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