established for every candidate by Remark 1, and does not rely on this or any other numerical example. Because every equation-driven matrix K⋆ shares the same deg… view at source ↗
Figure 5
Figure 5. Figure 5: Proposed observation and coefficient recovery framework under [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Observation geometries considered in this study. (a) Direct Chebyshev [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Identification margin dmargin as a function of the number of ob￾servations N for uniform and irregular sampling configurations. For irregular sampling, the solid curve reports dmargin averaged over S = 5 random seeds, and the dashed curve reports the worst-case margin among those seeds. A positive margin indicates that the projected discrepancy d of the true governing PDE is smaller than that of every comp… view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

37 extracted references · 15 canonical work pages

  1. [1]

    Communication in the Presence of Noise,

    C. E. Shannon, “Communication in the Presence of Noise,”Proceedings of the Institute of Radio Engineers, vol. 37, no. 1, pp. 10–21, 1949. [Online]. Available: https://doi.org/10.1109/JRPROC.1949.232969

    arXiv 1949

  2. [2]

    Sampling-50 years after Shannon,

    M. Unser, “Sampling-50 years after Shannon,”Proceedings of the IEEE, vol. 88, no. 4, pp. 569–587, 2000. [Online]. Available: https://doi.org/10.1109/5.843002

    work page doi:10.1109/5.843002 2000

  3. [3]

    Y . C. Eldar,Sampling Theory: Beyond Bandlimited Systems. Cambridge University Press, 12 2014. [Online]. Available: https://doi.org/10.1017/CBO9780511762321

    work page doi:10.1017/cbo9780511762321 2014

  4. [4]

    A generalized Nyquist-Shannon sampling theorem using the Koopman operator,

    Z. Zeng, J. Liu, and Y . Yuan, “A generalized Nyquist-Shannon sampling theorem using the Koopman operator,”IEEE Trans. Signal Process., vol. 72, pp. 3595–3610, 2024

    2024

  5. [5]

    Capture method for digital twin of formation processes of sand bars,

    D. Moteki, T. Murai, T. Hoshino, H. Yasuda, S. Muramatsu, and K. Hayasaka, “Capture method for digital twin of formation processes of sand bars,”Physics of Fluids, vol. 34, no. 3, 3 2022. [Online]. Available: https://doi.org/10.1063/5.0085574

    work page doi:10.1063/5.0085574 2022

  6. [6]

    On the occurrence of sandbars,

    D. Moteki, S. Seki, S. Muramatsu, K. Hayasaka, and H. Yasuda, “On the occurrence of sandbars,”Physics of Fluids, vol. 35, no. 1, 1 2023. [Online]. Available: https://doi.org/10.1063/5.0128760

    work page doi:10.1063/5.0128760 2023

  7. [7]

    Physics-informed neural networks for inversion of river flow and geometry with shallow water model,

    Y . Ohara, D. Moteki, S. Muramatsu, K. Hayasaka, and H. Yasuda, “Physics-informed neural networks for inversion of river flow and geometry with shallow water model,”Physics of Fluids, vol. 36, no. 10, p. 106633, 10 2024. [Online]. Available: https://doi.org/10.1063/5.0232852

    work page doi:10.1063/5.0232852 2024

  8. [8]

    Acceleration-induced laminarization of sheet flow over smooth steep slopes,

    H. Karisawa and H. Yasuda, “Acceleration-induced laminarization of sheet flow over smooth steep slopes,”Physics of Fluids, vol. 37, no. 8, p. 85198, 8 2025. [Online]. Available: https://doi.org/10.1063/5.0279763

    work page doi:10.1063/5.0279763 2025

  9. [9]

    S. L. Brunton and J. N. Kutz,Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, 5 2022. [Online]. Available: https://doi.org/10.1017/9781009089517

    doi:10.1017/9781009089517 2022

  10. [10]

    J. J. Bramburger,Data-Driven Methods for Dy- namic Systems. Philadelphia, PA: Society for Indus- trial and Applied Mathematics, 2024. [Online]. Available: https://epubs.siam.org/doi/abs/10.1137/1.9781611978162

    doi:10.1137/1.9781611978162 2024

  11. [11]

    Strang,Linear Algebra and Learning from Data

    G. Strang,Linear Algebra and Learning from Data. Wellesley, MA: Wellesley-Cambridge Press, 2019. [Online]. Available: https://epubs.siam.org/doi/abs/10.1137/1.9780692196380

    work page doi:10.1137/1.9780692196380 2019

  12. [12]

    Physics-informed dynamic mode decomposition,

    P. J. Baddoo, B. Herrmann, B. J. McKeon, J. Nathan Kutz, and S. L. Brunton, “Physics-informed dynamic mode decomposition,” Proceedings of the Royal Society A, vol. 479, no. 2271, p. 20220576, 3 2023. [Online]. Available: https://doi.org/10.1098/rspa.2022.0576

    arXiv 2023

  13. [13]

    Multi-Resolution Convolutional Dictionary Learning for Riverbed Dynamics Modeling,

    E. Kobayashi, H. Yasuda, K. Hayasaka, Y . Otake, S. Ono, and S. Muramatsu, “Multi-Resolution Convolutional Dictionary Learning for Riverbed Dynamics Modeling,” inICASSP 2023 - 2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). Institute of Electrical and Electronics Engineers (IEEE), 5 2023, pp. 1–5

    2023

  14. [14]

    Mauroy, I

    A. Mauroy, I. Mezi ´c, and Y . Susuki, Eds.,The Koopman Operator in Systems and Control, ser. Lecture Notes in Control and Information Sciences. Cham: Springer International Publishing, vol. 484, 2020. [Online]. Available: https://doi.org/10.1007/978-3-030-35713-9

    doi:10.1007/978-3-030-35713-9 2020

  15. [15]

    A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition,

    M. O. Williams, I. G. Kevrekidis, and C. W. Rowley, “A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition,”Journal of Nonlinear Science, vol. 25, no. 6, pp. 1307–1346, 12 2015. [Online]. Available: https://doi.org/10.1007/s00332-015-9258-5

    doi:10.1007/s00332-015-9258-5 2015

  16. [16]

    Data-driven spectral analysis of the Koopman operator,

    M. Korda, M. Putinar, and I. Mezi ´c, “Data-driven spectral analysis of the Koopman operator,”Applied and Computational Harmonic Analysis, vol. 48, no. 2, pp. 599–629, 3 2020. [Online]. Available: https://doi.org/10.1016/j.acha.2018.08.002

    work page doi:10.1016/j.acha.2018.08.002 2020

  17. [17]

    Self-supervised Koopman operator learning for distributed final synchronization prediction of networked nonlinear dynamics,

    F. Hu, H.-T. Zhang, C. Lv, and J. Wang, “Self-supervised Koopman operator learning for distributed final synchronization prediction of networked nonlinear dynamics,”IEEE Trans. Neural Netw. Learn. Syst., vol. 37, no. 7, pp. 3325–3335, 2026

    2026

  18. [18]

    Discovering Mathematical Expressions Through DeepSymNet: A Classification- Based Symbolic Regression Framework,

    M. Wu, W. Li, L. Yu, L. Sun, J. Liu, and W. Li, “Discovering Mathematical Expressions Through DeepSymNet: A Classification- Based Symbolic Regression Framework,”IEEE Transactions on Neural Networks and Learning Systems, vol. 36, no. 1, pp. 1356–1370, 2025. 13

    2025

  19. [19]

    Data-driven Discovery of Governing Differential Equations Across Physical Systems,

    S. Lou, H. Xu, W. Wang, L. Lu, H. Sun, Y . Liu, L. Zhang, D. Zhang, and Y . Chen, “Data-driven Discovery of Governing Differential Equations Across Physical Systems,” 6 2026. [Online]. Available: https://arxiv.org/abs/2606.09638

    Pith/arXiv arXiv 2026

  20. [20]

    Data-driven discovery of partial differential equations,

    S. H. Rudy, S. L. Brunton, J. L. Proctor, and J. N. Kutz, “Data-driven discovery of partial differential equations,” Science Advances, vol. 3, no. 4, 4 2017. [Online]. Available: https://doi.org/10.1126/sciadv.1602614

    doi:10.1126/sciadv.1602614 2017

  21. [21]

    Symbolic genetic algorithm for discovering open-form partial differential equations (SGA- PDE),

    Y . Chen, Y . Luo, Q. Liu, H. Xu, and D. Zhang, “Symbolic genetic algorithm for discovering open-form partial differential equations (SGA- PDE),”Physical Review Research, vol. 4, no. 2, p. 023174, 6 2022. [Online]. Available: https://doi.org/10.1103/PhysRevResearch.4.023174

    doi:10.1103/physrevresearch.4.023174 2022

  22. [22]

    DISCOVER: Deep identification of symbolically concise open-form partial differential equations via enhanced reinforcement learning,

    M. Du, Y . Chen, and D. Zhang, “DISCOVER: Deep identification of symbolically concise open-form partial differential equations via enhanced reinforcement learning,”Physical Review Research, vol. 6, no. 1, p. 013182, 1 2024. [Online]. Available: https://doi.org/10.1103/PhysRevResearch.6.013182

    work page doi:10.1103/physrevresearch.6.013182 2024

  23. [23]

    How Low Can You Go? Active Learning for Sparse Model Discovery in the Ultra-Low-Data Limit,

    A. Larra ˜naga, U. Fasel, and S. L. Brunton, “How Low Can You Go? Active Learning for Sparse Model Discovery in the Ultra-Low-Data Limit,” 6 2026. [Online]. Available: https://arxiv.org/abs/2606.12182

    Pith/arXiv arXiv 2026

  24. [24]

    Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,

    M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,” Journal of Computational Physics, vol. 378, pp. 686–707, 2 2019

    2019

  25. [25]

    Partial Differential Equations Meet Deep Neural Networks: A Survey,

    S. Huang, W. Feng, C. Tang, Z. He, C. Yu, and J. Lv, “Partial Differential Equations Meet Deep Neural Networks: A Survey,”IEEE Transactions on Neural Networks and Learning Systems, 2025

    2025

  26. [26]

    Koopman Spectral Linearization vs. Carleman Linearization: A Computational Comparison Study,

    D. Shi and X. Yang, “Koopman Spectral Linearization vs. Carleman Linearization: A Computational Comparison Study,” Mathematics, vol. 12, no. 14, p. 2156, 7 2024. [Online]. Available: https://doi.org/10.3390/math12142156

    work page doi:10.3390/math12142156 2024

  27. [27]

    Discrete Cosine Transform,

    N. Ahmed, T. Natarajan, and K. R. Rao, “Discrete Cosine Transform,” IEEE Transactions on Computers, vol. C-23, no. 1, pp. 90–93, 1974

    1974

  28. [28]

    Ochoa-Dominguez and K

    H. Ochoa-Dominguez and K. R. Rao,Discrete Cosine Transform, 2nd ed. CRC Press, 2019

    2019

  29. [29]

    Algorithms for the integration and derivation of Chebyshev series,

    R. Barrio, “Algorithms for the integration and derivation of Chebyshev series,”Applied Mathematics and Computation, vol. 150, no. 3, pp. 707–717, 3 2004. [Online]. Available: https://doi.org/10.1016/S0096- 3003(03)00301-1

    doi:10.1016/s0096- 2004

  30. [30]

    Numerical Spectrum Linking: Identi- fication of Governing PDE via Koopman-Chebyshev Approximation,

    P. Sisaykeo, and S. Muramatsu, “Numerical Spectrum Linking: Identi- fication of Governing PDE via Koopman-Chebyshev Approximation,” inICASSP 2026 - 2026 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). Institute of Electrical and Electronics Engineers (IEEE), 5 2026, pp. 1001–1005

    2026

  31. [31]

    An overview of Koopman-based control: From error bounds to closed-loop guarantees,

    R. Str ¨asser, K. Worthmann, I. Mezi ´c, J. Berberich, M. Schaller, and F. Allg ¨ower, “An overview of Koopman-based control: From error bounds to closed-loop guarantees,” 9 2025. [Online]. Available: https://arxiv.org/pdf/2509.02839

    arXiv 2025

  32. [32]

    Reduced Order Data-Driven Twin Models for Nonlinear PDEs by Randomized Koopman Orthogonal Decomposition and Explainable Deep Learning,

    D. A. Bistrian, “Reduced Order Data-Driven Twin Models for Nonlinear PDEs by Randomized Koopman Orthogonal Decomposition and Explainable Deep Learning,”Mathematics, vol. 13, no. 17, p. 2870, 9 2025. [Online]. Available: https://doi.org/10.3390/math13172870

    work page doi:10.3390/math13172870 2025

  33. [33]

    Chebyshev Polynomials,

    J. Mason and D. C. Handscomb, “Chebyshev Polynomials,” 9 2002. [Online]. Available: https://doi.org/10.1201/9781420036114

    doi:10.1201/9781420036114 2002

  34. [34]

    Shifted Chebyshev polynomials based solution of partial differential equations,

    P. Karunakar and S. Chakraverty, “Shifted Chebyshev polynomials based solution of partial differential equations,”SN Applied Sciences, vol. 1, no. 4, pp. 1–9, 4 2019. [Online]. Available: https://doi.org/10.1007/s42452-019-0292-z

    work page doi:10.1007/s42452-019-0292-z 2019

  35. [35]

    Representing derivatives of Chebyshev polynomials by Chebyshev polynomials and related questions,

    H. Prodinger, “Representing derivatives of Chebyshev polynomials by Chebyshev polynomials and related questions,”Open Mathematics, vol. 15, no. 1, pp. 1156–1160, 1 2017. [Online]. Available: https://doi.org/10.1515/math-2017-0096

    work page doi:10.1515/math-2017-0096 2017

  36. [36]

    Derivations and Identities for Chebyshev Polynomials,

    L. P. Bedratyuk and N. B. Lunio, “Derivations and Identities for Chebyshev Polynomials,”Ukrainian Mathematical Journal, vol. 73, no. 8, pp. 1175–1188, 1 2022. [Online]. Available: https://doi.org/10.1007/s11253-022-01985-8

    work page doi:10.1007/s11253-022-01985-8 2022

  37. [37]

    Data-Driven Koopman Based System Identification for Partially Observed Dynamical Systems with Input and Disturbance,

    P. Ketthong, J. Samkunta, N. T. Mai, M. A. S. Kamal, I. Murakami, and K. Yamada, “Data-Driven Koopman Based System Identification for Partially Observed Dynamical Systems with Input and Disturbance,”Sci, vol. 6, no. 4, p. 84, 12 2024. [Online]. Available: https://doi.org/10.3390/sci6040084

    work page doi:10.3390/sci6040084 2024