{"id":"d5d444b7-56ad-48fb-be65-1b31895ee8d8","arxiv_id":"2608.06660","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"CT-PIKAN applies coordinate-transformed physics-informed KANs with autograd-computed metrics to solve PDEs in curvilinear domains, but lacks quantitative validation and has a metric inconsistency in its wavy-domain example.","lead":"This paper combines coordinate transformation with physics-informed Kolmogorov-Arnold networks to solve PDEs on curved domains, using automatic differentiation to compute the metric terms. The method is tested on advection, Poisson, and heat equations in polar, wavy, and star-shaped domains, but the reported evidence is mostly visual and contains an internal contradiction for the wavy geometry.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"§3.3's wavy-domain mapping (Eqs. 30–31) has non-constant metrics; Eq. 40's constant-coefficient advection equation is not the correct pullback, so the key curvilinear demo solves a different problem.","rationale":"The reader's weakest assumption is exactly the load-bearing issue. Section 3.3 is the paper's only hyperbolic curvilinear example, and it uses an incorrect constant-coefficient transformed equation. The autograd-based metric framework itself is plausible, and the availability of a public repository is a point in the paper's favor, but this benchmark as written does not test the intended wavy-domain PDE. Combined with the absence of quantitative error tables throughout the results, the central claim of accurate and general curvilinear PDE solving is not supported by the current manuscript. I agree with the reader's REJECT verdict; the error would need to be corrected and quantitative comparisons added before the claim could be accepted.","tokens_in":15192,"tokens_out":7034,"duration_ms":64783,"concrete_test":"Re-derive the metrics for the map x=ξ+0.02 sin(6πη), y=η+0.02 sin(6πξ); evaluate J and a=ξ_x+ξ_y, b=η_x+η_y on a 100×100 grid over [0,1]^2. Then rerun the wavy-domain advection case with the variable-coefficient equation ut+c[a(ξ,η)u_ξ+b(ξ,η)u_η]=0, using the same architecture and training schedule as §3.3, and compare the resulting solution with Fig. 7 and with a reference solution. If the variable-coefficient solution differs appreciably from Fig. 7, the published result was obtained from the wrong PDE. Also check the caption's 'conformal mapping' claim: the map is not conformal because x_η=6πA cos(6πη)≠−y_ξ=−6πA cos(6πξ).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claimed advantage is that autograd computes geometric terms from the user's coordinate map, eliminating analytic derivation and enabling arbitrary smooth domains. Section 3.3's wavy-domain benchmark contradicts this. The collocation map (30)–(31), with Δx=Δy=1/(N_CP−1), corresponds to x(ξ,η)=ξ+A sin(6πη), y(ξ,η)=η+A sin(6πξ) for (ξ,η)∈[0,1]^2. Hence x_ξ=1, y_η=1, but x_η=6πA cos(6πη) and y_ξ=6πA cos(6πξ). These are not constant, and J=1−(6πA)^2 cos(6πη)cos(6πξ) varies from about 0.858 to 1.142 for A=0.02. The manuscript's assertion after Eq. (37) that 'these metrics as well as the determinant of the Jacobian matrix are constant' is therefore false. Consequently Eqs. (38)–(40), which collapse the transformed advection operator to ut+c(a u_ξ+b u_η) with constant a=ξ_x+ξ_y, b=η_x+η_y, do not represent the pullback of Eq. (28) to this wavy domain. The Figure 7 results, obtained under this equation, cannot be taken as a valid CT-PIKAN solution of the intended curvilinear advection problem. This is load-bearing because the wavy advection case is the only hyperbolic demonstration on a non-Cartesian domain and is featured as evidence for the method's generality. It is also textually inconsistent with the autograd claim: if autograd were used, the metrics would have been computed pointwise and the constant-coefficient simplification would not have appeared.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":15571,"tokens_out":5361,"duration_ms":48617,"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":[{"comment":"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.","section":"§3.3, Eqs. (30)–(31) and text after Eq. (37)"},{"comment":"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.","section":"§3, throughout"},{"comment":"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.","section":"§3.5, star-domain case, text after Eq. (56)"},{"comment":"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.","section":"§3.3, Fig. 7 caption"}],"minor_comments":[{"comment":"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.","section":"§2.2, Eq. (7)"},{"comment":"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.","section":"§3.3, paragraph after Eq. (29)"},{"comment":"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.","section":"§3.3, text near Eqs. (35)–(36)"},{"comment":"The figure shows CT-PIKAN and analytical solutions but no error plot or quantitative comparison; adding an error map would strengthen the validation.","section":"§3.1, Fig. 3"},{"comment":"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'.","section":"§3.5, text after Eq. (47)"},{"comment":"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.","section":"§2.4, Algorithm 2"}],"recommendation":"major_revision","confidential_remarks":"The Section 3.3 error is load-bearing and suggests that the wavy advection experiment may not have actually been run with the autograd metric pipeline; the authors should clarify and, if necessary, re-run that case with pointwise metric evaluation. The overall lack of quantitative results is also a serious deficiency for a journal submission. However, the central autograd idea is salvageable and the errors are fixable within the manuscript's scope, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is a reasonable engineering effort with one load-bearing mathematical error that breaks its key curvilinear advection demonstration. The autograd-based metric computation is a genuinely convenient idea, and the framework is clearly described, with several benchmarks across PDE classes and public code. But Section 3.3's claim that the metrics for the wavy mapping (30)-(31) are constant is simply false: for x=ξ+A sin(6πη), y=η+A sin(6πξ), the cross derivatives and Jacobian determinant are variable. Consequently Eqs. (38)-(40) do not represent the pullback of the advection equation to that domain, and Figure 7 solves a different problem. That is load-bearing because it is the only hyperbolic test on a non-Cartesian domain and the paper leans on it as evidence of generality.\n\nThe rest of the paper is more believable. The polar advection, annular Poisson, and wavy/star heat cases use correct Laplace-Beltrami formulations via autograd, and the plots suggest stable training. Credit where due: the code is public, the methodology section is transparent, and the authors correctly note they only need the forward map.\n\nSoft spots beyond the big one: no quantitative error norms anywhere, only figures, so claims of 'improved accuracy' over standard PIKAN are not supported. In the star-shaped heat cases, the inner cutout (ξ=0) has no boundary condition imposed, which is generally ill-posed for diffusion. The abstract and introduction overstate generality given these gaps.\n\nBottom line: the core idea is salvageable and worth a serious referee, but the manuscript as written should not be accepted. The authors need to fix the wavy-domain derivation, add error tables, and clarify the star-domain boundary setup. I'd send it to review with a clear 'major revision' signal.","headline":"A useful autograd-based PIKAN framework with a load-bearing math error in the wavy-domain advection demo that invalidates that specific result.","tokens_in":16069,"tokens_out":2924,"would_cite":false,"duration_ms":28844,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65N35","65M70","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"CT-PIKAN computes all geometric terms of a coordinate-transformed PDE solver automatically, eliminating hand-derived Jacobians and metric tensors for curved domains.","keywords":["Physics-Informed Kolmogorov-Arnold Network","Coordinate transformation","Curvilinear domains","Automatic differentiation","Metric tensor","Laplace-Beltrami operator","Partial differential equations","Physics-informed learning"],"falsifier":"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).","tokens_in":15007,"feed_emoji":"📐","tokens_out":8846,"duration_ms":76395,"temperature":0.7,"pith_summary":"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.","feed_headline":"Autograd metrics take physics-informed KANs to curved domains","feed_subtitle":"Autograd extracts the geometry terms, so curved PDE domains reduce to square training grids.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines the physics-informed neural network loss formulation that CT-PIKAN extends.","marker":"[7]"},{"why":"Introduces Kolmogorov-Arnold Networks with B-spline activations, the architecture underlying PIKAN.","marker":"[14]"},{"why":"Presents the physics-informed KAN (PIKAN) baseline that this work extends to curvilinear coordinates.","marker":"[15]"},{"why":"Represents coordinate-transformed PINNs that require hand-derived metric coefficients, the limitation the paper removes.","marker":"[23]"},{"why":"Supplies the automatic differentiation machinery used to compute Jacobians and metric tensors from the coordinate map.","marker":"[24]"},{"why":"Provides the curvilinear-coordinate identities, including the Laplace-Beltrami operator, used in the transformed PDEs.","marker":"[27]"},{"why":"Defines the sinusoidal wavy point distribution used to generate the curvilinear advection geometry.","marker":"[33–35]"}],"fun_headline_variants":["No more metric derivations: CT-PIKAN autograds curved PDEs","Autograd computes geometry, PIKAN solves curved PDEs","Curved PDEs without metric headaches: CT-PIKAN uses autograd","PIKAN meets curved domains via autograd geometry","Autograd eliminates PDE metric derivations in CT-PIKAN"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["No more metric derivations: CT-PIKAN autograds curved PDEs","Autograd computes geometry, PIKAN solves curved PDEs","Curved PDEs without metric headaches: CT-PIKAN uses autograd","PIKAN meets curved domains via autograd geometry","Autograd eliminates PDE metric derivations in CT-PIKAN"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000609,"raw_usage":{"total_tokens":2852,"prompt_tokens":981,"completion_tokens":1871,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":597,"completion_tokens_details":{"reasoning_tokens":1790}},"tokens_in":597,"tokens_out":1871,"duration_ms":13072,"temperature":1.0,"reasoning_tokens":1790,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:58:10.122875+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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).","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the physics-informed neural network loss formulation that CT-PIKAN extends."},{"cited_title":"Y., & Tegmark, M","cited_arxiv_id":null,"evidence_quote":"Introduces Kolmogorov-Arnold Networks with B-spline activations, the architecture underlying PIKAN."},{"cited_title":"S., Zhuang, X., Rabczuk, T., & Liu, Y","cited_arxiv_id":null,"evidence_quote":"Presents the physics-informed KAN (PIKAN) baseline that this work extends to curvilinear coordinates."},{"cited_title":"S., Son, S., Kim, Y., & Seok, J","cited_arxiv_id":null,"evidence_quote":"Represents coordinate-transformed PINNs that require hand-derived metric coefficients, the limitation the paper removes."},{"cited_title":"Pearlmutter, Alexey Andreyevich Radul, and Jeffrey Mark Siskind","cited_arxiv_id":null,"evidence_quote":"Supplies the automatic differentiation machinery used to compute Jacobians and metric tensors from the coordinate map."},{"cited_title":"Nodal integral methods for heat and mass transfer in curvilinear coordinates","cited_arxiv_id":null,"evidence_quote":"Provides the curvilinear-coordinate identities, including the Laplace-Beltrami operator, used in the transformed PDEs."}],"review_version":1}