Pith. sign in

REVIEW 3 major objections 5 minor 61 references

Discontinuity-aware KAN-based physics-informed neural networks

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read A physics-informed neural network built on discontinuous-activation layers and learnable localized viscosity resolves shock waves with substantially lower error and parameter counts than existing PINN variants.

desk verdict A useful engineering combination with honest ablations, but the headline accuracy sits inside a narrow shock-sensor tuning window and the Kolmogorov overclaim is unsupported. read the letter →

arxiv 2507.08338 v2 pith:CYAYAKLW submitted 2025-07-11 physics.comp-ph physics.flu-dyn

Guoqiang Lei , D. Exposito , Xuerui Mao This is my paper · ORCID
classification physics.comp-phphysics.flu-dyn MSC 68T0735L6565M7076L05 PACS 47.11.-j02.70.-c
keywords discontinuity-awarePINNKolmogorov-ArnoldnetworkartificialviscosityshockcapturingtransonicflowsupersonicRiemannproblemFourierfeatureembedding
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 DPINN, a physics-informed neural network designed for PDEs with discontinuous solutions such as shock waves. It combines four mechanisms: an adaptive Fourier-feature embedding to relieve spectral bias, a discontinuity-aware Kolmogorov-Arnold network (DKAN) whose activations mix Dynamic Tanh with B-splines to represent jumps, a mesh transformation for curved geometries, and a learnable artificial viscosity term whose action is restricted to shock regions by a sensor. In tests on the inviscid Burgers equation, two Riemann problems, and transonic and supersonic NACA0012 airfoil flows, DPINN captures sharp shock fronts with relative $L^2$ errors below about 4%, while standard MLP- and KAN-based PINNs give errors of 30–44% and viscosity-smoothed baselines give roughly 3.5–11%. A sympathetic reader would care because accurate shock resolution is a known failure mode of PINNs, and the method reaches it with roughly one-tenth the parameters of MLP-based models.

What carries the argument

The load-bearing object is the discontinuity-aware activation $\psi(x)=w_t\tanh(w_a(x-w_b))+w_s\sum_i w_{c_i}S_i(x)$, which lets a KAN layer represent jumps instead of only continuous functions, together with the learnable localized viscosity $\mu=w_\nu r(x)s(x)$ in the PDE loss. The sensor $s(x,y)=s_{\mathrm{shock}}(x,y)\,s_{\mathrm{stag}}(x,y)$ uses the dot product of normalized velocity with the pressure gradient to find shocks and a momentum-gradient term to suppress stagnation-point false positives; multiplying by the spectral radius $|u|+c$ scales dissipation to the local wave speed. Mesh transformation maps a body-fitted grid to a uniform computational domain so the network trains on coordinates where the solution varies more smoothly. These components work as a chain: the KAF layer supplies high frequencies, DKAN represents the jump, and the sensor-limited viscosity stabilizes training without smearing the final shock.

What would settle it

Run DPINN on a transonic airfoil at a condition outside the tuned set, for example $Ma=0.8$, $\alpha=2^\circ$, using the thresholds stated for the $Ma=0.7$, $\alpha=4^\circ$ case and without re-tuning the sensor parameters. If the $R_2$ error jumps to the 20–30% range or the learned viscosity spreads over smooth regions, the central claim that the method captures discontinuities generally is refuted for that setting.

Watch

Extended reading notes

Core claim

The central claim is that a PINN can resolve discontinuities accurately if the network's architecture and the loss function are both made discontinuity-aware rather than relying on smoothness assumptions. DPINN realizes this by replacing smooth activations with the DKAN activation $\psi(x)=w_t\tanh(w_a(x-w_b))+w_s\sum_i w_{c_i}S_i(x)$ (DyT plus B-spline basis), embedding inputs through a learnable hybrid-frequency KAF layer, mapping curved domains to uniform computational grids, and adding $\mu=w_\nu r(x)s(x)$ to the Euler equations, where $r$ is the spectral radius of the flux Jacobians and $s$ is a shock sensor built from pressure and momentum gradients. The paper reports that this combination resolves the moving shock in Burgers' equation with $R_2=0.88\%$ (against $44.19\%$ for MLP and $3.56\%$ for the best local-AV baseline), resolves transonic NACA0012 flow at $Ma=0.7,\alpha=4^\circ$ with $R_2=3.80\%$ (against $32.86\%$ for MLP), and captures bow and oblique shocks at $Ma=1.3$ with $R_2=4.63\%$, all with roughly an order of magnitude fewer parameters than the MLP models.

Load-bearing premise

The load-bearing premise is that the hand-tuned shock sensor reliably flags the true shock locations from the network's own predicted flow field; if the detection threshold is off, the viscosity is either wasted on smooth regions or missing at the shock, and the reported accuracy collapses (Appendix B shows a tenfold increase in $k_{\mathrm{shock},0}$ raising transonic $R_2$ from $3.80\%$ to $30.11\%$).

Editorial extensions

If this is right

  • On the 1D inviscid Burgers benchmark, DPINN reaches $R_2=0.88\%$ with 771 parameters, compared with $44.19\%$ for the MLP PINN (4835 parameters) and $3.56\%$ for MLP with local artificial viscosity.
  • For transonic NACA0012 flow at $Ma=0.7$, $\alpha=4^\circ$, DPINN yields $R_2=3.80\%$ and shock-front error $R_{\mathrm{front}}=0.39\%$, beating the local-AV MLP baseline ($10.73\%$ and $2.31\%$).
  • For supersonic flow at $Ma=1.3$, DPINN captures both the detached bow shock and trailing-edge oblique shocks with $R_2=4.63\%$, while MLP-based methods give $14.71\%$ or worse and the global-AV variant is outperformed by the plain MLP.
  • The learned viscosity coefficient ends up smaller than the fixed coefficient used in the AV baselines ($5.12\times10^{-4}$ vs $1.25\times10^{-3}$ in the transonic case), so the final solution is less modified by artificial dissipation.
  • Ablations show each component contributes: DKAN alone stabilizes Burgers but not the 2D transonic shock, and mesh transformation is indispensable for the airfoil cases.

Reading between the lines

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

  • The method's generality is only as good as the shock sensor: thresholds are hand-picked per problem, and Appendix B shows that increasing $k_{\mathrm{shock},0}$ tenfold degrades the transonic $R_2$ from $3.80\%$ to $30.11\%$, so a self-calibrating or learned sensor would be the natural next step for 3D unsteady flows.
  • The paper's parameter-efficiency claim does not mean time-to-solution efficiency: DPINN is reported 3–13 times slower in wall-clock training than the MLP baselines, so the practical win is cheaper inference and memory, not faster training.
  • The same recipe of discontinuous activations plus sensor-limited viscosity could plausibly transfer to other hyperbolic problems (e.g., multiphase or relativistic shocks), but nothing in the paper tests that transfer, so it remains an open extension.
  • A direct test of this transfer would be to run DPINN on a Sod or Lax problem with shock-sensor thresholds inherited from the Burgers case; the paper re-picks thresholds for every problem, implying this inheritance is not automatic.
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 manuscript presents DPINN, a physics-informed neural network for PDEs with discontinuous solutions. DPINN combines a Kolmogorov-Arnold Fourier embedding layer, a 'discontinuity-aware' KAN (DKAN) whose activation is a sum of a dynamic tanh and B-spline basis, a body-fitted mesh transformation, and a learnable local artificial viscosity whose action is gated by hand-set shock and stagnation sensors. The method is benchmarked on the 1D inviscid Burgers equation, the Sod and Lax Riemann problems, and transonic (Ma=0.7, alpha=0,4 degrees) and supersonic (Ma=1.3, alpha=0) flows around a NACA0012 airfoil, using L1, L2, and shock-front metrics. DPINN reports the lowest errors among the compared MLP/KAN variants with and without global/local AV, at substantially reduced parameter counts, and the paper includes ablations, wall-clock timings, and a sensitivity study of the sensor thresholds.

Significance. If the numerical results are reproducible, the paper provides a practical high-performance recipe for shock-capturing PINNs and a careful engineering study of the ingredients. The parameter counts and training/inference times are reported transparently, the ablations (Tables 1 and 2) isolate the contribution of each component, and Appendix B's sensitivity analysis is a welcome and honest assessment of the sensor's fragility. These strengths separate the paper from a mere claim of better numbers. However, the claimed theoretical basis, that DKAN extends the Kolmogorov representation theorem to discontinuous functions, is not established, and the reported accuracy gains are conditioned on sensor thresholds that are hand-tuned per problem and shown to be narrowly concentrated. The central scientific contribution is therefore the numerical recipe, not the theoretical generalization.

major comments (3)
  1. [Section 2.3 and Abstract] The claim that DKAN 'generalizes the Kolmogorov representation theorem to the discontinuous regime' is not supported by the manuscript. No theorem, proof, or precise statement of an approximation guarantee is given for the activation class in Eq. (9) (DyT plus B-splines); the citation to Ref. [49] is not connected to this specific construction. Furthermore, the paper's own ablation (Table 2, transonic alpha=4) shows DKAN without artificial viscosity yields R2=30.1%, no better than MLP, so DKAN alone is not 'discontinuity-aware' in the high-dimensional setting that motivates the paper. The authors should either supply a rigorous result or rephrase the theoretical contribution as a heuristic architecture choice.
  2. [Section 3.3, Appendix B] The headline accuracy of DPINN on the transonic alpha=4 case sits in a narrow basin with respect to the hand-set sensor threshold k_shock,0: multiplying it by 10 changes R2 from 3.80% to 30.11% and R_infinity from 0.39% to 19.40%, and multiplying by 0.1 yields R2=9.14% (Table 4). The thresholds are also re-picked for each test problem (Table 3: k_shock,0=0.015, 0.5, 1, 0.5, 0.5 for Burgers, transonic, supersonic, Sod, and Lax) with no stated selection rule, cross-validation, or adaptation. The conclusion acknowledges that 'the reliance on shock sensors constrains generalizability,' but the abstract and introduction state the method is superior without this caveat. Please provide a principled threshold-selection or adaptation mechanism, or explicitly characterize the reported results as conditioned on per-problem manual tuning and quantify the performance over a range of thresholds.
  3. [Section 3.1, Table 1] The role of the learnable local viscosity is inconsistent across benchmarks. On the Burgers equation, DKAN+Local AV is worse than DKAN alone (R2=1.31% vs 0.94%), and the authors attribute this to the viscosity not decaying to a negligible magnitude. In the 2D airfoil cases, however, local AV is essential (Table 2: DKAN alone gives R2=30.1% on transonic alpha=4). The manuscript should reconcile these findings, for example by reporting the final learned viscosity coefficient w_nu in each case and explaining why the learned viscosity decays in some settings but not others.
minor comments (5)
  1. [Table 4 and Eq. (17)] The metric R_infinity in Table 4 is not defined in Eq. (17); please define it (e.g., relative L_infinity norm) or remove it.
  2. [Sections 3.1 and 3.3] The 'local AV' baselines are not specified. Please state whether they use the same sensor function as DPINN (e.g., the sensor from Ref. [19]) and how the viscosity is localized, so that the comparison is apples-to-apples.
  3. [Eq. (9)] The summation index in the spline term 'w_s sum_i^{k+g} w_ci S_i(x)' is ambiguous; specify the range of i and the meaning of k+g (e.g., number of basis functions).
  4. [Eq. (23)] The term u/|u| is undefined at points where u=0. Although s_stag is designed to suppress stagnation regions, a sentence on how division by zero is handled (e.g., a small-velocity regularization) would improve clarity.
  5. [Appendix B] The notation R_infinity should be introduced; the caption of Table 4 says 'Relative Error (%)' but R_infinity is presumably the relative maximum norm. Please align notation with Eq. (17).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: DPINN's claimed accuracy is obtained by training networks against PDE residuals and is benchmarked against external high-fidelity references; the hand-set shock-sensor thresholds are disclosed hyperparameters with a sensitivity study, and no load-bearing self-citation or fitted-input-as-prediction step was found.

full rationale

The paper's derivation chain is self-contained rather than circular. The central training objective (Eq. 2, and its artificial-viscosity modification in Eq. 16) minimizes PDE, initial, and boundary residuals directly; the reported errors in Tables 1 and 2 are computed against independent WENO or high-fidelity finite-volume reference solutions, not against quantities used to fit the model. The learnable viscosity coefficient w_nu is a scalar trained through the residual loss, and the sensor functions in Eqs. 19 and 23 are explicit functions of the network's own predicted fields, with thresholds k_shock,0, k_shock,1, k_stag,0, k_stag,1 listed per test case in Table 3 and perturbed in the Appendix B sensitivity analysis. This is a disclosed manual hyperparameter choice with measured sensitivity, not a fitted parameter renamed as a prediction. No uniqueness theorem is imported from the authors' prior work, and the paper contains no load-bearing self-citations: the cited KAF, sensor, mesh-transformation, and optimizer works are independent references. The strongest concern—that the headline transonic accuracy of 3.80% degrades to 30.11% when k_shock,0 is multiplied by 10—is a robustness and generalization limitation, and the paper itself concedes that 'the reliance on shock sensors constrains generalizability.' That concession, plus the sensitivity table, makes the threshold dependence transparent rather than circular. The claim that DKAN 'generalizes the Kolmogorov representation theorem' is an overstatement, but it is not a circular derivation: the architecture is explicitly defined by Eq. 9 and evaluated by its numerical performance. Overall, no step in the paper reduces, by construction or by self-citation, to its own inputs.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claim rests on hand-tuned shock sensors, the unproven discontinuous extension of the Kolmogorov representation, and the standard PINN premise that small residual implies accurate solution. The viscosity coefficient w_nu is learned rather than prescribed, but it is still a fitted parameter. No new physical entities are introduced.

free parameters (4)
  • Local artificial viscosity coefficient w_nu = 5.12e-4 (transonic), 5.41e-4 (supersonic)
    Learned during training to balance stabilization and accuracy; depends on the shock sensor and the test case.
  • Shock sensor thresholds k_shock,0 and k_shock,1 = 0.015 and 1 (Burgers); 0.5 and 5 (transonic); 1 and 4 (supersonic)
    Hand-chosen to detect shocks for each test case; sensitivity analysis shows k_shock,0 x10 degrades transonic R2 to 30.11%.
  • Stagnation sensor thresholds k_stag,0 and k_stag,1 = 3 and 4 (transonic); 3 and 5 (supersonic)
    Hand-chosen to avoid misclassifying stagnation points as shocks; sensitivity analysis shows intermediate impact.
  • Number of Fourier embedding frequencies m = 64 (Burgers); 128 (Riemann and airfoils)
    Hyperparameter controlling spectral coverage; chosen by the authors, not derived.
assumptions (4)
  • domain assumption If the PINN loss is small, the network output approximates the PDE solution.
    Standard PINN premise invoked in Sec 2.1; no explicit convergence guarantee is given for discontinuous solutions.
  • ad hoc to paper DKAN with DyT and B-spline activations can approximate discontinuous functions, extending the Kolmogorov representation theorem.
    No theorem or convergence rate is proven; the paper cites Ref. [49] for the theoretical possibility but does not derive an extension.
  • domain assumption The shock sensor s(x) correctly identifies shock regions and excludes stagnation points from the predicted flow field.
    Heuristic from Ref. [19]; thresholds are manually selected per test case and are not derived from first principles.
  • domain assumption Adding the localized viscosity term mu grad^2 Q yields a solution close to the inviscid entropy solution when mu is small and localized.
    Classical artificial viscosity convergence is assumed; the learned w_nu magnitude is not proven to vanish.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Discontinuity-aware KAN-based physics-informed neural networks." pith.science (2026). https://pith.science/paper/CYAYAKLW

@misc{pith2026250708338,
  author       = {Pith},
  title        = {Pith review of: Discontinuity-aware KAN-based physics-informed neural networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CYAYAKLW}},
  note         = {Machine review of arXiv:2507.08338}
}
read the original abstract

Physics-informed neural networks (PINNs) have proven to be a promising method for the rapid solving of partial differential equations (PDEs) in both forward and inverse problems. However, due to the smoothness assumption of functions approximated by general neural networks, PINNs are prone to spectral bias and numerical instability and suffer from reduced accuracy when solving PDEs with sharp spatial transitions or fast temporal evolution. To address this limitation, a discontinuity-aware physics-informed neural network (DPINN) method is proposed. It incorporates an adaptive Fourier-feature embedding layer to mitigate spectral bias and capture steep gradients, a discontinuity-aware network that generalizes the Kolmogorov representation theorem to the discontinuous regime for the modeling of shock-wave properties, mesh transformation to accelerate convergence across complex geometries, and learnable local artificial viscosity to stabilize the algorithm near discontinuities. In numerical experiments regarding the inviscid Burgers' equation, Riemann problems, and transonic and supersonic airfoil flows, DPINN demonstrated superior accuracy in capturing discontinuities compared to existing methods.

Figures

Figures reproduced from arXiv: 2507.08338 by the authors.

Figure 1
Figure 1. Flowchart of the DPINN method applied to a two-dimensional (2D) unsteady flow. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Schematic of the adaptive Fourier embedding layer with red and blue channels representing high [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Left: Schematic of the DKAN architecture. Right: The discontinuous-aware activation function [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (16 more)
Figure 4
Figure 4. Figure 4: Schematic of the body-fitted coordinate system. (a) Physical space. (b) Computational space. [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: Evolution of the loss function with training epochs. [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: The solutions at different time steps for various methods, with markers placed at [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: Training loss evolution in the ablation study for the 1D Burgers’ equation. [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: Sod Shock Tube: solution at t = 0.2 [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: Lax Shock Tube: solution at t = 0.14. 3.3. Transonic flows around an airfoil The proposed method is then demonstrated on a 2D transonic NACA0012 airfoil flow. In this scenario, the flow locally accelerates to supersonic speeds, forming normal shock-waves on the airfoil…
Figure 10
Figure 10. Figure 10: The pressure coefficient distributions around the NACA0012 airfoil at [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]
Figure 11
Figure 11. Figure 11: Global pressure field for the transonic flow around the NACA0012 airfoil at [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]
Figure 12
Figure 12. Figure 12: Global pressure solution (left), absolute error (middle), and viscosity distribution (right) obtained [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]
Figure 13
Figure 13. Figure 13: Training evolution: (a) history of the learnable viscosity parameter [PITH_FULL_IMAGE:figures/full_fig_p018_13.png]
Figure 14
Figure 14. Figure 14: Results of the ablation study for 2D transonic flow over an airfoil. (a) Loss history; (b) pressure [PITH_FULL_IMAGE:figures/full_fig_p019_14.png]
Figure 15
Figure 15. Figure 15: Global pressure field for the supersonic flow around the NACA0012 airfoil at [PITH_FULL_IMAGE:figures/full_fig_p020_15.png]
Figure 16
Figure 16. Figure 16: Global pressure solution (left), absolute error (middle), and viscosity distribution (right) obtained [PITH_FULL_IMAGE:figures/full_fig_p020_16.png]
Figure 17
Figure 17. Figure 17: (a–c) Artificial viscosity distributions and (d–f) corresponding pressure fields for [PITH_FULL_IMAGE:figures/full_fig_p023_17.png]
Figure 18
Figure 18. Figure 18: Sod shock tube problem: velocity, pressure, and density contours (left to right) obtained from (a) [PITH_FULL_IMAGE:figures/full_fig_p024_18.png]
Figure 19
Figure 19. Figure 19: Lax shock tube problem: velocity, pressure, and density contours (left to right) obtained from (a) [PITH_FULL_IMAGE:figures/full_fig_p025_19.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

61 extracted references · 45 canonical work pages

  1. [49]

    Ismayilova, V

    A. Ismayilova, V. E. Ismailov, On the kolmogorov neural networks, Neural Networks 176 (2024) 106333

    work page 2024

  2. [1]

    D. V. Gaitonde, Progress in shock wave/boundary layer interactions, in: 43rd AIAA Fluid Dynamics Conference, 2013, p. 2607

    work page 2013

  3. [2]

    M. S. Longair, High energy astrophysics, Cambridge university press, 2011

    work page 2011

  4. [3]

    Jalili, S

    D. Jalili, S. Jang, M. Jadidi, G. Giustini, A. Keshmiri, Y. Mahmoudi, Physics-informed neural networks for heat transfer prediction in two-phase flows, International Journal of Heat and Mass Transfer 221 (2024) 125089

    work page 2024

  5. [4]

    Gottlieb, L

    D. Gottlieb, L. Lustman, S. A. Orszag, Spectral calculations of one-dimensional inviscid compressible flows, SIAM Journal on Scientific and Statistical Computing 2 (1981) 296– 310

    work page 1981

  6. [5]

    Gottlieb, C.-W

    D. Gottlieb, C.-W. Shu, On the gibbs phenomenon and its resolution, SIAM review 39 (1997) 644–668

    work page 1997

  7. [6]

    VonNeumann, R

    J. VonNeumann, R. D. Richtmyer, A method for the numerical calculation of hydro- dynamic shocks, Journal of applied physics 21 (1950) 232–237

    work page 1950

  8. [7]

    X.-D. Liu, S. Osher, T. Chan, Weighted essentially non-oscillatory schemes, Journal of computational physics 115 (1994) 200–212

    1994

Show all 61 references
  1. [8]

    S. K. Godunov, I. Bohachevsky, Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics, Matematičeskij sbornik 47 (1959) 271–306

    1959

  2. [9]

    Bassi, S

    F. Bassi, S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible navier–stokes equations, Journal of computa- tional physics 131 (1997) 267–279. 26

    1997

  3. [10]

    Pirozzoli, Numerical methods for high-speed flows, Annual review of fluid mechanics 43 (2011) 163–194

    S. Pirozzoli, Numerical methods for high-speed flows, Annual review of fluid mechanics 43 (2011) 163–194

    2011

  4. [11]

    Kramer, B

    B. Kramer, B. Peherstorfer, K. E. Willcox, Learning nonlinear reduced models from data with operator inference, Annual Review of Fluid Mechanics 56 (2024) 521–548

    2024

  5. [12]

    Raissi, P

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

    2019

  6. [13]

    L. Liu, S. Liu, H. Xie, F. Xiong, T. Yu, M. Xiao, L. Liu, H. Yong, Discontinuity computing using physics-informed neural networks, Journal of Scientific Computing 98 (2024) 22

    2024

  7. [14]

    S. Wang, Y. Teng, P. Perdikaris, Understanding and mitigating gradient flow patholo- gies in physics-informed neural networks, SIAM Journal on Scientific Computing 43 (2021) A3055–A3081

    2021

  8. [15]

    Rahaman, A

    N. Rahaman, A. Baratin, D. Arpit, F. Draxler, M. Lin, F. Hamprecht, Y. Bengio, A. Courville, On the spectral bias of neural networks, in: International conference on machine learning, PMLR, 2019, pp. 5301–5310

    2019

  9. [16]

    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

  10. [17]

    Abbasi, A

    J. Abbasi, A. D. Jagtap, B. Moseley, A. Hiorth, P. Ø. Andersen, Challenges and ad- vancements in modeling shock fronts with physics-informed neural networks: A review and benchmarking study, arXiv preprint arXiv:2503.17379 (2025)

    2025 arXiv

  11. [18]

    Wassing, S

    S. Wassing, S. Langer, P. Bekemeyer, Adopting computational fluid dynamics concepts for physics-informed neural networks, in: AIAA SCITECH 2025 Forum, 2025, p. 0269

    2025

  12. [19]

    Wassing, S

    S. Wassing, S. Langer, P. Bekemeyer, Physics-informed neural networks for transonic flows around an airfoil, arXiv preprint arXiv:2408.17364 (2024)

    2024 arXiv

  13. [20]

    O. Fuks, H. A. Tchelepi, Limitations of physics informed machine learning for nonlinear two-phase transport in porous media, Journal of Machine Learning for Modeling and Computing 1 (2020)

    2020

  14. [21]

    E. J. R. Coutinho, M. Dall’Aqua, L. McClenny, M. Zhong, U. Braga-Neto, E. Gildin, Physics-informed neural networks with adaptive localized artificial viscosity, Journal of Computational Physics 489 (2023) 112265. 27

    2023

  15. [22]

    R. G. Patel, I. Manickam, N. A. Trask, M. A. Wood, M. Lee, I. Tomas, E. C. Cyr, Thermodynamically consistent physics-informed neural networks for hyperbolic sys- tems, Journal of Computational Physics 449 (2022) 110754

    2022

  16. [23]

    W. Diab, O. Chaabi, W. Zhang, M. Arif, S. Alkobaisi, M. Al Kobaisi, Data-free and data-efficient physics-informed neural network approaches to solve the buckley–leverett problem, Energies 15 (2022) 7864

    2022

  17. [24]

    A. J. Huang, S. Agarwal, On the limitations of physics-informed deep learning: Illus- trations using first-order hyperbolic conservation law-based traffic flow models, IEEE Open Journal of Intelligent Transportation Systems 4 (2023) 279–293

    2023

  18. [25]

    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

  19. [26]

    Abbasi, P

    J. Abbasi, P. Ø. Andersen, Simulation and prediction of countercurrent spontaneous imbibition at early and late time using physics-informed neural networks, Energy & Fuels 37 (2023) 13721–13733

    2023

  20. [27]

    J. Wu, Y. Wu, G. Zhang, Y. Zhang, Variable linear transformation improved physics- informed neural networks to solve thin-layer flow problems, Journal of Computational Physics 500 (2024) 112761

    2024

  21. [28]

    De Ryck, S

    T. De Ryck, S. Mishra, R. Molinaro, wpinns: Weak physics informed neural networks for approximating entropy solutions of hyperbolic conservation laws, SIAM Journal on Numerical Analysis 62 (2024) 811–841

    2024

  22. [29]

    Kharazmi, Z

    E. Kharazmi, Z. Zhang, G. E. Karniadakis, Variational physics-informed neural net- works for solving partial differential equations, arXiv preprint arXiv:1912.00873 (2019)

    2019 arXiv

  23. [30]

    Chaumet, J

    A. Chaumet, J. Giesselmann, Improving weak pinns for hyperbolic conservation laws: Dual norm computation, boundary conditions and systems, arXiv preprint arXiv:2211.12393 (2022)

    2022 arXiv

  24. [31]

    A. G. Neelan, G. S. Krishna, V. Paramanantham, Physics-informed neural networks and higher-order high-resolution methods for resolving discontinuities and shocks: A comprehensive study, Journal of Computational Science 83 (2024) 102466

    2024

  25. [32]

    Ferrer-Sánchez, J

    A. Ferrer-Sánchez, J. D. Martín-Guerrero, R. R. de Austri-Bazan, A. Torres-Forné, J. A. Font, Gradient-annihilated pinns for solving riemann problems: Application to relativistic hydrodynamics, Computer Methods in Applied Mechanics and Engineering 424 (2024) 116906. 28

    2024

  26. [33]

    Y. Yang, Q. Yang, Y. Deng, Q. He, Moving sampling physics-informed neural networks induced by moving mesh pde, Neural Networks 180 (2024) 106706

    2024

  27. [34]

    Z. Mao, A. D. Jagtap, G. E. Karniadakis, Physics-informed neural networks for high- speed flows, Computer Methods in Applied Mechanics and Engineering 360 (2020) 112789

    2020

  28. [35]

    J. F. Urbán, P. Stefanou, J. A. Pons, Unveiling the optimization process of physics informed neural networks: How accurate and competitive can pinns be?, Journal of Computational Physics 523 (2025) 113656

    2025

  29. [36]

    Tancik, P

    M. Tancik, P. Srinivasan, B. Mildenhall, S. Fridovich-Keil, N. Raghavan, U. Singhal, R. Ramamoorthi, J. Barron, R. Ng, Fourier features let networks learn high frequency functions in low dimensional domains, Advances in neural information processing sys- tems 33 (2020) 7537–7547

    2020

  30. [37]

    Kiyani, K

    E. Kiyani, K. Shukla, J. F. Urbán, J. Darbon, G. E. Karniadakis, Which optimizer works best for physics-informed neural networks and kolmogorov-arnold networks?, arXiv preprint arXiv:2501.16371 (2025)

    2025

  31. [38]

    Rodriguez-Torrado, P

    R. Rodriguez-Torrado, P. Ruiz, L. Cueto-Felgueroso, M. C. Green, T. Friesen, S. Ma- tringe, J. Togelius, Physics-informed attention-based neural network for hyperbolic partial differential equations: application to the buckley–leverett problem, Scientific reports 12 (2022) 7557

    2022

  32. [39]

    S. J. Anagnostopoulos, J. D. Toscano, N. Stergiopulos, G. E. Karniadakis, Residual- based attention in physics-informed neural networks, Computer Methods in Applied Mechanics and Engineering 421 (2024) 116805

    2024

  33. [40]

    A. D. Jagtap, G. E. Karniadakis, Extended physics-informed neural networks (xpinns): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations, Communications in Computational Physics 28 (2020)

    2020

  34. [41]

    S. Wang, H. Wang, P. Perdikaris, On the eigenvector bias of fourier feature networks: From regression to solving multi-scale pdes with physics-informed neural networks, Computer Methods in Applied Mechanics and Engineering 384 (2021) 113938

    2021

  35. [42]

    O. P. Bruno, J. S. Hesthaven, D. V. Leibovici, Fc-based shock-dynamics solver with neural-network localized artificial-viscosity assignment, Journal of Computational Physics: X 15 (2022) 100110. 29

    2022

  36. [43]

    Mojgani, M

    R. Mojgani, M. Balajewicz, P. Hassanzadeh, Kolmogorov n–width and lagrangian physics-informed neural networks: A causality-conforming manifold for convection- dominated pdes, Computer Methods in Applied Mechanics and Engineering 404 (2023) 115810

    2023

  37. [44]

    A. D. Jagtap, K. Kawaguchi, G. E. Karniadakis, Adaptive activation functions accel- erate convergence in deep and physics-informed neural networks, Journal of Computa- tional Physics 404 (2020) 109136

    2020

  38. [45]

    Catalani, S

    G. Catalani, S. Agarwal, X. Bertrand, F. Tost, M. Bauerheim, J. Morlier, Neural fields for rapid aircraft aerodynamics simulations, Scientific Reports 14 (2024) 25496

    2024

  39. [46]

    J.Zhang, Y.Fan, K.Cai, K.Wang, Kolmogorov-arnoldfouriernetworks, arXivpreprint arXiv:2502.06018 (2025)

    2025 arXiv

  40. [47]

    A. N. Kolmogorov, On the representation of continuous functions of several variables by superpositions of continuous functions of a smaller number of variables, American Mathematical Society, 1961

    1961

  41. [48]

    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)

    2024 arXiv

  42. [50]

    J. Zhu, X. Chen, K. He, Y. LeCun, Z. Liu, Transformers without normalization, arXiv preprint arXiv:2503.10622 (2025)

    2025 arXiv

  43. [51]

    Y. Zang, G. Bao, X. Ye, H. Zhou, Weak adversarial networks for high-dimensional partial differential equations, Journal of Computational Physics 411 (2020) 109409

    2020

  44. [52]

    X. Meng, Z. Li, D. Zhang, G. E. Karniadakis, Ppinn: Parareal physics-informed neu- ral network for time-dependent pdes, Computer Methods in Applied Mechanics and Engineering 370 (2020) 113250

    2020

  45. [53]

    W. Cao, J. Song, W. Zhang, A solver for subsonic flow around airfoils based on physics- informed neural networks and mesh transformation, Physics of Fluids 36 (2024)

    2024

  46. [54]

    J. W. White, A new form of artificial viscosity, Journal of Computational Physics 11 (1973) 573–590

    1973

  47. [55]

    E. J. Caramana, M. J. Shashkov, P. P. Whalen, Formulations of artificial viscosity for multi-dimensional shock wave computations, Journal of Computational Physics 144 (1998) 70–97. 30

    1998

  48. [56]

    Margolin, N

    L. Margolin, N. Lloyd-Ronning, Artificial viscosity—then and now, Meccanica 58 (2023) 1039–1052

    2023

  49. [57]

    Jameson, W

    A. Jameson, W. Schmidt, E. Turkel, Numerical solution of the euler equations by finite volume methods using runge kutta time stepping schemes, in: 14th fluid and plasma dynamics conference, 1981, p. 1259

    1981

  50. [58]

    J. Song, W. Cao, F. Liao, W. Zhang, Vw-pinns: A volume weighting method for pde residuals in physics-informed neural networks, Acta Mechanica Sinica 41 (2025) 324140

    2025

  51. [59]

    Xiang, W

    Z. Xiang, W. Peng, X. Liu, W. Yao, Self-adaptive loss balanced physics-informed neural networks, Neurocomputing 496 (2022) 11–34

    2022

  52. [60]

    S. Fan, J. Qin, Y. Dong, Y. Jiang, X. Deng, Wcns3-mr-nn: A machine learning- based shock-capturing scheme with accuracy-preserving and high-resolution properties, Journal of Computational Physics 532 (2025) 113973

    2025

  53. [61]

    Y. Song, H. Wang, H. Yang, M. L. Taccari, X. Chen, Loss-attentional physics-informed neural networks, Journal of Computational Physics 501 (2024) 112781. 31

    2024

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.