Pith. sign in

REVIEW 5 major objections 5 minor 1 cited by

Neural Tangent Kernel Analysis to Probe Convergence in Physics-informed Neural Solvers: PIKANs vs. PINNs

T0 review · 5 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read Stable kernel spectra track cPIKAN convergence gains.

desk verdict A useful first NTK study of cPIKANs with a new kernel formula and broad experiments, but the causal claim that spectra explain convergence is overreached and the drift-bound proof has a real flaw. read the letter →

arxiv 2506.07958 v1 pith:XFZTZFIQ submitted 2025-06-09 cs.LG math-phmath.APmath.MPmath.SP

Salah A. Faroughi , Farinaz Mostajeran This is my paper
classification cs.LGmath-phmath.APmath.MPmath.SP
keywords neuraltangentkernelphysics-informednetworksKolmogorov-ArnoldChebyshevpolynomialsspectralbiasdomaindecompositionPDEsolversconvergenceanalysis
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 analyzes Chebyshev-based physics-informed Kolmogorov–Arnold networks (cPIKANs) through neural tangent kernel (NTK) theory, aiming to show that their training dynamics are more tractable than those of standard physics-informed neural networks (PINNs). It argues that the cPIKAN kernel stays more structured and better conditioned during training, and that this spectral structure helps explain why cPIKANs converge faster and more accurately on PDE benchmarks. The empirical centerpiece is a forced vibration problem where cPIKAN with 8 temporal subdomains achieves about 98% lower maximum error than a PINN with the same decomposition. A sympathetic reader would take the paper to be establishing kernel spectral statistics as a practical diagnostic for when domain decomposition and optimizer choice improve training.

What carries the argument

The carrying object is the empirical neural tangent kernel matrix of the physics-informed network, together with two derived diagnostics. The kernel's blocks are inner products of parameter gradients: $K_{uu}$ for network outputs, $K_{rr}$ for PDE residuals, and $K_{ur}$ for the cross-terms, and under gradient flow the stacked outputs follow $d\psi/d\tau = -K_{ntk}(\psi-G)$. The diagnostics are the eigenvalue spectrum $\lambda(K_{ij})$ and the spectral entropy $S = -\sum_i p_i \log p_i$ with $p_i = |\lambda_i|/\sum_j |\lambda_j|$, which measures whether learning directions are concentrated in a few dominant modes. This machinery translates architecture and training choices, such as Chebyshev degree, input scaling, optimizer, and subdomain count, into a kernel-conditioning picture in which stable, concentrated spectra indicate effective learning directions.

What would settle it

Train a cPIKAN on the Allen–Cahn equation with 8 temporal subdomains while measuring $\|\theta(\tau)-\theta(0)\|$ and the NTK spectral drift; if the kernel moves substantially during training while accuracy still improves, or if a deliberately diffuse initial spectrum still converges to low error, the spectral-stability explanation is falsified. A sharper test is to compute the alignment between the top NTK eigenvectors and the training residual $(K_{ntk}\psi - G)$ and check whether that alignment, not just eigenvalue concentration, tracks convergence.

Watch

Extended reading notes

Core claim

The central claim is that the NTK of a cPIKAN is analytically tractable and empirically informative. At initialization the expected kernel has a closed two-term form, $E[K_{ntk}(x,x')] = N[\sum_n C_n(x,x') + \sum_i \sum_n T_n(\tilde x_i)T_n(\tilde x'_i)D(x,x')]$, where $C_n$ captures Chebyshev-feature correlations after tanh activations and $D$ captures gradients through the activation function. Theorem 2 bounds kernel drift during gradient flow by $2B_1B_2\|\theta(\tau)-\theta(0)\|$, so the kernel stays nearly constant when parameter drift is small. In the physics-informed setting the kernel is the block matrix with output, residual, and cross blocks, and the prediction error evolves as $d\psi/d\tau = -K_{ntk}(\psi-G)$. Across the diffusion, Helmholtz, Allen–Cahn, and forced-vibration experiments, the paper reports that cPIKAN spectra remain broader and more persistent in residual blocks, that spectral entropy drops as temporal subdomains increase, and that these spectral trends accompany large error reductions, most notably about 98% lower maximum error than PINN in the 8-subdomain forced vibration case.

Load-bearing premise

The load-bearing premise is that finite-width cPIKANs trained with Adam, L-BFGS, or hybrids remain close enough to the NTK regime for kernel spectra and spectral entropy to describe the learning dynamics that produce the observed accuracy gains; the paper itself notes these metrics do not by themselves explain accuracy or generalization.

Editorial extensions

If this is right

  • Temporal domain decomposition is not just a parallelization trick: for cPIKANs it lowers spectral entropy of $K_{uu}$ and $K_{u_t u_t}$ and accompanies error drops from $5.09\times10^{-1}$ to $6.21\times10^{-3}$ on the Allen–Cahn equation.
  • Optimizer choice changes NTK evolution: L-BFGS and hybrid ADAM+LBFGS shift eigenvalues faster and reduce the Helmholtz relative $L^2$ error from $6.61\times10^{-2}$ with ADAM alone to about $5\times10^{-3}$.
  • cPIKAN retains higher-magnitude eigenvalues in the residual block $K_{rr}$ through training, which the paper interprets as better information flow from PDE residuals than PINNs achieve.
  • The spectral metrics suggest a practical rule: when the NTK spectrum is diffuse or slow to stabilize, adding temporal subdomains or switching to a second-order optimizer should improve convergence.
  • The NTK-based analysis extends to high-order PDEs, where additional derivative terms enlarge the kernel into a structured block matrix that still yields interpretable spectral-entropy trends.

Reading between the lines

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

  • If spectral entropy at initialization predicts final error, one could choose subdomain counts and optimizers from a single cheap NTK computation before training, without solving the PDE.
  • The paper's own future-outlook section concedes that spectral metrics do not fully explain accuracy or generalization, so the spectral-stability story is likely one mechanism among several rather than a complete theory.
  • The same spectral-entropy diagnostic could be applied to other KAN basis choices, such as wavelets or radial basis functions, to predict which variant benefits most from domain decomposition.
  • Because Theorem 2's drift bound rests on gradient and Hessian bounds that the experiments do not verify, the observed spectral stability could partly reflect well-scaled targets rather than a true NTK regime; measuring parameter drift during training would separate these explanations.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

Summary. The manuscript analyzes the Neural Tangent Kernel (NTK) of Chebyshev-based Kolmogorov-Arnold Networks in both supervised (cKAN) and physics-informed (cPIKAN) settings. It derives an expected NTK formula for a one-hidden-layer cKAN, states a drift bound for finite-width cKANs, and gives a generic block-form NTK for physics-informed losses. The main body presents four numerical PDE experiments (diffusion, Helmholtz, Allen-Cahn, forced Euler-Bernoulli vibration) comparing cPIKAN with PINN and bPIKAN under Adam, L-BFGS, and hybrid optimization, with temporal domain decomposition in the last two experiments. The paper claims that NTK eigenvalue spectra and spectral entropy reveal and directly explain cPIKAN convergence advantages over PINNs.

Significance. If the central claim were established, this would be a useful contribution: it is the first systematic NTK study of cPIKANs, it covers four benchmark PDEs, and it proposes spectral diagnostics for when domain decomposition helps. The paper's strengths include the explicit parameter-free expectation in Theorem 1, the clear empirical comparisons against PINN and bPIKAN baselines, and the reproducible tables of architecture, error, and runtime. However, the load-bearing causal claim that spectral behavior 'directly links' to convergence is not supported by the current theory or experiments; the cPIKAN-specific NTK is not actually derived beyond a generic block form, and the only drift theorem has a flawed proof. The empirical results remain valuable if reframed as descriptive observations rather than a proven mechanistic explanation.

major comments (5)
  1. [Appendix B, Eq. (B.4)] The proof of Theorem 2 replaces the integral of ||theta_dot(s)|| ds from 0 to tau with ||theta(tau) - theta(0)||. This is false in general because path length is at least the displacement, so Eq. (B.4) does not follow from Eq. (B.3). As written, the advertised bound (36) is not established. The theorem must either state the correct bound with the integrated gradient norm and prove that this integral is small under the stated assumptions, or add assumptions that make the replacement valid. The additional claim that ||theta(tau)-theta(0)|| vanishes as N -> infinity is also not proved and is independent of the previous boundedness assumptions.
  2. [Section 3.2, Lemma 1 and Eq. (45)] The physics-informed NTK for cPIKAN is not actually derived. Lemma 1 reproduces the generic block form of Wang et al. [60] and states that the proof follows the same reasoning, but no computation of the residual derivatives d(N[u])/dtheta or of the blocks K_rr and K_ru in terms of the Chebyshev structure is provided. Since the novelty claim is a cPIKAN-specific NTK analysis, this missing derivation is load-bearing. Theorem 1 is limited to a supervised one-hidden-layer cKAN and does not cover the multi-layer cPIKANs used in the experiments.
  3. [Abstract and Section 5] The central claim that spectral trends 'directly link kernel behavior to convergence rates' is not supported by the evidence. Section 5 concedes that the spectral metrics 'do not fully explain how the NTK influences model accuracy or generalization.' The experiments show correlations only: for example, Section 4.2 explicitly states that 'a well-conditioned NTK spectrum alone does not guarantee high solution accuracy.' To support the causal or predictive claim, the paper needs either a convergence theorem involving the NTK spectrum or a quantitative test, such as measuring the alignment between the NTK top eigenspace and the residual/target during training, and showing that this alignment predicts the observed error drops.
  4. [Experiments 4.1-4.4 and Theorem 2] The NTK-regime assumption is unverified for the actual training setups. The experiments use finite-width networks (widths 8-47, depths 2-4) trained with Adam, L-BFGS, or hybrid updates, while Theorem 2 assumes gradient flow and uniformly bounded gradients and Hessians. None of these conditions is checked. Moreover, the paper's own figures (Figs. 2, 4, 6, 8-9, 11) show eigenvalue spectra changing substantially during training, which indicates that the empirical NTK is not near-constant. The paper should either report kernel-drift magnitudes and verify the assumptions, or explicitly reframe the spectral analysis as a descriptive study of a time-varying kernel rather than a fixed kernel governing convergence.
  5. [Section 4.4, Eq. (55)] The spectral entropy metric is introduced without any theoretical connection to convergence or generalization. It is claimed to 'capture the transient and convergence behavior of the NTK spectrum,' but no argument or experiment establishes that reduced spectral entropy causes or predicts faster convergence. Since this metric is used to support the main claim in the forced-vibration experiment, it needs at least a formal definition of what it measures (e.g., effective rank versus variance concentration) and a demonstration that it tracks the convergence behavior beyond visual inspection.
minor comments (5)
  1. [Appendix A, Eq. (A.13)] The definition of D(x,x') in Eq. (A.13) contains an extra factor psi(x') compared with Eq. (32), making the two definitions inconsistent; this appears to be a typo that should be corrected.
  2. [Section 4.4, Eq. (55) and Figures 11-12] The text calls the quantity in Eq. (55) 'spectral entropy' but the figures and captions label it 'spectral bias'; the terminology should be made consistent and the relationship between the two concepts clarified.
  3. [Section 2.1, B-splines] The sentence 'B-splines are a common choice []' has an empty citation; a reference should be supplied.
  4. [Section 4.2, Figure 6 discussion] The sentence 'LBFGS and hybrid approaches demonstrate a more rapid and structured shift in the eigenvalues' is vague; it would be helpful to quantify the shift, for example by reporting the effective rank or the largest eigenvalue ratio at selected epochs.
  5. [Section 4.3, Table 3] The error reduction from 7.90e-3 (2 subdomains) to 6.21e-3 (4 subdomains) is small relative to the reduction from the full domain, so the text should avoid implying that the 4-subdomain improvement is of the same magnitude as the 2-subdomain improvement.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: NTK derivations are self-contained, and spectral/error links are empirical correlations, not fitted predictions.

full rationale

The paper's derivation chain is self-contained at the points that matter. Theorem 1 computes the expected NTK of a one-hidden-layer cKAN as an explicit expectation over i.i.d. Gaussian initialization, and Appendix A carries out the calculation without quoting a prior result; this is parameter-free and not fitted to the PDEs. Lemma 1 is an exact chain-rule identity for gradient flow (the same structure as the external reference [60]), so the cPIKAN NTK is not a fitted quantity. The empirical claims that cPIKAN spectra are more concentrated/stable and correlate with lower errors are post-hoc observations on the same training runs, which is an explanatory limitation (the paper itself concedes in Section 5 that these metrics 'do not fully explain how the NTK influences model accuracy or generalization') rather than a circular reduction. The only salient self-citation is [54], which supplies the scaled-cPIKAN architecture and prior NTK observation; but the present paper re-derives a nested-cKAN NTK and does not invoke [54] as a proof of the central spectral claims. I therefore find no step where a 'prediction' equals its input by construction. Note for correctness, not circularity: Appendix B replaces the integral of ||theta_dot(s)|| ds with ||theta(tau)-theta(0)||, which is not generally valid; this is a proof flaw, not an input-output equivalence.

Assumptions & free parameters 2 free parameters · 6 assumptions · 1 invented entities

The theoretical development introduces no fitted constants into the central NTK formulas; however, the empirical claims depend on hand-chosen architectures, sampling sizes, and scaling choices inherited from prior work. The core assumptions are standard NTK linearization conditions that are stated but not verified for the finite networks actually trained, plus domain-level assumptions about PDE exactness.

free parameters (2)
  • Domain scaling parameters (from Scaled-cPIKAN [54]) = not reported
    All experiments use the scaled cPIKAN architecture; the scaling choices affect the NTK spectra and final errors but no values or selection procedure are given.
  • Per-experiment architecture and sampling sizes (N_l, N_n, k, N_r, N_d) = reported per experiment (e.g., (2,8,5), 2000/800)
    These hyperparameters are chosen by hand and directly determine the reported relative L2 errors and spectral behavior; no sensitivity analysis is provided.
assumptions (6)
  • standard math Chebyshev polynomial identities and boundedness on [-1,1]
    Used throughout Theorem 1 and Section 2.1 for NTK expectations.
  • domain assumption tanh maps inputs and hidden activations into [-1,1], preserving smoothness
    Architecture definition Eq. (7) and Theorem 1 rely on this normalization to keep Chebyshev evaluations stable.
  • domain assumption Well-posedness of the four PDEs and availability of analytic ground-truth solutions
    Relative L2 errors are computed against closed-form solutions for diffusion, Helmholtz, Allen-Cahn, and Euler-Bernoulli beam; no verification beyond assumed exactness.
  • domain assumption Gradient flow is a valid continuous-time model for Adam and L-BFGS training
    Lemma 1 and Theorem 2 are derived under gradient flow, but experiments use Adam, L-BFGS, and hybrid schemes; the approximation is not validated.
  • domain assumption Uniform boundedness of gradients and Hessians (Assumption II in Theorem 2)
    Required for the NTK drift bound; not established for the finite-width cKANs/cPIKANs used in the experiments.
  • domain assumption Parameter drift vanishes in the infinite-width limit for cKANs
    Invoked in Theorem 2 to make the NTK constant; no cKAN-specific proof of this vanishing is provided.
invented entities (1)
  • Spectral entropy of the NTK
    purpose: Scalar diagnostic (Eq. 55) summarizing how concentrated the NTK eigenvalue distribution is; used to argue domain decomposition simplifies learning dynamics.
    The paper introduces this metric and interprets lower values as better conditioning, but does not validate it against independent predictors or show it predicts accuracy beyond the cases already fitted by post hoc observation.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Neural Tangent Kernel Analysis to Probe Convergence in Physics-informed Neural Solvers: PIKANs vs. PINNs." pith.science (2026). https://pith.science/paper/XFZTZFIQ

@misc{pith2026250607958,
  author       = {Pith},
  title        = {Pith review of: Neural Tangent Kernel Analysis to Probe Convergence in Physics-informed Neural Solvers: PIKANs vs. PINNs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XFZTZFIQ}},
  note         = {Machine review of arXiv:2506.07958}
}
read the original abstract

Physics-informed Kolmogorov-Arnold Networks (PIKANs), and in particular their Chebyshev-based variants (cPIKANs), have recently emerged as promising models for solving partial differential equations (PDEs). However, their training dynamics and convergence behavior remain largely unexplored both theoretically and numerically. In this work, we aim to advance the theoretical understanding of cPIKANs by analyzing them using Neural Tangent Kernel (NTK) theory. Our objective is to discern the evolution of kernel structure throughout gradient-based training and its subsequent impact on learning efficiency. We first derive the NTK of standard cKANs in a supervised setting, and then extend the analysis to the physics-informed context. We analyze the spectral properties of NTK matrices, specifically their eigenvalue distributions and spectral bias, for four representative PDEs: the steady-state Helmholtz equation, transient diffusion and Allen-Cahn equations, and forced vibrations governed by the Euler-Bernoulli beam equation. We also conduct an investigation into the impact of various optimization strategies, e.g., first-order, second-order, and hybrid approaches, on the evolution of the NTK and the resulting learning dynamics. Results indicate a tractable behavior for NTK in the context of cPIKANs, which exposes learning dynamics that standard physics-informed neural networks (PINNs) cannot capture. Spectral trends also reveal when domain decomposition improves training, directly linking kernel behavior to convergence rates under different setups. To the best of our knowledge, this is the first systematic NTK study of cPIKANs, providing theoretical insight that clarifies and predicts their empirical performance.

Figures

Figures reproduced from arXiv: 2506.07958 by the authors.

Figure 1
Figure 1. Comparison of the predicted solutions for the diffusion equation in Experiment 4.1, demonstrating that cPIKAN yields [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Evolution of the NTK eigenvalue spectra during training for the diffusion equation in Experiment 4.1. The NTK [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Comparison of the predicted solutions for the Helmholtz equation in Experiment 4.2, showing that cPIKAN produces [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Evolution of the NTK eigenvalue spectra during training for the Helmholtz equation in Experiment 4.2. The spectra [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: Comparison of predicted solutions for the Helmholtz equation using different optimization strategies in Experiment 4.2. [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]
Figure 6
Figure 6. Figure 6: Evolution of the NTK eigenvalue spectra during training for the Helmholtz equation in Experiment 4.2, using three [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: Comparison of the predicted solutions for the Allen–Cahn equation (Experiment 4.3) using the cPIKAN method with [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: Evolution of the NTK eigenvalue spectra during training of the cPIKAN method for solving the Allen–Cahn equation [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: Evolution of the NTK eigenvalue spectra during training of the cPIKAN method for solving the Allen–Cahn equation [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 10
Figure 10. Figure 10: Comparison of predicted solutions for the forced vibration equation (Experiment 4.4) using cPIKAN with varying [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]
Figure 11
Figure 11. Figure 11: Comparison of the training dynamics and spectral entropy behavior of the cPIKAN method for the forced vibration [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]
Figure 12
Figure 12. Figure 12: Comparison of training dynamics and spectral bias between (a) cPIKAN and (b) PINN for the forced vibration problem [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Deformable Medical Image Registration with KAN-based Implicit Neural Representations

    cs.CV 2025-09 conditional novelty 6.0 of 10

    KAN-based implicit neural networks with randomized basis sampling outperform existing INR registration methods on three medical imaging datasets at lower computational cost.

Reference graph

Works this paper leans on

104 extracted references · 64 canonical work pages · cited by 1 Pith paper

  1. [60]

    S. Wang, X. Yu, P. Perdikaris, When and why pinns fail to train: A neural tangent kernel perspective, Journal of Computational Physics 449 (2022) 110768

    2022

  2. [1]

    D. R. Gaston, C. J. Permann, J. W. Peterson, A. E. Slaughter, D. Andrš, Y. Wang, M. P. Short, D. M. Perez, M. R. Tonks, J. Ortensi, et al., Physics-based multiscale coupling for full core nuclear reactor simulation, Annals of Nuclear Energy 84 (2015) 45–54

    2015

  3. [2]

    D. A. Cullen, K. C. Neyerlin, R. K. Ahluwalia, R. Mukundan, K. L. More, R. L. Borup, A. Z. Weber, D. J. Myers, A. Kusoglu, New roads and challenges for fuel cells in heavy-duty transportation, Nature energy 6 (5) (2021) 462–474

    2021

  4. [3]

    T. Zhou, R. Gani, K. Sundmacher, Hybrid data-driven and mechanistic modeling approaches for multi- scale material and process design, Engineering 7 (9) (2021) 1231–1238

    2021

  5. [4]

    C. Xi, L. Rongchao, T. Ye, T. Yongqi, L. Ao, M. Daofeng, Z. Haibo, Engineering design and numerical design for chemical looping combustion reactor: A review, Energy Reviews (2024) 100100

    2024

  6. [5]

    J. Lee, D. Jun, B. Chun, S. M. Mousavi, B. J. Lee, S. A. Faroughi, Large eddy simulation of the effects of radiative heat loss on combustion instability prediction, Acta Astronautica 217 (2024) 312–322

    2024

  7. [6]

    D. I. Fotiadis, A. I. Sakellarios, V. T. Potsika, Multiscale Modelling in Biomedical Engineering, John Wiley & Sons, 2023

    2023

  8. [7]

    H. Yu, L. Zhang, W. Wang, K. Yang, Z. Zhang, X. Liang, S. Chen, S. Yang, J. Li, X. Liu, Lithium- ion battery multi-scale modeling coupled with simplified electrochemical model and kinetic monte carlo model, Iscience 26 (9) (2023)

    2023

Show all 104 references
  1. [8]

    X. Gao, B. Knueven, J. D. Siirola, D. C. Miller, A. W. Dowling, Multiscale simulation of integrated energy system and electricity market interactions, Applied Energy 316 (2022) 119017. 24

    2022

  2. [9]

    N. M. Pawar, R. Soltanmohammadi, S. Faroughi, S. A. Faroughi, Geo-guided deep learning for spatial downscaling of solute transport in heterogeneous porous media, Computers & Geosciences 188 (2024) 105599

    2024

  3. [10]

    S. K. Mahjour, G. Liguori, S. A. Faroughi, Selection of representative general circulation models under climatic uncertainty for western north america, Journal of Water and Climate Change 15 (2) (2024) 686–702

    2024

  4. [11]

    S. K. Mahjour, J. P. Tiefenbacher, S. A. Faroughi, Select representative general circulation model-runs using enveloped-based technique, Journal of Climate (2025)

    2025

  5. [12]

    Heinlein, A

    A. Heinlein, A. A. Howard, D. Beecroft, Multifidelity domain decomposition-based physics-informed neural networks and operators for time-dependent problems, Mathematical Optimization for Machine Learning: Proceedings of the MATH+ Thematic Einstein Semester 2023 (2025) 79

    2025

  6. [13]

    X. Zhou, Y. Liu, M. Ali, M. He, A multilevel-multiphysics modeling and simulation approach for multi- chip electronics, Applied Thermal Engineering (2025) 125738

    2025

  7. [14]

    B. G. Van Willigen, M. B. van der Hout-van der Jagt, W. Huberts, F. N. van de Vosse, A multiscale mathematical model for fetal gas transport and regulatory systems during second half of pregnancy, International Journal for Numerical Methods in Biomedical Engineering 41 (1) (20...

    2025

  8. [15]

    S. A. Faroughi, N. M. Pawar, C. Fernandes, M. Raissi, S. Das, N. K. Kalantari, S. Kourosh Mahjour, Physics-guided, physics-informed, and physics-encoded neural networks and operators in scientific com- puting: Fluid and solid mechanics, Journal of Computing and Information Sci...

    2024

  9. [16]

    D.Kim, J.Lee, Areviewofphysicsinformedneuralnetworksformultiscaleanalysisandinverseproblems, Multiscale Science and Engineering 6 (1) (2024) 1–11

    2024

  10. [17]

    O.-H. E. Oladayo, O. Joshua, Stability analysis of explicit finite difference methods for neutral stochastic differential equations with multiplicative noise, Asian Research Journal of Current Science 7 (1) (2025) 12–21

    2025

  11. [18]

    Ö. Oruç, A. Esen, F. Bulut, Numerical solution of the rosenau-kdv-rlw equation via combination of a polynomial scaling function collocation and finite difference method, Mathematical Methods in the Applied Sciences (2025)

    2025

  12. [19]

    Weiss, M

    J. Weiss, M. Knezevic, Effects of element type on accuracy of microstructural mesh crystal plasticity finite element simulations and comparisons with elasto-viscoplastic fast fourier transform predictions, Computational Materials Science 240 (2024) 113002

    2024

  13. [20]

    Raissi, P

    M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics-informed neural networks: A deep learning frame- work for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Computational physics 378 (2019) 686–707

    2019

  14. [21]

    Raissi, G

    M. Raissi, G. E. Karniadakis, Hidden physics models: Machine learning of nonlinear partial differential equations, Journal of Computational Physics 357 (2018) 125–141

    2018

  15. [22]

    G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, L. Yang, Physics-informed machine learning, Nature Reviews Physics 3 (6) (2021) 422–440

    2021

  16. [23]

    Mostajeran, R

    F. Mostajeran, R. Mokhtari, Deepbhcp: Deep neural network algorithm for solving backward heat conduction problems, Computer Physics Communications 272 (2022) 108236

    2022

  17. [24]

    Hanrahan, M

    S. Hanrahan, M. Kozul, R. D. Sandberg, Studying turbulent flows with physics-informed neural networks and sparse data, International Journal of Heat and Fluid Flow 104 (2023) 109232

    2023

  18. [25]

    S. Jang, M. Jadidi, S. Rezaeiravesh, A. Revell, Y. Mahmoudi, Physics-informed neural network for tur- bulent flow reconstruction in composite porous-fluid systems, Machine Learning: Science and Technology 5 (3) (2024) 035030

    2024

  19. [26]

    Yazdani, M

    S. Yazdani, M. Tahani, Data-driven discovery of turbulent flow equations using physics-informed neural networks, Physics of Fluids 36 (3) (2024)

    2024

  20. [27]

    Gafoor CTP, S

    A. Gafoor CTP, S. Kumar Boya, R. Jinka, A. Gupta, A. Tyagi, S. Sarkar, D. N. Subramani, A physics- informed neural network for turbulent wake simulations behind wind turbines, Physics of Fluids 37 (1) (2025)

    2025

  21. [28]

    Mahmoudabadbozchelou, G

    M. Mahmoudabadbozchelou, G. E. Karniadakis, S. Jamali, nn-pinns: Non-newtonian physics-informed neural networks for complex fluid modeling, Soft Matter 18 (1) (2022) 172–185. 25

    2022

  22. [29]

    Thakur, M

    S. Thakur, M. Raissi, A. M. Ardekani, Viscoelasticnet: A physics informed neural network framework for stress discovery and model selection, Journal of Non-Newtonian Fluid Mechanics 330 (2024) 105265

    2024

  23. [30]

    E.Kharazmi, Z.Zhang, G.E.Karniadakis, hp-vpinns: Variationalphysics-informedneuralnetworkswith domain decomposition, Computer Methods in Applied Mechanics and Engineering 374 (2021) 113547

    2021

  24. [31]

    Arzani, S

    A. Arzani, S. T. Dawson, Data-driven cardiovascular flow modelling: examples and opportunities, Jour- nal of the Royal Society Interface 18 (175) (2021) 20200802

    2021

  25. [32]

    Zhang, B

    X. Zhang, B. Mao, Y. Che, J. Kang, M. Luo, A. Qiao, Y. Liu, H. Anzai, M. Ohta, Y. Guo, et al., Physics-informed neural networks (pinns) for 4d hemodynamics prediction: An investigation of optimal framework based on vascular morphology, Computers in Biology and Medicine 164 (20...

    2023

  26. [33]

    Rasht-Behesht, C

    M. Rasht-Behesht, C. Huber, K. Shukla, G. E. Karniadakis, Physics-informed neural networks (pinns) for wave propagation and full waveform inversions, Journal of Geophysical Research: Solid Earth 127 (5) (2022) e2021JB023120

    2022

  27. [34]

    J. Zou, C. Liu, Y. Wang, C. Song, U. b. Waheed, P. Zhao, Accelerating the convergence of physics- informed neural networks for seismic wave simulation, Geophysics 90 (2) (2025) T23–T32

    2025

  28. [35]

    J. Zou, C. Liu, P. Zhao, C. Song, Seismic wavefields modeling with variable horizontally-layered velocity models via velocity-encoded pinn, IEEE Transactions on Geoscience and Remote Sensing (2024)

    2024

  29. [36]

    A. D. Jagtap, Z. Mao, N. Adams, G. E. Karniadakis, Physics-informed neural networks for inverse problems in supersonic flows, Journal of Computational Physics 466 (2022) 111402

    2022

  30. [37]

    W. Wu, M. Daneker, M. A. Jolley, K. T. Turner, L. Lu, Effective data sampling strategies and boundary condition constraints of physics-informed neural networks for identifying material properties in solid mechanics, Applied mathematics and mechanics 44 (7) (2023) 1039–1068

    2023

  31. [38]

    H. Hu, L. Qi, X. Chao, Physics-informed neural networks (pinn) for computational solid mechanics: Numerical frameworks and applications, Thin-Walled Structures (2024) 112495

    2024

  32. [39]

    R. d. O. Teloli, R. Tittarelli, M. Bigot, L. Coelho, E. Ramasso, P. Le Moal, M. Ouisse, A physics-informed neural networks framework for model parameter identification of beam-like structures, Mechanical Sys- tems and Signal Processing 224 (2025) 112189

    2025

  33. [40]

    Mishra, R

    S. Mishra, R. Molinaro, Estimates on the generalization error of physics-informed neural networks for approximating pdes, IMA Journal of Numerical Analysis 43 (1) (2023) 1–43

    2023

  34. [41]

    Krishnapriyan, A

    A. Krishnapriyan, A. Gholami, S. Zhe, R. Kirby, M. W. Mahoney, Characterizing possible failure modes inphysics-informedneuralnetworks, Advancesinneuralinformationprocessingsystems34(2021)26548– 26560

    2021

  35. [42]

    S. Wang, H. Wang, P. Perdikaris, On the eigenvector bias of fourier feature networks: From regression to solvingmulti-scalepdeswithphysics-informedneuralnetworks, ComputerMethodsinAppliedMechanics and Engineering 384 (2021) 113938

    2021

  36. [43]

    J. Yu, L. Lu, X. Meng, G. E. Karniadakis, Gradient-enhanced physics-informed neural networks for forward and inverse pde problems, Computer Methods in Applied Mechanics and Engineering 393 (2022) 114823

    2022

  37. [44]

    X. Li, Y. Liu, Z. Liu, Physics-informed neural network based on a new adaptive gradient descent algo- rithm for solving partial differential equations of flow problems, Physics of Fluids 35 (6) (2023)

    2023

  38. [45]

    Basir, I

    S. Basir, I. Senocak, Critical investigation of failure modes in physics-informed neural networks, in: AiAA SCITECH 2022 Forum, 2022, p. 2353

    2022

  39. [46]

    Y. Shi, M. Beer, Physics-informed neural network classification framework for reliability analysis, Expert Systems with Applications 258 (2024) 125207

    2024

  40. [47]

    A. N. Kolmogorov, On the representations of continuous functions of many variables by superposition of continuous functions of one variable and addition, in: Dokl. Akad. Nauk USSR, Vol. 114, 1957, pp. 953–956

    1957

  41. [48]

    Arnold, On the representation of functions of several variables by superpositions of functions of fewer variables, Mat

    V. Arnold, On the representation of functions of several variables by superpositions of functions of fewer variables, Mat. Prosvesh. 3 (1958) 41–61

    1958

  42. [49]

    Z. Liu, Y. Wang, S. Vaidya, F. Ruehle, J. Halverson, M. Soljačić, T. Y. Hou, M. Tegmark, Kan: Kolmogorov-arnold networks, arXiv preprint arXiv:2404.19756 (2024). 26

    2024 arXiv

  43. [50]

    A. N. Kolmogorov, On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition, Doklady Akademii Nauk SSSR 114 (1957) 953–956

    1957

  44. [51]

    V. I. Arnold, On functions of three variables, Doklady Akademii Nauk SSSR 114 (1957) 679–681

    1957

  45. [52]

    Shukla, J

    K. Shukla, J. D. Toscano, Z. Wang, Z. Zou, G. E. Karniadakis, A comprehensive and fair comparison be- tween mlp and kan representations for differential equations and operator networks, Computer Methods in Applied Mechanics and Engineering 431 (2024) 117290

    2024

  46. [53]

    Mostajeran, S

    F. Mostajeran, S. A. Faroughi, Epi-ckans: Elasto-plasticity informed kolmogorov-arnold networks using chebyshev polynomials, arXiv preprint arXiv:2410.10897 (2024)

    2024 arXiv

  47. [54]

    Mostajeran, S

    F. Mostajeran, S. A. Faroughi, Scaled-cpikans: Spatial variable and residual scaling in chebyshev-based physics-informed kolmogorov-arnold networks, Journal of Computational Physics 537 (2025) 114116

    2025

  48. [55]

    S. SS, K. AR, A. KP, et al., Chebyshev polynomial-based kolmogorov-arnold networks: An efficient architecture for nonlinear function approximation, arXiv preprint arXiv:2405.07200 (2024)

    2024 arXiv

  49. [56]

    C. Guo, L. Sun, S. Li, Z. Yuan, C. Wang, Physics-informed kolmogorov-arnold network with chebyshev polynomials for fluid mechanics, arXiv preprint arXiv:2411.04516 (2024)

    2024 arXiv

  50. [57]

    Jacot, F

    A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks, Advances in neural information processing systems 31 (2018)

    2018

  51. [58]

    M.Seleznova, G.Kutyniok, Analyzingfiniteneuralnetworks: Canwetrustneuraltangentkerneltheory?, in: Mathematical and Scientific Machine Learning, PMLR, 2022, pp. 868–895

    2022

  52. [59]

    Seleznova, G

    M. Seleznova, G. Kutyniok, Neural tangent kernel beyond the infinite-width limit: Effects of depth and initialization, in: International Conference on Machine Learning, PMLR, 2022, pp. 19522–19560

    2022

  53. [61]

    M. H. Saadat, B. Gjorgiev, L. Das, G. Sansavini, Neural tangent kernel analysis of pinn for advection- diffusion equation, arXiv preprint arXiv:2211.11716 (2022)

    2022 arXiv

  54. [62]

    Arnold, On the representation of continuous functions of three variables by superpositions of contin- uous functions of two variables, Math

    V. Arnold, On the representation of continuous functions of three variables by superpositions of contin- uous functions of two variables, Math. Sb.(NS) 48 (90) (1959) 3–74

    1959

  55. [63]

    V. I. Arnold, On functions of three variables, Collected Works: Representations of Functions, Celestial Mechanics and KAM Theory, 1957–1965 (2009) 5–8

    2009

  56. [64]

    Lorentz, Approximation of functions.-holt, rinehart and wilson, Inc., New York (1966)

    G. Lorentz, Approximation of functions.-holt, rinehart and wilson, Inc., New York (1966)

    1966

  57. [65]

    G. G. Lorentz, M. von Golitschek, Y. Makovoz, Constructive approximation: advanced problems, Vol. 304, Citeseer, 1996

    1996

  58. [66]

    D. A. Sprecher, On the structure of continuous functions of several variables, Transactions of the Amer- ican Mathematical Society 115 (1965) 340–355

    1965

  59. [67]

    D. A. Sprecher, An improvement in the superposition theorem of kolmogorov, Journal of Mathematical Analysis and Applications 38 (1) (1972) 208–213

    1972

  60. [68]

    B. L. Fridman, An improvement in the smoothness of the functions in an kolmogorov’s theorem on superpositions, in: Doklady Akademii Nauk, Vol. 177, Russian Academy of Sciences, 1967, pp. 1019– 1022

    1967

  61. [69]

    Hecht-Nielsen, Kolmogorov’s mapping neural network existence theorem, in: Proceedings of the international conference on Neural Networks, Vol

    R. Hecht-Nielsen, Kolmogorov’s mapping neural network existence theorem, in: Proceedings of the international conference on Neural Networks, Vol. 3, IEEE press New York, NY, USA, 1987, pp. 11–14

    1987

  62. [70]

    Kůrková, Kolmogorov’s theorem is relevant, Neural computation 3 (4) (1991) 617–622

    V. Kůrková, Kolmogorov’s theorem is relevant, Neural computation 3 (4) (1991) 617–622

    1991

  63. [71]

    Kůrková, Kolmogorov’s theorem and multilayer neural networks, Neural networks 5 (3) (1992) 501– 506

    V. Kůrková, Kolmogorov’s theorem and multilayer neural networks, Neural networks 5 (3) (1992) 501– 506

    1992

  64. [72]

    D. A. Sprecher, A numerical implementation of kolmogorov’s superpositions, Neural networks 9 (5) (1996) 765–772

    1996

  65. [73]

    M. Köppen, On the training of a kolmogorov network, in: Artificial Neural Networks—ICANN 2002: International Conference Madrid, Spain, August 28–30, 2002 Proceedings 12, Springer, 2002, pp. 474– 479. 27

    2002

  66. [74]

    Braun, M

    J. Braun, M. Griebel, On a constructive proof of kolmogorov’s superposition theorem, Constructive approximation 30 (2009) 653–675

    2009

  67. [75]

    Cybenko, Approximation by superpositions of a sigmoidal function, Mathematics of control, signals and systems 2 (4) (1989) 303–314

    G. Cybenko, Approximation by superpositions of a sigmoidal function, Mathematics of control, signals and systems 2 (4) (1989) 303–314

    1989

  68. [76]

    Hornik, Approximation capabilities of multilayer feedforward networks, Neural networks 4 (2) (1991) 251–257

    K. Hornik, Approximation capabilities of multilayer feedforward networks, Neural networks 4 (2) (1991) 251–257

    1991

  69. [77]

    A. R. Barron, Universal approximation bounds for superpositions of a sigmoidal function, IEEE Trans- actions on Information theory 39 (3) (1993) 930–945

    1993

  70. [78]

    Telgarsky, Benefits of depth in neural networks, in: Conference on learning theory, PMLR, 2016, pp

    M. Telgarsky, Benefits of depth in neural networks, in: Conference on learning theory, PMLR, 2016, pp. 1517–1539

    2016

  71. [79]

    Z. Lu, H. Pu, F. Wang, Z. Hu, L. Wang, The expressive power of neural networks: A view from the width, Advances in neural information processing systems 30 (2017)

    2017

  72. [80]

    Schmidt-Hieber, Nonparametric regression using deep neural networks with relu activation function (2020)

    J. Schmidt-Hieber, Nonparametric regression using deep neural networks with relu activation function (2020)

    2020

  73. [81]

    Poggio, H

    T. Poggio, H. Mhaskar, L. Rosasco, B. Miranda, Q. Liao, Why and when can deep-but not shallow- networks avoid the curse of dimensionality: a review, International Journal of Automation and Comput- ing 14 (5) (2017) 503–519

    2017

  74. [82]

    D. A. Sprecher, S. Draghici, Space-filling curves and kolmogorov superposition-based neural networks, Neural Networks 15 (1) (2002) 57–67

    2002

  75. [83]

    Fakhoury, E

    D. Fakhoury, E. Fakhoury, H. Speleers, Exsplinet: An interpretable and expressive spline-based neural network, Neural Networks 152 (2022) 332–346

    2022

  76. [84]

    J.He, Onthe optimalexpressive powerof reludnns anditsapplicationinapproximation withkolmogorov superposition theorem, arXiv preprint arXiv:2308.05509 (2023)

    2023 arXiv

  77. [85]

    Y. Sun, L. Xiao, H. Zhao, On the performance and generalization of kolmogorov-arnold networks for high-dimensional function approximation, Neural Networks 138 (2021) 23–38

    2021

  78. [86]

    Chang, G

    B. Chang, G. Li, X. Bai, Bridging Kolmogorov-Arnold representation and deep architectures: A survey and analysis, IEEE Transactions on Neural Networks and Learning Systems 33 (10) (2022) 5170–5183

    2022

  79. [87]

    Bozorgasl, H

    Z. Bozorgasl, H. Chen, Wav-kan: Wavelet kolmogorov-arnold networks, 2024, arXiv preprint arXiv:2405.12832

    2024 arXiv

  80. [88]

    Li, Kolmogorov-arnold networks are radial basis function networks, arXiv preprint arXiv:2405.06721 (2024)

    Z. Li, Kolmogorov-arnold networks are radial basis function networks, arXiv preprint arXiv:2405.06721 (2024)

    2024 arXiv

  81. [89]

    T. J. Rivlin, Chebyshev polynomials, Courier Dover Publications, 2020

    2020

  82. [90]

    Schmidt-Hieber, The kolmogorov–arnold representation theorem revisited, Neural networks 137 (2021) 119–126

    J. Schmidt-Hieber, The kolmogorov–arnold representation theorem revisited, Neural networks 137 (2021) 119–126

    2021

  83. [91]

    Z. Hu, K. Shukla, G. E. Karniadakis, K. Kawaguchi, Tackling the curse of dimensionality with physics- informed neural networks, Neural Networks 176 (2024) 106369

    2024

  84. [92]

    Arora, S

    S. Arora, S. S. Du, W. Hu, Z. Li, R. R. Salakhutdinov, R. Wang, On exact computation with an infinitely wide neural net, Advances in neural information processing systems 32 (2019)

    2019

  85. [93]

    J. Lee, L. Xiao, S. Schoenholz, Y. Bahri, R. Novak, J. Sohl-Dickstein, J. Pennington, Wide neural networks of any depth evolve as linear models under gradient descent, Advances in neural information processing systems 32 (2019)

    2019

  86. [94]

    A. M. Mood, Introduction to the theory of statistics. (1950)

    1950

  87. [95]

    Ross, Probability and statistics for engineers and scientists, Elsevier, New Delhi 16 (2009) 32–33

    S. Ross, Probability and statistics for engineers and scientists, Elsevier, New Delhi 16 (2009) 32–33

    2009

  88. [96]

    C. M. Bishop, N. M. Nasrabadi, Pattern recognition and machine learning, Vol. 4, Springer, 2006

    2006

  89. [97]

    Bottou, F

    L. Bottou, F. E. Curtis, J. Nocedal, Optimization methods for large-scale machine learning, SIAM review 60 (2) (2018) 223–311

    2018

  90. [98]

    C. E. Shannon, A mathematical theory of communication, The Bell system technical journal 27 (3) (1948) 379–423. 28

    1948

  91. [99]

    T. M. Cover, Elements of information theory, John Wiley & Sons, 1999

    1999

  92. [100]

    D. S. Ornstein, B. Weiss, Entropy and data compression schemes, IEEE Transactions on information theory 39 (1) (1993) 78–83

    1993

  93. [101]

    K. J. Balakrishnan, N. A. Touba, Relationship between entropy and test data compression, IEEE Trans- actions on computer-aided design of integrated circuits and systems 26 (2) (2007) 386–395

    2007

  94. [102]

    K. P. Murphy, Machine learning: a probabilistic perspective, MIT press, 2012

    2012

  95. [103]

    S. A. Sepúlveda-Fontaine, J. M. Amigó, Applications of entropy in data analysis and machine learning: A review, Entropy 26 (12) (2024) 1126

    2024

  96. [104]

    ψ(x)·ψ(x ′)· Pk m=0 T ′ m(tanh(hj(x)))·T ′ m(tanh(hj(x′))) # . (A.12) We denote the inner expectation asD(x,x′), defined by, D(x,x ′) :=E h,h′

    I. Goodfellow, Deep learning (2016). 29 Appendix A. Proof of Theorem 1 The Neural Tangent Kernel between two inputsxandx′ is defined as the inner product of the gradients of the network output with respect to the parameters, Kntk(x,x ′) = ∂f(x;θ(0)) ∂θ , ∂f(x ′;θ(0)) ∂θ = X i,...

    2016

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.