Pith. sign in

REVIEW 3 major objections 6 minor 50 references

FiberKAN: Kolmogorov-Arnold Networks for Nonlinear Fiber Optics

T0 review · 3 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

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

Pith's one-line read Kolmogorov-Arnold networks can rediscover and characterize solutions of the nonlinear Schrödinger equation governing fiber optics, including implicit and non-analytic cases, with accuracy exceeding MLPs at the same parameter count.

desk verdict A competent early KAN-for-fiber-optics paper whose explicit-solution results are solid but whose 'implicit discovery' claim is undercut by feeding the target as an input. read the letter →

arxiv 2504.18833 v1 pith:MS2AVIAD submitted 2025-04-26 physics.optics

Xiaotian Jiang , Min Zhang , Xiao Luo , Zelai Yu , Yiming Meng , Danshi Wang This is my paper · ORCID
classification physics.optics
keywords Kolmogorov-ArnoldnetworksnonlinearfiberopticsSchrödingerequationscientificdiscoverysymbolicregressionself-steepeningintrapulseRamanscatteringsupercontinuumgeneration
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proposes FiberKAN, a Kolmogorov-Arnold Network framework for scientific discovery in nonlinear fiber optics. It argues that replacing MLPs' fixed node activations with learnable edge activations lets a small, interpretable network recover explicit analytical solutions (power attenuation, dispersion, solitons), an implicit self-steepening solution, and characterize non-analytical intrapulse Raman scattering and supercontinuum dynamics. Across these tasks the KANs match or beat MLPs of comparable trainable-parameter count, and after symbolization the soliton solution is recovered to errors around 1e-11. The claim matters because it points to a route toward automated extraction of physical laws from optical data, not just black-box fitting.

What carries the argument

The central object is the Kolmogorov-Arnold Network (KAN) and its multiplicative variant MultKAN, which implement the Kolmogorov-Arnold representation theorem by placing learnable univariate activation functions (B-splines) on network edges instead of fixed nonlinearities on nodes. In FiberKAN this machinery is driven through a four-step cycle—pretraining, pruning, training, symbolization—during which redundant edges and nodes are eliminated, the remaining splines are visualized and fitted to elementary functions, and the symbolic coefficients are fine-tuned to machine precision. The multiplication nodes of MultKAN are what let the network represent separable products such as sech(ω) cos(β_k ω^k z / k!) directly, and the absence of node nonlinearities is what makes each edge interpretable in physical terms.

What would settle it

Re-run the Section III-D self-steepening experiment and inspect the post-pruning network: if the edge from the input I to the output I survives with a weight near 1 and the other edges account for only a small correction, or if predictions outside the training range match the identity mapping rather than sech(t-3sIz), then the implicit-discovery claim is falsified.

Watch

Extended reading notes

Core claim

The central claim is that Kolmogorov-Arnold Networks, because their activation functions live on edges and are learnable splines, are capable of discovering and characterizing the dynamics of nonlinear fiber-optic systems described by the nonlinear Schrödinger equation. In the explicit-solution cases, a two-layer MultKAN with 608 parameters rediscovers the fundamental soliton h(z,t)=sech(t)exp(iz/2), and after symbolization the discovered solution matches the analytical one to errors on the order of 1e-11; the same procedure recovers power attenuation, GVD, and TOD laws. For the self-steepening effect, the network identifies the implicit relation I(z,t)=f(t-3sIz) by using I itself as an input feature. For intrapulse Raman scattering, which lacks an analytical pulse-like solution, the KAN reaches a mean maximum error of 0.0303 against 0.0561 for an MLP of 618 parameters, and in supercontinuum generation it learns high-frequency components that the equivalent-scale MLP essentially misses.

Load-bearing premise

The self-steepening discovery assumes that when the output intensity is also given as input, the network genuinely learns the nonlinear relation I(z,t)=sech(t-3sIz) rather than the simpler route I_out=I_in, because the training intentionally removes the regularization that would push the network away from that trivial copy; if the low held-out error actually comes from the identity mapping plus a small correction, the implicit-discovery claim collapses.

Editorial extensions

If this is right

  • FiberKAN can turn a few thousand sampled pulse-propagation points into closed-form symbolic laws (e.g., the soliton solution) without a candidate library, because each spline is individually symbolized and fine-tuned.
  • For effects with no analytic solution, such as intrapulse Raman scattering, KAN offers an interpretable black-box model with lower error and lower variance than an MLP of comparable parameter count.
  • Trained KAN structures transfer between similar dispersion regimes: the GVD model for u(z,ω) is adapted to TOD dynamics in about 100 iterations with only one activation function changing.
  • In supercontinuum generation, KANs preserve high-frequency spectral components that MLPs of similar size fail to reproduce, implying better suitability for broadband nonlinear dynamics.

Reading between the lines

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

  • Beyond the paper, the same pretrain-prune-train-symbolize pipeline could be applied to other envelope equations (coupled-mode, Gross-Pitaevskii, or Lugiato-Lefever) whenever the target dynamics admit decompositions into a few univariate splines; the main obstacle will be the absence of known symbolic targets to validate against.
  • The self-steepening identity-path failure is a general caution for implicit-law discovery: when an output variable is also an input, one should penalize or forbid the copy map either architecturally or in the loss, otherwise a low test loss can reflect the trivial mapping rather than a discovered law.
  • If the KAN advantage at equal parameter count persists outside these benchmarks, the framework offers a practical route to data-efficient optical channel modeling, because the symbolic edge functions can be reused and lightly retrained when fiber type or wavelength changes.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper introduces FiberKAN, a KAN-based framework for 'scientific discovery and dynamic characterization' of nonlinear fiber-optic pulse propagation. For the explicit-solution cases (power attenuation, GVD, TOD, and the fundamental soliton), the authors generate datasets from the known analytical solutions, train MultKANs with a pretrain-prune-train-symbolize pipeline, and report recovery of the symbolic formulas with errors down to about 10^-11. For self-steepening, they attempt to discover the implicit intensity relation I(z,t)=sech(t-3sIz) by feeding I as an input feature. For intrapulse Raman scattering and supercontinuum generation, they train KANs and MLPs with comparable parameter counts and report lower maximum errors for the KAN. Additional sections evaluate success rates, computational cost, noise resistance, prior-knowledge embedding, hypothesis testing, transfer learning, and comparisons with PINNs and symbolic regression.

Significance. If the broad claim were fully supported, FiberKAN would be a useful demonstration that KANs can serve as interpretable surrogate models and symbolic-regression front-ends for fiber optics. The explicit-solution experiments are internally consistent, use held-out test losses, and the soliton recovery to 10^-11 is a clean positive result; the IRS statistical comparison (mean maximum error 0.0303 versus 0.0561 over 100 seeds) is a reasonable, if modest, evidence of a KAN advantage; and the computational-cost analysis is honest about KAN's slower training. However, the only implicit-solution experiment is underdetermined because the target intensity is provided as an input, so the central 'explicit, implicit, and non-analytical' claim is not currently supported. The 'discovery' demonstrations are also benchmark-style recoveries of formulas used to generate the data, which tempers the scientific-discovery framing. The manuscript does not provide code or data, limiting reproducibility.

major comments (3)
  1. [Section III-D and Section IV-A, Table I] The self-steepening 'implicit dynamics discovery' is underdetermined by the supervised task. The network receives I(z,t) as an input feature and is trained to output I(z,t), so the identity map f(z,t,I)=I is an exact zero-error solution on both training and test points. Removing the L1 regularization (Table I, lam_l1=0) does not suppress this solution; it only avoids penalizing non-identity structure, and Section IV-A explicitly reports that in most failed cases the KAN discovers exactly this I=I mapping. Because the data points satisfy I=sech(t-3sIz) by construction, the identity and the claimed implicit relation coincide on the data manifold, so the reported held-out error cannot distinguish the two. This is load-bearing: it is the only implicit-solution demonstration and the basis for the abstract's 'explicit, implicit, and non-analytical' taxonomy.
  2. [Section II-B and Section III-C] The role of the symbolization step in the 'discovery' claim is underspecified. The manuscript states that each activation function is fitted to the 'closest' symbol from a candidate library and that coefficients are then fine-tuned, but it does not specify the candidate library, the similarity metric, or the selection criterion. Since the training data in Parts A-D are generated from the very formulas that are later 'discovered,' the reported 10^-11 accuracy (Section III-C) reflects, at least in part, the symbolic-regression postprocessor rather than the KAN alone. Reporting the pre-symbolization errors and the full symbolization protocol would clarify how much of the discovery is attributable to the KAN and would make the method reproducible.
  3. [Section III (opening paragraph) and Section VI] The manuscript frames the method as discovering dynamics 'that have not yet been discovered and characterized' (Abstract; Section I), but every 'discovery' experiment in Parts A-D trains on data generated by exactly the analytical solution that is later recovered. This makes the experiments controlled recovery benchmarks, not discoveries of unknown dynamics. I recommend either adding a case with a genuinely unknown or numerically generated law, or explicitly reframing the claim as recovery/validation of known solutions.
minor comments (6)
  1. [Section III-E, final paragraph] The sentence 'the maximum error of KAN in the entire domain is almost one order of magnitude higher than that of MLP' appears to contradict Fig. 9(e) and (f) and the surrounding claim; please correct the typo and report the actual maximum errors.
  2. [Section I, third paragraph] There is a duplicated phrase: 'their performances remains constrained by the constrained by fundamental properties of MLPs' should read 'their performance remains constrained by fundamental properties of MLPs.'
  3. [Section IV-A, Figure 10] The 'success rate' reported in Fig. 10 is never defined; please specify the criterion for a successful discovery (e.g., correct symbolic form with coefficient tolerance) and state how it is applied across the 100 trials.
  4. [Section V-D] The comparison with PINN is qualitative ('several orders of magnitude higher', 'comparable accuracy'); please provide a direct numerical comparison under controlled settings or soften the claim.
  5. [General (reproducibility)] No code or data availability statement is provided; since the method depends on many unspecified implementation details (spline initialization, pruning thresholds, symbol library), release of code would be important for reproducibility.
  6. [Figure 10] The labels on the bars in Fig. 10 are difficult to read in the current reproduction; larger fonts or a table would help.

Circularity Check

1 steps flagged · score 6.0 of 10

Self-steepening 'implicit discovery' is underdetermined by construction: I is both input and target, so identity is a zero-error solution; the paper's own failure statistics confirm the trivial shortcut.

  1. self definitional [Section III-D (Self-steepening Effect: Implicit Dynamics Discovery); Table I; Section IV-A]
    "To discover the implicit law, I is also applied as an input feature, and the regularization term is removed to prevent the network from learning the direct mapping of I = I. ... In most failed cases, KANs discover the direct mapping relationship of I = I, indicating that the dynamics of implicit solution is more difficult to identify when the amount of data is small."

    The supervised task is to output I from inputs (z, t, I). The identity map f(z,t,I)=I satisfies the training and test data with exactly zero error by construction, because the target is one of the inputs. Removing the L1 term does not remove this solution; it merely avoids the sparsity penalty that would suppress it. The claimed implicit solution I=sech(t-3sIz) is itself an equation whose left-hand side is the input I, so on the data manifold it is indistinguishable from the identity: both produce the same output for every training point. The paper's own statistics show that the network frequently finds I=I, confirming the ambiguity.

full rationale

The remaining demonstrations are benchmark-style validations rather than circular derivations: datasets for Parts A-D are generated from known analytical solutions, and the KAN pipeline (pretrain, prune, train, symbolize) fits splines and then fits the closest symbol from a library, which is a legitimate recovery test. The explicit-soliton, attenuation, dispersion, and IRS results are not constructed from their targets as inputs and are supported by held-out errors. The IRS and supercontinuum comparisons to MLPs are external empirical benchmarks. Self-citations to the authors' earlier PINN work ([19], [21], [48]) are used for comparison and not as load-bearing authority. The single construction-level issue is the self-steepening implicit case, where the target I is provided as an input feature, making I=I a zero-error solution and the claimed implicit-form discovery underdetermined; the paper itself reports this failure mode. This affects one of the three central claim categories (implicit solutions) while leaving the explicit and non-analytical claims intact, hence a partial circularity score of 6 rather than a higher score.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The paper draws on the KAN framework, the NLSE, and standard analytical solutions from the literature; no new physical entities or first-principles derivations are introduced. Its contribution is an empirical demonstration, so the axiom ledger consists of the mathematical grounding of KAN, the accepted physics model, and the assumed accuracy of the data-generation step.

free parameters (5)
  • Spline grid intervals G = 5
    Chosen by hand for all experiments (Table I); affects B-spline capacity and symbolization success.
  • Spline order k = 3
    Chosen by hand (Table I); cubic B-splines are a default choice, not derived from the problem.
  • L1 regularization coefficient lam_l1 = 10^-4 (Parts A-C), 0 (Parts D-E)
    Toggled by task to avoid the identity mapping in the implicit case; this choice directly affects whether the intended relation is discovered.
  • Initial injected noise to spline = 0.3
    Hyperparameter (Table I) used to perturb initial activation functions; likely influences which local optimum is reached.
  • Symbolic coefficients after symbolization = close to α=0.5, β_k, etc.
    The final formulas are produced by fitting each spline to a candidate symbol and fine-tuning coefficients to the training data (Section II-B); they are not derived from first principles.
assumptions (6)
  • standard math Kolmogorov-Arnold representation theorem
    Justifies that any continuous multivariate function can be represented as sums of univariate functions; the expressive power of KAN rests on this theorem (Eq. 1-4, Section II-A).
  • domain assumption The generalized NLSE governs pulse propagation in fiber
    The paper assumes the NLSE (Eq. 6) and the listed effects correctly describe the fiber-optic dynamics, citing Agrawal [11].
  • domain assumption The analytical solutions used to generate training data are correct
    Training data for Parts A-D is generated from known analytical solutions such as sech(t)exp(-αz), sech(ω)cos(β_k ω^k z / k!), and the soliton sech(t)exp(iz/2); the discovery claim is only as strong as these standard results.
  • domain assumption SSFM step number 100 L/LNL gives sufficient accuracy
    Non-analytical datasets (Part E) are generated by SSFM with step number set to 100 L/LNL for sufficient accuracy, without a convergence study; errors at this step size are treated as negligible.
  • domain assumption The symbol candidate library is sufficient
    Symbolization selects the closest function from a candidate library of basic symbols such as identity, exp, sech, cos, sin, and cubic; the discovered symbolic form is constrained by this library (Section II-B).
  • standard math B-spline activations can represent the needed univariate functions
    B-spline basis functions with order k and grid G are assumed capable of approximating the target univariate functions well enough before symbolization.

how reviews work

0 comments
Cite this review

Pith. "Pith review of FiberKAN: Kolmogorov-Arnold Networks for Nonlinear Fiber Optics." pith.science (2026). https://pith.science/paper/MS2AVIAD

@misc{pith2026250418833,
  author       = {Pith},
  title        = {Pith review of: FiberKAN: Kolmogorov-Arnold Networks for Nonlinear Fiber Optics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MS2AVIAD}},
  note         = {Machine review of arXiv:2504.18833}
}
read the original abstract

Scientific discovery and dynamic characterization of the physical system play a critical role in understanding, learning, and modeling the physical phenomena and behaviors in various fields. Although theories and laws of many system dynamics have been derived from rigorous first principles, there are still a considerable number of complex dynamics that have not yet been discovered and characterized, which hinders the progress of science in corresponding fields. To address these challenges, artificial intelligence for science (AI4S) has emerged as a burgeoning research field. In this paper, a Kolmogorov-Arnold Network (KAN)-based AI4S framework named FiberKAN is proposed for scientific discovery and dynamic characterization of nonlinear fiber optics. Unlike the classic multi-layer perceptron (MLP) structure, the trainable and transparent activation functions in KAN make the network have stronger physical interpretability and nonlinear characterization abilities. Multiple KANs are established for fiber-optic system dynamics under various physical effects. Results show that KANs can well discover and characterize the explicit, implicit, and non-analytical solutions under different effects, and achieve better performance than MLPs with the equivalent scale of trainable parameters. Moreover, the effectiveness, computational cost, interactivity, noise resistance, transfer learning ability, and comparison between related algorithms in fiber-optic systems are also studied and analyzed. This work highlights the transformative potential of KAN, establishing it as a pioneering paradigm in AI4S that propels advancements in nonlinear fiber optics, and fosters groundbreaking innovations across a broad spectrum of scientific and engineering disciplines.

Figures

Figures reproduced from arXiv: 2504.18833 by the authors.

Figure 1
Figure 1. Comparison between the structures of (a) MLP, (b) KAN, and (c) MultKAN. KAN(x) Φ1 Φ2 MLP(x) σ1 W1 W2 MultKAN(x) ψ1 ψ2 Φ1 Φ2 M1 M2 Addition nodes Multiplication nodes (a) (b) (c) [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. (a) Nonlinear dynamics in fiber optics and SSFM-based solution. (b) Schematic of KANs for nonlinear dynamics in fiber optics, taking the dynamics learning of pulse propagation under TOD effect as an example. (c) Details of the univariate activation function ϕi,j,k, which is parameterized as a B-spline and allows switching between coarse-grained and fine-grained grids. ( ) ( ) 2 +  − − + − =   2 2 2 3 2 0 26 z … view at source ↗
Figure 3
Figure 3. (a) The learning details of KANs for discovering and characterizing the power attenuation effect. (b) Loss curves of KAN for learning power attenuation. (c) Comparison between the analytical solution and generated solution obtained by KAN, as well as their absolute errors. z t ω u(z,t) z t ω u(z,t) z t ω u(z,t) z t ω u(z,t) z t ω (a) u(z,t) (b) (c) Pretrain Prune Train Symbolize z z z t t t u(z,t) upred(z,t) Error ×… view at source ↗
Figures from the paper (12 more)
Figure 5
Figure 5. Figure 5: (a) The learning details of KANs for discovering and characterizing the TOD effect. (b) Loss curves of KAN for learning TOD effect. (c) Comparison between the analytical solution and generated solution obtained by KAN, as well as their absolute errors. z ω u(z,ω) z ω u…
Figure 6
Figure 6. Figure 6: (a) The learning details of KANs for discovering and characterizing the fundamental soliton. (b) Loss curves of KAN for learning the fundamental soliton. (c) Comparison between the analytical solution and generated solution obtained by KAN, as well as their absolute er…
Figure 7
Figure 7. Figure 7: (a) The learning details of KANs for discovering and characterizing the SS effect. (b) Loss curves of KAN for learning SS effect. (c) Comparison between the analytical solution and generated solution obtained by KAN, as well as their absolute errors [PITH_FULL_IMAGE:f…
Figure 8
Figure 8. Figure 8: (a) The initial and trained KAN structures for characterizing the IRS effect. (b) Pulse evolution generated by KAN and the corresponding error distribution relative to the numerical solution. (c) Pulse evolution generated by MLP and the corresponding error distribution…
Figure 9
Figure 9. Figure 9: (a) Numerical solution of supercontinuum. Supercontinuum generated by (b) KAN and (c) MLP. (d) Predicted waveform comparison at 0.41LD. (e) Error between KAN and numerical solution. (f) Error between MLP and numerical solution. Supercontinuum generated by KAN Supercont…
Figure 11
Figure 11. Figure 11: Comparison of statistical tests between KAN and MLP in [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]
Figure 12
Figure 12. Figure 12: Maximum prediction error of KANs and MLPs with different number [PITH_FULL_IMAGE:figures/full_fig_p013_12.png]
Figure 13
Figure 13. Figure 13: (a) The learning details of KANs for discovering and characterizing the [PITH_FULL_IMAGE:figures/full_fig_p014_13.png]
Figure 15
Figure 15. Figure 15: Prior knowledge embedding of KANs for learning u(z,ω) under the TOD effect: (a) Molecular structure presetting; (b) Auxiliary variables presetting. z ω u(z,ω) z ω u(z,ω) z ω u(z,ω) Train Symbolize Initial state (Loss=2.95e-5) (Loss=4.46e-7) Pretrain Train & Initial st…
Figure 16
Figure 16. Figure 16: Hypothesis testing of KANs for discovering [PITH_FULL_IMAGE:figures/full_fig_p015_16.png]
Figure 17
Figure 17. Figure 17: Transfer learning of KANs for learning u(z,ω) under the TOD effect from u(z,ω) under the GVD effect. Train Initialize Symbolize z ω u(z,ω) z ω u(z,ω) z ω u(z,ω) z ω u(z,ω) (Loss=3.87e-7) (Loss=2.79e-4) [PITH_FULL_IMAGE:figures/full_fig_p016_17.png]
Figure 18
Figure 18. Figure 18: Comparison between the analytical solution and generated solution [PITH_FULL_IMAGE:figures/full_fig_p017_18.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

50 extracted references · 35 canonical work pages

  1. [1]

    Scientific discovery in the age of artificial intelligence,

    H. Wang, T. Fu, Y. Du, et al., “Scientific discovery in the age of artificial intelligence,” Nature, vol. 620, no. 7972, pp. 47–60, Aug. 2023

    work page 2023

  2. [2]

    Magnani, Abduction, reason and science: Processes of discovery and explanation, New York, USA: Springer, 2001

    L. Magnani, Abduction, reason and science: Processes of discovery and explanation, New York, USA: Springer, 2001

    work page 2001

  3. [3]

    Artificial intelligence for science in quantum, atomistic, and continuum systems,

    X. Zhang, L. Wang, J. Helwig, et al., “Artificial intelligence for science in quantum, atomistic, and continuum systems,” arXiv:2307.08423, 2023

    arXiv 2023

  4. [4]

    Highly accurate protein structure prediction with AlphaFold,

    J. Jumper, R. Evans, A. Pritzel, et al., “Highly accurate protein structure prediction with AlphaFold,” Nature, vol. 596, no. 7873, pp. 583–589, Aug. 2021

    work page 2021

  5. [5]

    Deep potential molecular dynamics: a scalable model with the accuracy of quantum mechanics,

    L. Zhang, J. Han, H. Wang, R. Car, and W. E, “Deep potential molecular dynamics: a scalable model with the accuracy of quantum mechanics,” Phys. Rev. Lett., vol. 120, no. 14, Art. no. 143001, Apr. 2018

    work page 2018

  6. [6]

    Th e molecular epidemiology of pain: a new discipline for drug discovery,

    M. B. Max and W. F. Stewart, “Th e molecular epidemiology of pain: a new discipline for drug discovery,” Nat. Rev. Drug Discov., vol. 7, no. 8, pp. 647–658, Aug. 2008

    work page 2008

  7. [7]

    Physics -AI symbiosis,

    B. Jalali, Y. Zhou, A. Kadambi, and V. Roychowdhury, “Physics -AI symbiosis,” Mach. Learn. Sci. Techn., vol. 3, no. 4, pp. 041001, 2022

    work page 2022

  8. [8]

    Physics-informed machine learning,

    G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang, “Physics-informed machine learning,” Nat. Rev. Phys., vol. 3, no. 6, pp. 422-440, 2021

    work page 2021

Show all 50 references
  1. [9]

    Analog optical computing,

    D. R. Solli, and B. Jalali, “Analog optical computing,” Nat. Photon., vol. 9, no. 11, pp. 704-706, 2015

    2015

  2. [10]

    Brain-inspired automated visual object discovery and detection,

    L. Chen, S. Singh, T. Kailath, and V. Roychowdhury, “Brain-inspired automated visual object discovery and detection,” In Proceedings of the National Academy of Sciences, vol. 116, no. 1, pp. 96-105, 2019

    2019

  3. [11]

    G. P. Agrawal, Nonlinear Fiber Optics , 2nd ed. Berlin, Germany: Springer, 2000

    2000

  4. [12]

    Nonlinear fiber optics: its history and recent progress ,

    G. P. Agrawal, “Nonlinear fiber optics: its history and recent progress ,” J. Opt. Soc. Am. B, vol. 28, no. 12, pp. A1-A10, 2011

    2011

  5. [13]

    Optical transmission fiber design evolution,

    M. J. Li, and D. A. Nolan, “Optical transmission fiber design evolution,” J. Lightw. Technol., vol. 26, no. 9, pp. 1079-1092, 2008

    2008

  6. [14]

    G. P. Agrawal, Fiber-optic communication systems . Hoboken, New Jersey, USA: John Wiley & Sons. 2012

    2012

  7. [15]

    Data -driven optical fiber channel modeling: A deep learning approach,

    D. Wang, Y. Song, J. Li, et al. , “Data -driven optical fiber channel modeling: A deep learning approach,” J. Lightw. Technol., vol. 38, no. 17, pp. 4730–4743, Sept. 2020

    2020

  8. [16]

    Fast and accurate optical fiber channel modeling using generative adversarial network,

    H. Yang, Z. Niu, S. Xiao, J. Fang, Z. Liu, D. Fainsin, and L. Yi, “Fast and accurate optical fiber channel modeling using generative adversarial network,” J. Lightw. Technol., vol. 39, no. 5, pp. 1322–1333, Mar. 2020

    2020

  9. [17]

    Multi -span long-haul fiber transmission model based on cascaded neural networks with multi-head attention mechanism,

    Y. Zang, Z. Yu, K. Xu, M. Chen, S. Yang, and H. Chen, “Multi -span long-haul fiber transmission model based on cascaded neural networks with multi-head attention mechanism,” J. Lightw. Technol., vol. 40, no. 19, pp. 6347–6358, Oct. 2022

    2022

  10. [18]

    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,” J. Comput. Phys., vol. 378, pp. 686–707, Feb. 2019

    2019

  11. [19]

    Physics- Informed Neural Network for Nonlinear Dynamics in Fiber Optics,

    X. Jiang, D. Wang, Q. Fan, M. Zhang, C. Lu, and A. P. T. Lau, “Physics- Informed Neural Network for Nonlinear Dynamics in Fiber Optics,” Laser Photon. Rev., vol. 16, no. 9, Art. no. 2100483, Sept. 2022

    2022

  12. [20]

    Principle-driven fiber transmission model based on PINN neural network,

    Y. Zang, Z. Yu, K. Xu, X. Lan, M. Chen, S. Yang, and H. Chen, “Principle-driven fiber transmission model based on PINN neural network,” J. Lightw. Technol., vol. 40, no. 2, pp. 404–414, Jan. 2021

    2021

  13. [21]

    Physics -informed neural network for optical fiber parameter estimation from the nonlinear Schrödinger equation,

    X. Jiang, D. Wang, X. Chen, and M. Zhang, “Physics -informed neural network for optical fiber parameter estimation from the nonlinear Schrödinger equation,” J. Lightw. Technol. , vol. 40, no. 21, pp. 7095 – 7105, Nov. 2022

    2022

  14. [22]

    Haykin, Neural Networks: A Comprehensive Foundation , Upper Saddle River, NJ, USA: Prentice Hall PTR, 1998

    S. Haykin, Neural Networks: A Comprehensive Foundation , Upper Saddle River, NJ, USA: Prentice Hall PTR, 1998

    1998

  15. [23]

    Multilayer feedforward networks are universal approximators,

    K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural Netw., vol. 2, no. 5, pp. 359–366, Jan. 1989

    1989

  16. [24]

    Sparse autoencoders find highly interpretable features in language models,

    R. Huben, H. Cunningham, L. R. Smith, A. Ewart, and L. S harkey, “Sparse autoencoders find highly interpretable features in language models,” presented at the 12th Int. Conf. Learn. Represent., May 2023

    2023

  17. [25]

    On the spectral bias of neural networks,

    N. Rahaman, A. Baratin, D. Arpit, F. Draxler, M. Lin, F. Hamprecht, ... and A. Courville, “On the spectral bias of neural networks,” in Proc. Int. Conf. Mach. Learn., Long Beach, CA, USA, pp. 5301–5310, 2019

    2019

  18. [26]

    Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators,

    L. Lu, P. Jin, G. Pang, Z. Zhang, and G. E. Karniadakis, “Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators,” Nat. Mach. Intell., vol. 3, no. 3, pp. 218–229, Mar. 2021

    2021

  19. [27]

    Fourier neural operator for parametric partial differential equations,

    Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart, and A. Anandkumar, “Fourier neural operator for parametric partial differential equations,” arXiv preprint arXiv:2010.08895, Oct. 2020

    2010 arXiv

  20. [28]

    Learning the solution operator of parametric partial differential equations with physics -informed DeepONets,

    S. Wang, H. Wang, and P. Perdikaris, “Learning the solution operator of parametric partial differential equations with physics -informed DeepONets,” Sci. Adv., vol. 7, no. 40, p. eabi8605, Oct. 2021

    2021

  21. [29]

    KAN: Kolmogorov-Arnold networks,

    Z. Liu, Y. Wang, S. Vaidya, F. Ruehle, J. Halverson, M. Soljačić, et al., “KAN: Kolmogorov-Arnold networks," arXiv:2404.19756, 2024

    2024 arXiv

  22. [30]

    On the representation of continuous functions of several variables as superpositions of continuous functions of a smaller number of variables,

    A. N. Kolmogorov, “On the representation of continuous functions of several variables as superpositions of continuous functions of a smaller number of variables,” Dokl. Akad. Nauk, vol. 108, no. 2, 1956

    1956

  23. [31]

    WAV-KAN: Wavelet Kolmogorov-Arnold networks,

    Z. Bozorgasl and H. Chen, “WAV-KAN: Wavelet Kolmogorov-Arnold networks,” arXiv:2405.12832, 2024

    2024 arXiv

  24. [32]

    Chebyshev polynomial-based Kolmogorov-Arnold networks: An efficient architecture for nonlinear function approximation,

    S. S. Sidharth, R. Gokul, K. P. Anas, and A. R. Keerthana, “Chebyshev polynomial-based Kolmogorov-Arnold networks: An efficient architecture for nonlinear function approximation, ” arXiv:2405.07200, 2024

    2024 arXiv

  25. [33]

    Kolmogorov -Arnold Fourier Networks,

    J. Zhang, Y. Fan, K. Cai, and K. Wang, “Kolmogorov -Arnold Fourier Networks,” arXiv:2502.06018, 2025

    2025 arXiv

  26. [34]

    Relu -kan: New kolmogorov-arnold networks that only need matrix addition, dot multiplication, and relu,

    Q. Qiu, T. Zhu, H. Gong, L. Chen, and H. Ning, “Relu -kan: New kolmogorov-arnold networks that only need matrix addition, dot multiplication, and relu,” arXiv:2406.02075, 2024

    2024 arXiv

  27. [35]

    Convolutional Kolmogorov -Arnold networks ,

    A. D. Bodner, A. S. Tepsich, J. N. Spolski, and S. Pourteau, “Convolutional Kolmogorov -Arnold networks ,” arXiv:2406.13155, 2024

    2024 arXiv

  28. [36]

    TKAN: Temporal Kolmogorov -Arnold networks,

    R. Genet and H. Inzirillo, “TKAN: Temporal Kolmogorov -Arnold networks,” arXiv:2405.07344, 2024

    2024 arXiv

  29. [37]

    U-KAN makes strong backbone for medical image segmentation and generation ,

    C. Li, X. Liu, W. Li, C. Wang, H. Liu, Y. Liu, et al., “U-KAN makes strong backbone for medical image segmentation and generation ,” arXiv:2406.02918, 2024

    2024 arXiv

  30. [38]

    Approximation by superpositions of a sigmoidal function,

    G. Cybenko, “Approximation by superpositions of a sigmoidal function,” Math. Control Signals Syst., vol. 2, pp. 303–314, 1989

    1989

  31. [39]

    Error bounds for deep ReLU networks using the Kolmogorov –Arnold superposition theorem,

    H. Montanelli and H. Yang, “Error bounds for deep ReLU networks using the Kolmogorov –Arnold superposition theorem, ” Neural Netw. , vol. 129, pp. 1–6, 2020

    2020

  32. [40]

    Space-filling curves and Kolmogorov superposition-based neural networks,

    D. A. Sprecher and S. Draghici, “Space-filling curves and Kolmogorov superposition-based neural networks,” Neural Netw., vol. 15, no. 1, pp. 57–67, 2002

    2002

  33. [41]

    ExSpliNet: An interpretable and expressive spline-based neural network,

    D. Fakhoury, E. Fakhoury, and H. Speleers, “ExSpliNet: An interpretable and expressive spline-based neural network,” Neural Netw., vol. 152, pp. 332–346, 2022

    2022

  34. [42]

    KAN 2.0: Kolmogorov-Arnold networks meet science,

    Z. Liu, P. Ma, Y. Wang, W. Matusik, and M. Tegmark, “KAN 2.0: Kolmogorov-Arnold networks meet science,” arXiv:2408.10205, 2024

    2024 arXiv

  35. [43]

    On the Convergence of (Stochastic) Gradient Descent for Kolmogorov-Arnold Networks,

    Y. Gao, and V. Y. Tan, “On the Convergence of (Stochastic) Gradient Descent for Kolmogorov-Arnold Networks,” arXiv:2410.08041, 2024

    2024 arXiv

  36. [44]

    On the study of frequency control and spectral bias in Wavelet-Based Kolmogorov Arnold networks: A path to physics-informed KANs,

    J. D. Meshir, A. Palafox, and E. A. Guerrero, “On the study of frequency control and spectral bias in Wavelet-Based Kolmogorov Arnold networks: A path to physics-informed KANs,” arXiv:2502.00280, 2025

    2025 arXiv

  37. [45]

    The role of Occam's razor in knowledge discovery,

    P. Domingos, “The role of Occam's razor in knowledge discovery,” Data mining and knowledge discovery, vol. 3, pp. 409-425, 1999

    1999

  38. [46]

    Soliton robustness in optical fibers,

    C. R. Menyuk, “Soliton robustness in optical fibers,” J. Opt. Soc. Am. B, vol. 10, no. 9, pp. 1585–1591, 1993

    1993

  39. [47]

    J. M. Dudley, and J. R. Taylor, Supercontinuum generation in optical fibers, Cambridge, UK: Cambridge university press, 2010

    2010

  40. [48]

    Predicting ultrafast nonlinear dynamics in fiber optics by enhanced physics-informed neural network,

    X. Jiang, M. Zhang, Y. Song, H. Chen, D. Huang, and D. Wang, “Predicting ultrafast nonlinear dynamics in fiber optics by enhanced physics-informed neural network,” J. Lightw. Technol., vol. 42, no. 5, pp. 1381–1394, Mar. 2024

    2024

  41. [49]

    Measuring catastrophic forgetting in neural networks,

    R. Kemker, M. McClure, A. Abitino, T. Hayes, and C. Kanan, “Measuring catastrophic forgetting in neural networks,” In Proceedings of the AAAI conference on artificial intelligence, vol. 32, no. 1, Apr. 2018

    2018

  42. [50]

    AI Feynman: A physics -inspired method for symbolic regression,

    S. M. Udrescu, and M. Tegmark, “AI Feynman: A physics -inspired method for symbolic regression,” Sci. Adv., vol. 6, no. 16, pp. eaay2631, 2020

    2020

Pith tools

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