Pith. sign in

REVIEW 3 major objections 5 minor 32 references

Physics-Informed Neural Networks for the Korteweg-de Vries Equation for Internal Solitary Wave Problem: Forward Simulation and Inverse Parameter Estimation

T0 review · 3 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 The paper claims that physics-informed neural networks can learn the parameter-to-solution operator for KdV internal solitary waves and invert layer height and density from sparse noisy data, reporting forward errors as low as $10^{-4}$…

desk verdict Straightforward PINN application with a load-bearing inverse-benchmark error: the reported ground-truth parameters make the KdV coefficients imaginary, so the central inverse claims can't be checked as written. read the letter →

arxiv 2506.14236 v1 pith:RVGEIC2X submitted 2025-06-17 physics.flu-dyn

Ming Kang , Hang Li , Qiwen Tan , Zhan Wang , Ruipeng Li , Junfang Zhao , Hui Xiang , Dixia Fan This is my paper · ORCID
classification physics.flu-dyn
keywords physics-informedneuralnetworksKorteweg-deVriesequationinternalsolitarywavesoperatorlearninginverseparameterestimationtwo-layerstratifiedflowsensorplacementnoiserobustness
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

This paper tries to establish that a single physics-informed neural network can serve both as a forward operator for internal solitary waves in a two-layer fluid and as an inverse solver for the physical parameters that are hard to measure in the field. For the forward problem, the network learns a mapping from the density ratio, layer depth, and amplitude to the wave profile, with mean absolute errors decreasing from about $10^{-2}$ to about $10^{-4}$ as the number of training parameter sets grows from 10 to 1000. For the inverse problem, the network recovers the layer height ratio $h_1/h_2$ and the lower-layer density $\rho_1$ with relative errors below 1% when the upper-layer density is fixed, and it remains usable with as few as one or five sensors and with moderate Gaussian noise. A sympathetic reader would care because direct density and depth measurements in stratified oceans are scarce and expensive, so a method that can infer them from sparse wave observations would change how internal-wave field data are collected and interpreted.

What carries the argument

The load-bearing mechanism is the weighted PINN loss that combines a KdV residual loss, a boundary-condition loss, and a data-fidelity loss, evaluated through automatic differentiation. For operator learning, the input parameters are expanded into a pseudo-sequential sequence with spatial coordinates so that the network learns a spatially structured map $G:(\rho, h_1, a)\to\eta(X)$; for inverse estimation, the unknown physical parameters are added to the trainable variables and optimized together with the network weights against the same composite loss. The exact sech$^2$ solitary-wave solution of the KdV equation supplies both the training data for the operator and the synthetic observations for the inverse experiments, so the physics constraint and the data are generated by the same model.

What would settle it

Generate test data from a different internal-wave model, such as a higher-order strongly nonlinear theory or a variable-coefficient KdV equation, or use field measurements with the same nominal parameters, and run the paper's inversion procedure; if the recovered $h_1/h_2$ and $\rho_1$ errors remain below 1%, the physics-regularized inversion is robust to model mismatch, whereas larger errors would show the reported accuracy depends on the exact-form synthetic data.

Watch

Extended reading notes

Core claim

The central claim is that a PINN architecture with 4 hidden layers of 20 tanh neurons, trained by Adam with loss weights $w_{\text{pde}}=0.1$, $w_{\text{bc}}=1$, and $w_{\text{data}}=10$, can solve both the operator-learning and inverse problems for the KdV equation describing internal solitary waves in a two-layer fluid. For operator learning, the network takes the parameter vector $s=(\rho_2/\rho_1, h_1, a)$ together with a pseudo-sequential spatial input $X$ and outputs the solitary-wave profile $\eta(X)$, trained on exact sech$^2$ solutions; experiments show mean absolute errors near $10^{-4}$ for 1000 training parameter sets and demonstrate that the boundary-condition loss is necessary for correct far-field decay. For the inverse problem, the network treats $h_1/h_2$ and $\rho_1$ as trainable variables and recovers them from synthetic data sampled from the exact solution, with sub-1% relative error when $\rho_2$ is fixed, errors below 3% with five sensors, errors below 8% for $\rho_1$ with a single sensor at the wave core, and noise robustness up to about 20% noise for $h_1/h_2$ and 40% for $\rho_1$.

Load-bearing premise

The inverse results assume the observed data are exactly generated by the same sech$^2$ solution of the KdV equation used in the physics loss, with no model error, and that Adam converges to a good minimum of the non-convex loss landscape.

Editorial extensions

If this is right

  • A single trained network can predict the internal solitary wave profile for a new parameter set without solving the KdV equation again, reducing the cost of parametric studies.
  • Inverting $h_1/h_2$ and $\rho_1$ from wave observations becomes feasible with very few sensors, provided one layer density or another parameter is known from prior measurement.
  • The boundary-condition loss is essential for extrapolating predictions to far-field regions, otherwise the network produces non-decaying unphysical solutions.
  • Sensor placement matters strongly: measurements in the central wave-core region carry the most information for parameter recovery, guiding where field instruments should be deployed.
  • The method tolerates moderate Gaussian noise better for the density $\rho_1$ than for the height ratio $h_1/h_2$, suggesting parameter-specific uncertainty in field applications.

Reading between the lines

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

  • Beyond the paper's synthetic benchmark, the same operator-learning strategy could be tested on data from a more complete internal-wave model, such as a variable-coefficient KdV equation or a strongly nonlinear theory, to see whether the physics prior remains sufficient when the exact sech$^2$ form is only an approximation.
  • The paper's parameter-coupling result implies a practical field protocol: measure at least one layer parameter (for example, the upper-layer density) with an in-situ instrument, and the PINN can then recover the remaining parameters from sparse wave-height time series.
  • The learned operator may serve as a fast surrogate for uncertainty quantification or data assimilation in oceanographic models, but this would require the network to be retrained or calibrated on real data rather than the exact analytical solution.
  • Because the inverse experiments use a single optimization run, a natural extension would be to repeat training with multiple random seeds and report the spread of recovered parameters, which would indicate whether the sub-1% errors are stable or depend on the initial guess.
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

3 major / 5 minor

Summary. The paper applies Physics-Informed Neural Networks (PINNs) to the Korteweg-de Vries (KdV) equation for internal solitary waves in a two-layer fluid. It addresses two problems: (1) operator learning, where a network maps physical parameters (density ratio, lower-layer depth, amplitude) plus the spatial coordinate to the sech^2 solitary-wave profile, reporting MAE values down to roughly 4e-4 in Table I; and (2) inverse parameter estimation, where the network recovers layer height ratio h1/h2 and lower-layer density rho1 from synthetic observations generated by the exact analytical solution, with additional studies of data sparsity, sensor placement, and Gaussian noise. The abstract claims prediction errors as low as 10^{-4} and successful inversion of nonlinear coefficients.

Significance. If the forward operator-learning results hold, the paper makes a modest but useful contribution: it demonstrates that a PINN with an explicit boundary-condition loss can learn a parametric map from KdV parameters to solitary-wave profiles, and the boundary-loss ablation in Fig. 2 is a clear practical finding. The inverse-problem half of the paper, however, is currently not credible as written because the stated ground-truth parameters make the linear wave speed c0 imaginary, so the synthetic data used to benchmark all inverse experiments do not exist as described. The paper provides no code, no repeated runs or error bars, and no baseline comparisons, so the claimed sub-1% and sparse-sensor accuracies cannot be independently verified.

major comments (3)
  1. [III.B.1] The ground-truth parameters used for the inverse benchmark are (h1/h2=4.13, rho1=0.977, rho2=1.0, h2=1, a=-0.77). Substituting these into Eq. (2) gives c0 = sqrt(9.81*4.13*(0.977-1.0)/(0.977*1+1.0*4.13)) = i*0.427, so c0 is imaginary. Since c1 and c2 are both proportional to c0, all coefficients in Eq. (1) are complex, and Eq. (3) with X=x-ct is not a real evolution equation on the stated domain. Consequently the exact sech^2 solution in Eqs. (4)-(5) used to generate the synthetic observations in Secs. III.B.1-III.B.4 is not a real-valued function of (x,t), and the reported sub-1% inversion for Case (b) and all downstream sparsity, sensor-placement, and noise experiments are not reproducible as described. This parameter set also violates the declared ranges in Eq. (6): rho2/rho1=1.023 is outside [0.9,1], a/h2=-0.77 is outside [-0.4,-0.05], and rho2>rho1 represents unstable stratification. The entire inverse section must be rerun with physically valid parameters satisfying Eq. (6) and rho1>rho2, and all conclusions derived from the current experiments must be updated.
  2. [III.B.2] The text claims that the PINN approach succeeds in sparse-data scenarios by 'outperforming conventional inversion methods under comparable conditions', but no baseline method is implemented or cited quantitatively anywhere in the paper. Because the synthetic data are generated from the closed-form solution (4)-(5), a direct nonlinear least-squares fit of h1/h2 and rho1 to the same observations would be a natural and much cheaper null model; without such a comparison, the claimed advantage over conventional methods is unsupported. In addition, all quantitative results in Table I and Figs. 4-9 are single-run values with no random seeds, restarts, or error bars; given the non-convex PINN loss landscape, repeated runs (mean +/- std over several seeds) are necessary to support the precision claims.
  3. [III.A] The operator-learning architecture relies on a 'pseudo-sequential sequence expansion between the input layer and the network', but the paper never defines this component mathematically. There is no equation for how the parameter vector s=(rho,h1,a) and the spatial coordinate X are expanded into the pseudo-sequence, no statement of the sequence ordering or length, and no description of how the output layer produces eta(X) from the sequence. Because the forward results in Table I and Fig. 3 depend on this architectural modification, the method cannot be reimplemented from the paper as written. Please provide a precise specification, including tensor shapes and the explicit transformation from the input to the hidden sequence.
minor comments (5)
  1. [Abstract] The abstract states 'prediction errors as low as 10^{-4} when using 1000 training points', but Table I reports MAE values around 4e-4 to 8e-4 for 1000 training parameter sets, not 1000 individual points; please correct the wording to 'training parameter sets'.
  2. [Eq. (7)] The boundary loss is written for |x_bc| -> infinity, while the experiments sample boundary points at x = +/-40. Please specify the truncated numerical domain and clarify whether eta is enforced to vanish exactly at the numerical boundaries or only weakly through the loss term.
  3. [III.B.4] The symbol eta is used both for the wave elevation in Eq. (1) and for the noise-level percentage eta% in Fig. 9. Rename the noise parameter (e.g., nu%) to avoid notational confusion.
  4. [Fig. 3 caption] The caption of Fig. 3 appears garbled ('Number of training points for ah ,, 1rho') and does not clearly identify which parameter set corresponds to each row; also, the Table I caption should define NX explicitly and state that the test parameters are held out from training.
  5. [III.A] The conclusion that there is 'a clear logarithmic relationship between data quantity and solution accuracy' is not supported by any fitted curve or statistical analysis; only three training-set sizes (10, 100, 1000) are reported, so this statement should be softened or substantiated.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the inverse benchmark uses external exact-solution ground truth, and the operator-learning task is supervised regression on a known closed-form solution family.

full rationale

The paper's derivation chain is self-contained against external benchmarks. The forward operator network is trained on parameter sets and exact KdV sech^2 solutions generated by Eqs. (4)-(5), and test errors are evaluated on held-out parameter sets from the same family; this is standard supervised surrogate modeling, not a prediction equivalent to its inputs by construction, since the network never sees the test parameter sets during training. The inverse section generates synthetic observations by sampling the exact analytical solution for known ground-truth parameters and then recovers those parameters from sparse data using the KdV equation as a physics regularizer; the output parameters are not defined in terms of the data, and the benchmark is an independent inversion test. No load-bearing self-citation or imported uniqueness theorem appears—references 26 and 27 are external operator-learning frameworks cited for context, not to justify the present claims. Separately, and outside the circularity question, the stated ground-truth parameters in Sec. III.B.1 (h1/h2=4.13, rho1=0.977, rho2=1.0, a=-0.77) appear inconsistent with Eq. (2), which yields a negative radicand for c0, and with the declared ranges in Eq. (6); this is a reproducibility/correctness concern, but it does not make the derivation circular.

Assumptions & free parameters 3 free parameters · 4 assumptions · 1 invented entities

The central claims rest on the KdV model, the exact sech^2 solution family, and the assumption that Adam finds a good minimum of a non-convex loss. The paper adds hand-chosen loss weights, architecture, and iteration counts, plus one vaguely described architectural component. No code or data are released, so the numerical results cannot be independently reproduced.

free parameters (3)
  • loss weights w_pde, w_bc, w_data = 0.1, 1, 10
    Chosen by hand without ablation; materially affect the balance between physics, boundary, and data terms in the total loss.
  • network architecture and training schedule = 4 hidden layers, 20 neurons, tanh; 20,000 or 25,000 Adam steps
    Hand chosen; no sensitivity analysis is provided, and convergence depends on these choices.
  • collocation point counts and learning rate
    Not reported in the text; these are required to reproduce the numerical results and to assess the stability of the reported errors.
assumptions (4)
  • domain assumption The KdV equation (1) with coefficients (2) accurately models small-amplitude internal solitary waves in a two-layer fluid with a rigid lid, for the parameter ranges in Eq. (6).
    The paper states this in Section II.A and cites Kodaira et al., but does not quantify the model error against higher-order theory or field data.
  • standard math The exact solution eta(X) = a sech^2(X/lambda) with lambda and speed c from Eq. (5) is the correct solution of Eq. (3) and serves as ground truth.
    This is a known closed-form solution of the KdV equation; the paper uses it to generate training and test data.
  • domain assumption The Adam optimizer converges to a global minimum of the non-convex PINN loss landscape.
    No convergence guarantee is provided, and the paper does not test multiple random seeds or restarts.
  • domain assumption Gaussian noise with standard deviation sigma = sigma_data * eta% represents realistic sensor noise.
    A standard assumption, but no comparison with other noise distributions or real sensor data is made.
invented entities (1)
  • Pseudo-sequential sequence expansion between the input layer and the network
    purpose: Reinforce the input states with spatial dependencies at X_i for the operator learning problem.
    No formal definition, no ablation study, and no comparison against a plain fully connected network are provided. The component is described only in one paragraph in Section III.A.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Physics-Informed Neural Networks for the Korteweg-de Vries Equation for Internal Solitary Wave Problem: Forward Simulation and Inverse Parameter Estimation." pith.science (2026). https://pith.science/paper/RVGEIC2X

@misc{pith2026250614236,
  author       = {Pith},
  title        = {Pith review of: Physics-Informed Neural Networks for the Korteweg-de Vries Equation for Internal Solitary Wave Problem: Forward Simulation and Inverse Parameter Estimation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RVGEIC2X}},
  note         = {Machine review of arXiv:2506.14236}
}
abstract

Physics-informed neural networks (PINNs) have emerged as a transformative framework for addressing operator learning and inverse problems involving the Korteweg-de Vries (KdV) equation for internal solitary waves. By integrating physical constraints with data-driven optimization, PINNs overcome the critical challenges of parameter unmeasurability in the KdV equation for internal solitary waves in two-layer fluid systems. This work addresses two problems: (1) Operator learning constructs a mapping from parameters to solutions, enabling wave evolution predictions from unknown parameters. Comparative studies demonstrate prediction errors as low as $10^{-4}$ when using 1000 training points. (2) Inverse problem solving leverages sparse and potentially noisy observational data with physics-regularized constraints to invert nonlinear coefficients successfully. Compared to conventional approaches, this end-to-end differentiable paradigm unifies operator learning and inverse problem-solving while overcoming mesh discretization errors and high-dimensional parameter space iteration costs. The method shows effectiveness for internal wave problems in stratified fluids, providing both accurate forward modeling and robust parameter inversion capabilities, even under noise.

Figures

Figures reproduced from arXiv: 2506.14236 by the authors.

Figure 1
Figure 1. FIG. 1: Schematic diagram of solving an internal solitary wave in a two-layer fluid system by PINNs. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Impact of the boundary condition loss on the PINN-predicted solution. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Learning the solution operator: A comparison between the exact solution and the predicted solution [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Convergence of parameter estimation over 25,000 training iterations for different scenarios using dense data [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Comparison of final field results for the parameter estimation of [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Effect of the number of spatial measurement points ( [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Comparisons of predicted solutions (lines) versus exact solution (dots) along slices at (a) [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Impact of single sensor location ( [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Impact of Gaussian noise level ( [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

32 extracted references · 16 canonical work pages

  1. [1]

    merlin.mbs aapmrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

    FUNCTION id.bst "merlin.mbs aapmrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translat...

    2010

  2. [2]

    merlin.mbs aipauth4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

    FUNCTION id.bst "merlin.mbs aipauth4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translat...

    2010

  3. [3]

    merlin.mbs aipnum4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

    FUNCTION id.bst "merlin.mbs aipnum4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translati...

    2010

  4. [4]

    merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

    FUNCTION id.bst "merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translati...

    2010

  5. [5]

    merlin.mbs apsrmp4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

    FUNCTION id.bst "merlin.mbs apsrmp4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translati...

    2010

  6. [6]

    Zhang , author H

    author author M. Zhang , author H. B. \ Hu , author B. B. \ Guo , author Q. Y. \ Liang , author F. Zhang , author X. P. \ Chen , author Z. L. \ Xie , \ and\ author P. Du ,\ title title Predicting shear stress distribution on structural surfaces under internal solitary wave loading: A deep learning perspective , \ @noop journal journal Physics of Fluids \ ...

    work page 2023

  7. [7]

    Cheng , author P

    author author L. Cheng , author P. Du , author C. Wang , author Z. L. \ Xie , author H. B. \ Hu , author X. P. \ Chen , author Z. Y. \ Li , \ and\ author Z. M. \ Yuan ,\ title title Tuning control parameters of underwater vehicle to minimize the influence of internal solitary waves , \ @noop journal journal Ocean Engineering \ volume 310 ,\ pages 118681 (...

    work page 2024

  8. [8]

    author author R. Z. \ Wang , author C. H. \ Zhi , \ and\ author Y. X. \ You ,\ title title Numerical study of internal solitary wave loads on a submerged slender body with multi-parameter coupling , \ @noop journal journal Physics of Fluids \ volume 36(11) ,\ pages 113361 ( year 2024 ) NoStop

    work page 2024

Show all 32 references
  1. [9]

    author author J. M. H. \ Li , author X. D. \ Huang , author Y. Shi , author Y. X. \ Yang , \ and\ author W. Zhao ,\ title title Effects of internal solitary waves on three-dimensional sound propagation and DOA estimation in the South China Sea , \ @noop journal journal Applied...

    2023

  2. [10]

    author author Y. D. \ Zheng , author J. Lin , \ and\ author X. Chen ,\ title title Application of fluctuations in the sound field in inversion of internal solitary wave phase speed , \ @noop journal journal Ocean Engineering \ volume 305 ,\ pages 117867 ( year 2024 ) NoStop

    2024

  3. [11]

    Brandt \ and\ author K

    author author A. Brandt \ and\ author K. R. \ Shipley ,\ title title Laboratory experiments on mass transport by large amplitude mode-2 internal solitary waves , \ @noop journal journal Physics of Fluids \ volume 26(4) ,\ pages 046601 ( year 2014 ) NoStop

    2014

  4. [12]

    Deepwell \ and\ author M

    author author D. Deepwell \ and\ author M. Stastna ,\ title title Mass transport by mode-2 internal solitary-like waves , \ @noop journal journal Physics of Fluids \ volume 28(5) ,\ pages 056606 ( year 2016 ) NoStop

    2016

  5. [13]

    author author K. R. \ Helfrich \ and\ author W. K. \ Melville ,\ title title Long nonlinear internal waves , \ @noop journal journal Annual Review of Fluid Mechanics \ volume 38 ,\ pages 395--425 ( year 2006 ) NoStop

    2006

  6. [14]

    Choi ,\ title title High-order strongly nonlinear long wave approximation and solitary wave solution

    author author W. Choi ,\ title title High-order strongly nonlinear long wave approximation and solitary wave solution. P art 2. internal waves , \ @noop journal journal Journal of Fluid Mechanics \ volume 952 ,\ pages A41 ( year 2022 ) NoStop

    2022

  7. [15]

    author author T. B. \ Benjamin ,\ title title Internal waves of finite amplitude and permanent form , \ @noop journal journal Journal of Fluid Mechanics \ volume 25 ,\ pages 97--116 ( year 1966 ) NoStop

    1966

  8. [16]

    Grue , author A

    author author J. Grue , author A. Jensen , author P. Rusas , \ and\ author J. K. \ Sveen ,\ title title Properties of large-amplitude internal waves , \ @noop journal journal Journal of Fluid Mechanics \ volume 380 ,\ pages 257--278 ( year 1999 ) NoStop

    1999

  9. [17]

    Kodaira , author T

    author author T. Kodaira , author T. Waseda , author M. Miyata , \ and\ author W. Choi ,\ title title Internal solitary waves in a two-fluid system with a free surface , \ @noop journal journal Journal of Fluid Mechanics \ volume 804 ,\ pages 201--223 ( year 2016 ) NoStop

    2016

  10. [18]

    Xuan , author H

    author author P. Xuan , author H. Du , author S. D. \ Wang , author P. Peng , \ and\ author G. Wei ,\ title title The applicability of nonlinear theories of the internal solitary wave and its loads on slender body by experimental methods , \ @noop journal journal Applied Ocean...

    2024

  11. [19]

    Cuomo , author V

    author author S. Cuomo , author V. S. \ Di Cola , author F. Giampaolo , author G. Rozza , author M. Raissi , \ and\ author F. Piccialli ,\ title title Scientific Machine Learning Through Physics -- Informed Neural Networks : Where we are and What 's Next , \ 10.1007/s10915-022...

    doi

  12. [20]

    Raissi , author P

    author author M. Raissi , author P. Perdikaris , \ and\ author G. E. \ Karniadakis ,\ 10.48550/arXiv.1711.10561 title Physics Informed Deep Learning ( Part I ): Data-driven Solutions of Nonlinear Partial Differential Equations , \ ( year 2017 a ),\ http://arxiv.org/abs/1711.10...

    doi arXiv

  13. [21]

    Raissi , author P

    author author M. Raissi , author P. Perdikaris , \ and\ author G. E. \ Karniadakis ,\ 10.48550/arXiv.1711.10566 title Physics Informed Deep Learning ( Part II ): Data-driven Discovery of Nonlinear Partial Differential Equations , \ ( year 2017 b ),\ http://arxiv.org/abs/1711.1...

    doi arXiv

  14. [22]

    Raissi , author P

    author author M. Raissi , author P. Perdikaris , \ and\ author G. Karniadakis ,\ title title Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations , \ https://doi.org/10.1016/j.jc...

    doi

  15. [23]

    Moseley , author A

    author author B. Moseley , author A. Markham , \ and\ author T. Nissen-Meyer ,\ 10.48550/arXiv.2006.11894 title Solving the wave equation with physics-informed deep learning , \ ( year 2020 ),\ note comment: 13 pages, 9 figures ,\ http://arxiv.org/abs/2006.11894 arXiv:2006.118...

    2006 doi arXiv

  16. [24]

    Cai , author Z

    author author S. Cai , author Z. Wang , author S. Wang , author P. Perdikaris , \ and\ author G. E. \ Karniadakis ,\ title title Physics-informed neural networks for heat transfer problems , \ 10.1115/1.4050542 journal journal Journal of Heat Transfer \ volume 143 ,\ pages 060...

    doi

  17. [25]

    author author R. A. \ Bafghi \ and\ author M. Raissi ,\ 10.48550/arXiv.2311.03626 title PINNs-TF2 : Fast and User-Friendly Physics-Informed Neural Networks in TensorFlow V2 , \ ( year 2023 ),\ note comment: Accepted at Machine Learning and the Physical Sciences Workshop, NeurI...

    2023 doi arXiv

  18. [26]

    Kadeethum , author T

    author author T. Kadeethum , author T. M. \ Jørgensen , \ and\ author H. M. \ Nick ,\ https://arxiv.org/abs/2005.09638 title Physics-informed neural networks for solving inverse problems of nonlinear biot's equations: Batch training , \ ( year 2020 ),\ http://arxiv.org/abs/200...

    2005 arXiv

  19. [27]

    Cai , author Z

    author author S. Cai , author Z. Mao , author Z. Wang , author M. Yin , \ and\ author G. E. \ Karniadakis ,\ title title Physics-informed neural networks ( PINNs ) for fluid mechanics: A review , \ 10.1007/s10409-021-01148-1 journal journal Acta Mechanica Sinica \ volume 37 ,\...

    doi

  20. [28]

    Jin , author M

    author author H. Jin , author M. Mattheakis , \ and\ author P. Protopapas ,\ title title Physics-informed neural networks for quantum eigenvalue problems , \ in\ 10.1109/IJCNN55064.2022.9891944 booktitle 2022 International Joint Conference on Neural Networks (IJCNN) \ ( year 2...

    2022

  21. [29]

    Zhang , author M

    author author E. Zhang , author M. Dao , author G. E. \ Karniadakis , \ and\ author S. Suresh ,\ title title Analyses of internal structures and defects in materials using physics-informed neural networks , \ 10.1126/sciadv.abk0644 journal journal Science Advances \ volume 8 ,...

    doi

  22. [30]

    Escapil-Inchausp \'e \ and\ author G

    author author P. Escapil-Inchausp \'e \ and\ author G. A. \ Ruz ,\ title title H- Analysis and data-parallel physics-informed neural networks , \ 10.1038/s41598-023-44541-5 journal journal Scientific Reports \ volume 13 ,\ pages 17562 ( year 2023 ) NoStop

    doi

  23. [31]

    Lu , author P

    author author L. Lu , author P. Jin , author G. Pang , author Z. Zhang , \ and\ author G. E. \ Karniadakis ,\ title title Learning nonlinear operators via deeponet based on the universal approximation theorem of operators , \ 10.1038/s42256-021-00302-5 journal journal Nature M...

    doi

  24. [32]

    Li , author N

    author author Z. Li , author N. Kovachki , author K. Azizzadenesheli , author B. Liu , author K. Bhattacharya , author A. Stuart , \ and\ author A. Anandkumar ,\ https://arxiv.org/abs/2010.08895 title Fourier neural operator for parametric partial differential equations , \ ( ...

    2010 arXiv

Pith tools

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