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REVIEW 4 major objections 6 minor 38 references

CT-PIKAN: Coordinate-Transformed Physics-Informed Kolmogorov-Arnold Network with Autograd-Based Metric Evaluation for Solving PDEs in Curvilinear Domains

T0 review · 4 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read CT-PIKAN computes all geometric terms of a coordinate-transformed PDE solver automatically, eliminating hand-derived Jacobians and metric tensors for curved domains.

desk verdict A useful autograd-based PIKAN framework with a load-bearing math error in the wavy-domain advection demo that invalidates that specific result. read the letter →

arxiv 2608.06660 v1 pith:UJH6FNI3 submitted 2026-08-07 math.NA cs.NA

Mohammad E. Heravifard , Kazem Hejranfar This is my paper · ORCID
classification math.NAcs.NA MSC 65N3565M7068T07
keywords Physics-InformedKolmogorov-ArnoldNetworkCoordinatetransformationCurvilineardomainsAutomaticdifferentiationMetrictensorLaplace-BeltramioperatorPartialdifferentialequationslearning
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 CT-PIKAN, a way of solving partial differential equations on curved or irregular two-dimensional domains with physics-informed Kolmogorov-Arnold networks. Its central idea is to map the physical domain into a unit square and to compute the geometry terms required by the transformed equation—Jacobian, metric tensor, inverse metric, and the transformed differential operators—automatically from the coordinate mapping using automatic differentiation. This replaces the usual step of hand-deriving metric coefficients for each new geometry. If the claim holds, the same code can handle polar, wavy, annular, and star-shaped domains for elliptic, parabolic, and hyperbolic PDEs while keeping the mesh-free, data-free training of physics-informed networks. The paper supports the claim with numerical experiments on advection, Poisson, and heat equations.

What carries the argument

The load-bearing mechanism is the differentiable coordinate transformation $\Phi:(\xi,\eta)\mapsto(x,y)$ from the physical domain to $[0,1]^2$, implemented as a function the automatic differentiator can differentiate. From this single map the framework obtains $x_\xi,x_\eta,y_\xi,y_\eta$, the Jacobian determinant $J$, the metric tensor $G=J^TJ$, its inverse $G^{-1}$, and $\sqrt{|G|}$, and then assembles the transformed PDE operators, including the Laplace-Beltrami form $\frac{1}{\sqrt{|G|}}\nabla_\xi\cdot(\sqrt{|G|}\,G^{-1}\nabla_\xi u)$ for second-order terms. The network itself is a spline-based Kolmogorov-Arnold network whose edge activations are weighted sums of a SiLU basis and B-splines. Training happens entirely in computational coordinates, with uniform collocation sampling and boundary conditions on the square, after which the solution is mapped back to the physical domain.

What would settle it

Evaluate the Jacobian and metric terms of the map $x(\xi,\eta)=L_x\xi+A\sin(6\pi\eta)$, $y(\xi,\eta)=L_y\eta+A\sin(6\pi\xi)$ at two different points; $J=L_xL_y-A^2(6\pi)^2\cos(6\pi\eta)\cos(6\pi\xi)$ and the off-diagonal metrics differ between points, contradicting the constant-metric claim that leads to Eq. (40).

Watch

Extended reading notes

Core claim

The paper's central claim is that the geometric overhead of solving PDEs on curved domains can be removed by treating the coordinate map as a differentiable function and reading off all metric quantities with automatic differentiation. In CT-PIKAN, a spline-based KAN is trained in a unit-square computational domain; the physical-domain PDE is rewritten using the Jacobian $J$, the metric tensor $G=J^T J$, and the Laplace-Beltrami operator, with every one of these quantities computed on the fly from $\Phi(\xi,\eta)$. The paper reports that this formulation solves the advection equation on Cartesian, polar, and sinusoidal wavy domains, the Poisson equation on an annulus, and the heat equation on wavy-channel and star-shaped domains, all without analytical metric derivations.

Load-bearing premise

The wavy-domain advection example assumes the transformation metrics $x_\xi,x_\eta,y_\xi,y_\eta$ and the Jacobian determinant are constant, but for the sinusoidal map in Eqs. (30)-(31) they vary with position, so the constant-coefficient transformed equation used in that section is not valid.

Editorial extensions

If this is right

  • A new curvilinear geometry requires only a smooth, invertible mapping function; all geometric coefficients are derived by automatic differentiation, so no per-geometry analytical work is needed.
  • Because training is done on the unit square, collocation points can be sampled uniformly and boundary conditions are imposed on a regular boundary, removing sampling distortion and boundary-condition difficulty associated with curved domains.
  • The same CT-PIKAN formulation covers elliptic, parabolic, and hyperbolic PDEs, as demonstrated by the Poisson, heat, and advection benchmarks.
  • For time-dependent problems, time enters as an additional network input and only the spatial sub-vector is passed through the Laplace-Beltrami operator, so the method extends directly to transient simulations.
  • The learned solution is produced in the computational domain and mapped back to physical coordinates, giving a mesh-free, data-free solver on non-Cartesian geometries.

Reading between the lines

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

  • Editorial inference: if the constant-metric simplification in the wavy-advection section is corrected to variable coefficients, the reported accuracy for that benchmark may change; the paper does not show the variable-metric version.
  • Editorial inference: the autograd-based pipeline will face a new bottleneck when extended to fourth-order PDEs, since higher-order metric derivatives may amplify numerical noise; that is a testable stability question not addressed here.
  • Editorial inference: uniform sampling in computational coordinates does not guarantee uniform physical resolution when the map stretches or compresses areas; an adaptive sampling scheme that accounts for $\sqrt{|G|}$ would be a natural extension.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper proposes CT-PIKAN, a coordinate-transformed physics-informed Kolmogorov–Arnold network for solving PDEs on curvilinear domains. The key idea is to let the user provide a smooth mapping from a physical domain to a unit square, and to compute all geometric quantities (Jacobian, metric tensor, inverse metric, determinant) automatically via automatic differentiation, thereby eliminating hand-derived metric coefficients. The method is demonstrated on the 2D advection equation in Cartesian, polar, and wavy domains, the Poisson equation on an annulus, and the transient heat equation on wavy and star-shaped domains. The paper also makes a public implementation available.

Significance. The autograd-based metric evaluation is a useful and timely idea, since it promises to lower the barrier for applying PINN/PIKAN methods to curvilinear geometries. If the claims were fully validated, the paper would provide a practical, geometry-agnostic solver. The availability of source code is a concrete strength. However, the numerical validation is almost entirely qualitative: no error norms, convergence tables, or comparisons to reference solutions are reported, and a key demonstration case (the wavy-domain advection problem) contains a direct mathematical error that undermines the central claim of generality. The contribution is therefore plausible but not adequately established in the current form.

major comments (4)
  1. [§3.3, Eqs. (30)–(31) and text after Eq. (37)] The claim that the transformation metrics xξ, xη, yξ, yη and the Jacobian determinant are constant is false. The mapping defined by Eqs. (30)–(31) with Δx=Δy=1/(N_CP−1), Lx=Ly=1 is x(ξ,η)=ξ+A sin(6πη), y(ξ,η)=η+A sin(6πξ). Its partial derivatives are xξ=1, yη=1, xη=6πA cos(6πη), yξ=6πA cos(6πξ), and J=1−(6πA)^2 cos(6πη)cos(6πξ), all of which vary over the domain. Consequently, the simplified constant-coefficient equation (40) is not the pullback of Eq. (28) for this wavy mapping. The Figure 7 results therefore do not demonstrate CT-PIKAN solving the intended curvilinear advection problem, and the section is internally inconsistent with the paper's autograd-based methodology, which would compute these metrics pointwise.
  2. [§3, throughout] No quantitative error measures are reported anywhere in the results. The paper states in the introduction that the method demonstrates 'improved accuracy, convergence, and robustness compared to standard PIKAN formulations', but no numerical error norms, convergence rates, or comparisons against analytical solutions are given for any test case. Figures alone cannot support quantitative claims of accuracy, especially for a solver whose main selling point is geometric generality. This makes the validation insufficient to assess the method's actual performance.
  3. [§3.5, star-domain case, text after Eq. (56)] The paper states that for the star-shaped domain the inner edge ξ=0 corresponds to an 'interior annular cutout' and that 'no boundary condition is imposed there'. If the physical domain excludes the disk of radius rinner, then the inner circle is a genuine boundary and the transient heat equation is ill-posed without a condition on it. If the inner circle is only an artificial coordinate cutout and the physical domain actually includes the center, then the mapping as written (with rinner=0.5) does not cover that region. Either way, the setup needs clarification and justification; as written, the star-domain heat results are ambiguous.
  4. [§3.3, Fig. 7 caption] The caption describes the results as obtained with a 'conformal mapping', but the mapping defined by Eqs. (30)–(31) is not conformal: conformality would require xξ=yη and xη=−yξ, and while xξ=yη=1, we have xη=6πA cos(6πη) and yξ=6πA cos(6πξ), which do not satisfy xη=−yξ everywhere. This inaccurate terminology should be corrected.
minor comments (6)
  1. [§2.2, Eq. (7)] The symbol J is used for both the Jacobian matrix and its determinant, which is confusing; use distinct notations such as J for the matrix and |J| or J for the determinant.
  2. [§3.3, paragraph after Eq. (29)] The discussion of imposing boundary conditions on the wavy geometry is vague: it mentions analytical relations for BCs and ICs but does not specify the actual boundary conditions used for the advection problem in this section.
  3. [§3.3, text near Eqs. (35)–(36)] The derivation of the inverse derivative relations is written in a way that is easy to misread; the placement of parentheses in Eqs. (35)–(36) should be made more explicit.
  4. [§3.1, Fig. 3] The figure shows CT-PIKAN and analytical solutions but no error plot or quantitative comparison; adding an error map would strengthen the validation.
  5. [§3.5, text after Eq. (47)] The phrase 'thermal energy decay → E(t)=∫Ω u² dΩ' in the Fig. 14 caption is awkwardly worded; also, no justification is given for using this functional as the 'thermal energy'.
  6. [§2.4, Algorithm 2] The notation N is used for the PDE operator in Algorithm 2, while the main text uses F in Eq. (19) and (22); unifying these notations would improve readability.

Circularity Check

0 steps flagged · score 1.0 of 10

No load-bearing circularity: CT-PIKAN minimizes PDE residuals with user-supplied mappings and autograd-computed metrics; Section 3.3 contains a factual metric error, not a circular derivation.

full rationale

I traced the claimed derivation chain. The baseline PIKAN is a data-free residual minimization of the advection PDE, and the CT-PIKAN extension replaces Cartesian derivatives with chain-rule expressions whose Jacobians and metric tensors are obtained by automatic differentiation from the user-supplied coordinate mapping. No physical parameter is fitted from output data and then renamed as a prediction; the coordinate map is an input, not a learned quantity, so computing x_xi, x_eta, y_xi, y_eta via autograd is not circular. The only citation involving the present authors is Ref. [35], used to motivate the sinusoidal collocation-point construction in Eqs. (30)-(31); it is not load-bearing for the central autograd-metric claim and does not constrain the outcome of the method. The assertion in Section 3.3 that the metrics x_xi, x_eta, y_xi, y_eta and the Jacobian determinant are constant for the mapping implied by Eqs. (30)-(31) is mathematically incorrect: with x = xi + A sin(6 pi eta) and y = eta + A sin(6 pi xi), one has x_eta = 6 pi A cos(6 pi eta), y_xi = 6 pi A cos(6 pi xi), and J = 1 - (6 pi A)^2 cos(6 pi eta) cos(6 pi xi), which are not constant. Consequently the constant-coefficient transformed equation Eq. (40) is not the correct pullback of Eq. (28) to that wavy domain. That is a benchmark-validity and correctness concern, not a circularity concern. Because the central derivation is self-contained and the only self-citation is minor and non-load-bearing, the circularity score is low.

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

The central claim relies on standard differential geometry for the transformed operators and on the network's ability to minimize the residual. No new physical entities are introduced. The only ad hoc assumption is the constant-metric claim in the wavy advection case, which contradicts the paper's own mapping.

assumptions (4)
  • standard math Kolmogorov-Arnold representation theorem
    Invoked in Section 2.1, Eq. (1), to justify decomposing multivariate functions into univariate spline-based functions.
  • domain assumption Existence of a smooth, bijective coordinate transformation with nonzero Jacobian
    Section 2.2, around Eqs. (6)-(7). The framework assumes such a mapping exists for the physical domain. For the wavy and star-shaped maps this is plausible but not proven in the paper.
  • standard math Transformed differential operators follow the chain rule and the Laplace-Beltrami formula
    Section 2.2, Eqs. (8)-(12). Standard differential geometry results adopted without proof.
  • ad hoc to paper Metrics for the wavy transformation are constant
    Section 3.3, after Eq. (37). The text asserts constancy for the sinusoidal mapping, but Eqs. (30)-(31) define a mapping where x and y each depend on both coordinates through sine terms, so the metric terms vary with position. This assumption is specific to the paper's demonstration and is mathematically incorrect.

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Cite this review

Pith. "Pith review of CT-PIKAN: Coordinate-Transformed Physics-Informed Kolmogorov-Arnold Network with Autograd-Based Metric Evaluation for Solving PDEs in Curvilinear Domains." pith.science (2026). https://pith.science/paper/UJH6FNI3

@misc{pith2026260806660,
  author       = {Pith},
  title        = {Pith review of: CT-PIKAN: Coordinate-Transformed Physics-Informed Kolmogorov-Arnold Network with Autograd-Based Metric Evaluation for Solving PDEs in Curvilinear Domains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UJH6FNI3}},
  note         = {Machine review of arXiv:2608.06660}
}
read the original abstract

Physics-Informed Kolmogorov-Arnold Networks have recently emerged as an effective class of neural solvers for partial differential equations, combining the expressive power of spline-based Kolmogorov-Arnold representations with physics-informed learning. However, existing PIKAN formulations are primarily developed for Cartesian domains and cannot naturally accommodate the geometric complexity introduced by curvilinear domains. In this work, we propose Coordinate-Transformed Physics-Informed Kolmogorov-Arnold Networks (CT-PIKAN), a geometry-aware framework for solving PDEs on arbitrarily shaped domains through coordinate transformation. A smooth mapping transforms the physical domain into a regular computational domain, while the transformed governing equations are enforced directly within the physics-informed loss. Unlike conventional transformed PINN approaches that require manually derived metric coefficients, CT-PIKAN employs automatic differentiation to evaluate Jacobians, metric tensors, and transformed differential operators directly from the coordinate mapping, eliminating analytical derivations and improving implementation flexibility. To establish the proposed framework, a data-free B-spline-based PIKAN is first constructed and validated on the two-dimensional advection equation. The CT-PIKAN methodology is subsequently assessed on representative elliptic, parabolic, and hyperbolic benchmark problems, including the Poisson, heat, and advection equations formulated in polar and wavy curvilinear coordinates. The proposed framework provides a general and extensible methodology for integrating differential geometry with physics-informed Kolmogorov-Arnold networks, enabling efficient and accurate PDE solutions on complex domains.

Figures

Figures reproduced from arXiv: 2608.06660 by the authors.

Figure 1
Figure 1. Schematic of the proposed CT-PIKAN architecture. [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. The training process of the proposed CT-PIKAN. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. CT-PIKAN & analytical solutions for 2D advection equation with a sinusoidal I.C. [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: CT-PIKAN solutions for 2D advection equation on polar coordinates with sinu [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: 2D spatial & 3D spatio-temporal distribution of collocation points for solving 2D [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: Collocation points for a curvilinear (wavy) domain. [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: CT-PIKAN solutions for 2D advection equation on a curvilinear (wavy) domain [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 8
Figure 8. Figure 8: CT-PIKAN solutions & PDE residuals for 2D Poisson equation on an annular [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 9
Figure 9. Figure 9: 3D view of the solution & metric determinant for CT-PIKAN solver. [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: Training convergence & an ablation study (loss vs. epoch) for CT-PIKAN solver. [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]
Figure 11
Figure 11. Figure 11: Visualization of collocation points in the physical curvilinear (wavy) domain & [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]
Figure 12
Figure 12. Figure 12: Transient temperature evolution of 2D Heat equation using CT-PIKAN solver. [PITH_FULL_IMAGE:figures/full_fig_p020_12.png]
Figure 13
Figure 13. Figure 13: 3D view of collocation points & CT-PIKAN solution for 2D Heat equation. [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]
Figure 14
Figure 14. Figure 14: CT-PIKAN solutions for transient 2D Heat equation on curvilinear (wavy and [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]
Figure 15
Figure 15. Figure 15: Evaluation of CT-PIKAN solver performance (I.C. & B.C. errors & PDE residual). [PITH_FULL_IMAGE:figures/full_fig_p021_15.png]
Figure 16
Figure 16. Figure 16: Geometric quantities calculated via CT-PIKAN solver. [PITH_FULL_IMAGE:figures/full_fig_p022_16.png]
Figure 17
Figure 17. Figure 17: CT-PIKAN solutions for 2D Heat equation on curvilinear (star-shaped) domain. [PITH_FULL_IMAGE:figures/full_fig_p022_17.png]

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Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages

  1. [1]

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

    work page 2021

  2. [2]

    Evans, L. C. Partial Differential Equations (2nd ed.). American Mathematical Society. 2010

    work page 2010

  3. [3]

    Numerical Mathematics (2nd ed.)

    Quarteroni, A., Sacco, R., & Saleri, F. Numerical Mathematics (2nd ed.). Springer. 2007

    work page 2007

  4. [4]

    C., Taylor, R

    Zienkiewicz, O. C., Taylor, R. L., & Zhu, J. Z. The Finite Element Method: Its Basis and Fundamentals (7th ed.). Elsevier. 2013

    work page 2013

  5. [5]

    H., & Peri, M

    Ferziger, J. H., & Peri, M. Computational Methods for Fluid Dynamics (3rd ed.). Springer. 2002

    work page 2002

  6. [6]

    F., Soni, B

    Thompson, J. F., Soni, B. K., & Weatherill, N. P. Handbook of Grid Generation. CRC Press. 1999

    work page 1999

  7. [7]

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

    work page 2019

  8. [8]

    E., Kevrekidis, I

    Karniadakis, G. E., Kevrekidis, I. G., Lu, L., Perdikaris, P., Wang, S., & Yang, L. Physics-informed machine learning. Nature Reviews Physics. 2021

    work page 2021

Show all 38 references
  1. [9]

    Understanding and mitigating gradient flow pathologies in physics-informed neural networks

    Wang, S., Teng, Y., & Perdikaris, P. Understanding and mitigating gradient flow pathologies in physics-informed neural networks. SIAM Journal on Scientific Computing. 2021

    2021

  2. [10]

    S., Gholami, A., Zhe, S., Kirby, R

    Krishnapriyan, A. S., Gholami, A., Zhe, S., Kirby, R. M., & Mahoney, M. W. Charac- terizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems (NeurIPS). 2021

    2021

  3. [11]

    Respecting causality is all you need for train- ing physics-informed neural networks

    Wang, S., Sankaran, S., & Perdikaris, P. Respecting causality is all you need for train- ing physics-informed neural networks. Computer Methods in Applied Mechanics and Engineering. 2022

    2022

  4. [12]

    Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks

    Sukumar, N., & Srivastava, A. Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks. Computer Methods in Applied Me- chanics and Engineering. 2022

    2022

  5. [13]

    Finite Basis Physics-Informed Neural Networks (FBPINNs): A scalable domain decomposition approach for solving differential equations

    Moseley, B., Markham, A., & Nissen-Meyer, T. Finite Basis Physics-Informed Neural Networks (FBPINNs): A scalable domain decomposition approach for solving differential equations. Advances in Computational Mathematics. 2023

    2023

  6. [14]

    Y., & Tegmark, M

    Liu, Z., Wang, Y., Vaidya, S., Ruehle, F., Halverson, J., Soljai, M., Hou, T. Y., & Tegmark, M. KAN: Kolmogorov-Arnold Networks. arXiv preprint. 2024

    2024

  7. [15]

    S., Zhuang, X., Rabczuk, T., & Liu, Y

    Wang, Y., Sun, J., Bai, J., Anitescu, C., Eshaghi, M. S., Zhuang, X., Rabczuk, T., & Liu, Y. KolmogorovArnold-Informed Neural Network: A Physics-Informed Deep Learn- ing Framework for Solving Forward and Inverse Problems Based on KolmogorovArnold Networks. Computer Methods in ...

    2025

  8. [16]

    A., & Stinis, P

    Jacob, B., Howard, A. A., & Stinis, P. SPIKANs: Separable Physics-Informed Kolmogorov-Arnold Networks. Machine Learning: Science and Technology. 2025

    2025

  9. [17]

    A physics-informed deep learn- ing framework for solving forward and inverse problems based on KolmogorovArnold Networks

    Wang, Yizheng, Jia Sun, Jinshuai Bai, Cosmin Anitescu, Mohammad Sadegh Eshaghi, Xiaoying Zhuang, Timon Rabczuk, and Yinghua Liu. A physics-informed deep learn- ing framework for solving forward and inverse problems based on KolmogorovArnold Networks. Computer Methods in Applie...

    2025

  10. [18]

    KAN-ODEs: KolmogorovArnold net- work ordinary differential equations for learning dynamical systems and hidden physics

    Koenig, Benjamin C., Suyong Kim, and Sili Deng. KAN-ODEs: KolmogorovArnold net- work ordinary differential equations for learning dynamical systems and hidden physics. Computer Methods in Applied Mechanics and Engineering. 2024

    2024

  11. [19]

    Adaptive training of grid-dependent physics-informed kolmogorov-arnold networks

    Rigas, Spyros, Michalis Papachristou, Theofilos Papadopoulos, Fotios Anagnostopou- los, and Georgios Alexandridis. Adaptive training of grid-dependent physics-informed kolmogorov-arnold networks. IEEE Access. 2024

    2024

  12. [20]

    Physics-informed kolmogorov-arnold networks for power system dynamics

    Shuai, Hang, and Fangxing Li. Physics-informed kolmogorov-arnold networks for power system dynamics. IEEE Open Access Journal of Power and Energy. 2025

    2025

  13. [21]

    D., & Karniadakis, G

    Jagtap, A. D., & Karniadakis, G. E. Extended Physics-Informed Neural Networks (XPINNs): A generalized space-time domain decomposition based deep learning frame- work for nonlinear partial differential equations. Communications in Computational Physics. 2020

    2020

  14. [22]

    PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irreg- ular Domain

    Gao, H., Sun, L., & Wang, J.-X. PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irreg- ular Domain. arXiv preprint. 2020

    2020

  15. [23]

    S., Son, S., Kim, Y., & Seok, J

    Hwang, H. S., Son, S., Kim, Y., & Seok, J. Mitigating Coordinate Transformation for Solving Partial Differential Equations with Physics-Informed Neural Networks. Proceed- ings of the 13th International Conference on Ubiquitous and Future Networks (ICUFN). IEEE. 2022

    2022

  16. [24]

    Pearlmutter, Alexey Andreyevich Radul, and Jeffrey Mark Siskind

    Baydin, Atilim Gunes, Barak A. Pearlmutter, Alexey Andreyevich Radul, and Jeffrey Mark Siskind. Automatic differentiation in machine learning: A survey. Journal of Ma- chine Learning Research. 2018

    2018

  17. [25]

    A mathematical model for automatic differentia- tion in machine learning

    Bolte, Jérôme, and Edouard Pauwels. A mathematical model for automatic differentia- tion in machine learning. Advances in Neural Information Processing Systems. 2020

    2020

  18. [26]

    Auto- matic differentiation in ML: Where we are and where we should be going

    Van Merriënboer, Bart, Olivier Breuleux, Arnaud Bergeron, and Pascal Lamblin. Auto- matic differentiation in ML: Where we are and where we should be going. Advances in Neural Information Processing Systems. 2018

    2018

  19. [27]

    Nodal integral methods for heat and mass transfer in curvilinear coordinates

    Jarrah, Ibrahim Ahmad Mohammad. "Nodal integral methods for heat and mass transfer in curvilinear coordinates." PhD diss., University of Illinois at Urbana-Champaign, 2022

    2022

  20. [28]

    Pseudo-Spectral Methods for Linear Advection and Dispersive Problems

    Fazio, Riccardo, and Salvatore Iacono. "Pseudo-Spectral Methods for Linear Advection and Dispersive Problems." IAENG International Journal of Applied Mathematics 39, no. 1 (2009). 25

    2009

  21. [29]

    A residual- based artificial viscosity finite difference method for scalar conservation laws

    Stiernström, Vidar, Lukas Lundgren, Murtazo Nazarov, and Ken Mattsson. "A residual- based artificial viscosity finite difference method for scalar conservation laws." Journal of Computational Physics 430 (2021): 110100

    2021

  22. [30]

    Numerical study of 1D and 2D advection-diffusion-reaction equations using Lucas and Fibonacci polynomials

    Ali, Ihteram, Sirajul Haq, Kottakkaran Sooppy Nisar, and Shams Ul Arifeen. "Numerical study of 1D and 2D advection-diffusion-reaction equations using Lucas and Fibonacci polynomials." Arabian Journal of Mathematics 10, no. 3 (2021): 513-526

    2021

  23. [31]

    Numerical Solution of TwoDimensional Nonlinear Unsteady AdvectionDiffusionReaction Equations with Variable Coefficients

    Tsega, Endalew Getnet. "Numerical Solution of TwoDimensional Nonlinear Unsteady AdvectionDiffusionReaction Equations with Variable Coefficients." International Journal of Mathematics and Mathematical Sciences 2024, no. 1 (2024): 5541066

    2024

  24. [32]

    Nodal integral method for convection- diffusion transport using linear and higher order quadrilateral elements

    Sharma, Rishabh Prakash, and Neeraj Kumar. "Nodal integral method for convection- diffusion transport using linear and higher order quadrilateral elements." Numerical Heat Transfer, Part B: Fundamentals 74, no. 3 (2018): 623-645

    2018

  25. [33]

    High-order accurate methods for unsteady vortical flows on curvilinear meshes

    Visbal, Migual, and Datta Gaitonde. "High-order accurate methods for unsteady vortical flows on curvilinear meshes." In 36th AIAA Aerospace Sciences Meeting and Exhibit, p

  26. [34]

    On the use of higher-order finite-difference schemes on curvilinear and deforming meshes

    Visbal, Miguel R., and Datta V. Gaitonde. "On the use of higher-order finite-difference schemes on curvilinear and deforming meshes." Journal of Computational Physics 181, no. 1 (2002): 155-185

    2002

  27. [35]

    A spectral difference lattice Boltzmann method for solution of inviscid compressible flows on structured grids

    Hejranfar, Kazem, and Ali Ghaffarian. "A spectral difference lattice Boltzmann method for solution of inviscid compressible flows on structured grids." Computers & Mathe- matics with Applications 72, no. 5 (2016): 1341-1368

    2016

  28. [36]

    Galerkin approach for estimating boundary data in Poisson equation on annular domain with application to heat transfer coefficient estimation in coiled tubes

    Bazán, Fermin SV, and J. R. Quiroz. "Galerkin approach for estimating boundary data in Poisson equation on annular domain with application to heat transfer coefficient estimation in coiled tubes." Numerical Algorithms 81, no. 1 (2019): 79-98

    2019

  29. [37]

    Chebyshev spectral neural networks for solving partial differential equations

    Yin, Pengsong, Shuo Ling, and Wenjun Ying. "Chebyshev spectral neural networks for solving partial differential equations." arXiv preprint arXiv:2407.03347 (2024)

    2024 arXiv

  30. [38]

    Solution to the Problem of Convective Heat Transfer in a Toroidal Channel Using a Curvilinear Coordinate System

    Razuvanov, N. G., E. A. Belavina, O. N. Polyanskaya, and I. A. Belayev. "Solution to the Problem of Convective Heat Transfer in a Toroidal Channel Using a Curvilinear Coordinate System." Thermal Engineering 69, no. 8 (2022): 585-595. 26

    2022

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Reviewed August 10, 2026 · model on record in the stance chip above.