{"id":"183c5134-1ea2-4dec-b4c2-dfc4e349b90e","arxiv_id":"2607.20378","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":5,"one_line_summary":"PG-KINN pairs a KAN trial space with a Petrov–Galerkin test space for forward and inverse PDEs, but the inverse benchmark data is inconsistent with the governing equation and the accuracy claims are not supported by the paper's own tables.","lead":"PG-KINN combines Kolmogorov–Arnold Networks with a Petrov–Galerkin weak form to solve forward and inverse PDEs, reporting accuracy gains over MLP- and KAN-based baselines. The paper's inverse benchmark appears to use manufactured data that does not satisfy the governing equation, and its own tables contradict the 'consistently outperforms' claim.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Inverse conductivity benchmark in Eq. (33) is inconsistent: T=cos(15πxy) and f=(15π)^2(x^2+y^2) k T do not satisfy -∇·(k∇T)=f unless k is constant, so the inverse results are built on a manufactured non-solution.","rationale":"The stress test confirms the reader's central objection. The inverse experiment is critical because the paper's novelty includes 'an inverse formulation whose residual-based loss we prove to be free of the zero-modulus collapse' and the abstract claims robust identification. But Eq. 33 does not satisfy the PDE for nonconstant k; the missing advection-like term −∇k·∇T is of the same order as the retained term for textured coefficients. This invalidates Table 4 and Proposition 1's applicability to that experiment. The concrete test is a simple differentiation check and a rerun with a consistent f. I agree with the reader's weakest assumption on this point. Additional supporting evidence: Table 2 shows PIKAN_BINN (0.00023) outperforms PG-KINN (0.00035) on the plate displacement, and Table 5 lists PG-KINN and PIKAN_BINN both at 0.00473 on the flower while the text says PG-KINN 'does not match' PIKAN_BINN; these internal contradictions further undermine the 'consistently outperforms' claim, but the inverse inconsistency is the most load-bearing single concern. No code or error bars are provided, so independent reproduction is impossible. The verdict should remain REJECT (i.e., UNCHANGED relative to the reader). The check is feasible by any reviewer with automatic differentiation.","tokens_in":16337,"tokens_out":4486,"duration_ms":36353,"concrete_test":"Evaluate the strong-form residual r(x,y) = −∇·(k∇T) − f for Eq. (33) with a nonconstant k, e.g., k = 1 + x + y or one of the three textures. With T and f as defined, r = −∇k·∇T, which is O(|∇k|) and not zero. If the benchmark instead replaces f with the exact −∇·(k∇T) and reruns the inverse training, the reported errors and rankings in Table 4 will change. This single check determines whether the inverse results are valid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In §5.3.1 the manufactured inverse benchmark defines T(x,y)=cos(15πxy) and f=[(15πx)^2+(15πy)^2] T k (Eq. 33), claiming this solves −∇·(k∇T)=f for a textured, spatially varying k. The identity is false. Direct differentiation gives ΔT = −(15π)^2(x^2+y^2)T, hence −∇·(k∇T) = −kΔT − ∇k·∇T = (15π)^2(x^2+y^2)kT − ∇k·∇T. Eq. 33 omits the −∇k·∇T term. For any nonconstant k (the Picasso/Caspar/Van Gogh textures are strongly nonconstant), the residual −∇·(k∇T)−f = −∇k·∇T is not zero. Thus the 'measured' field T is not a solution of the stated forward problem, the premise of Proposition 1 ('exact data generated by the true coefficient κ⋆') does not hold, and the inverse loss at κ⋆ is not zero. Table 4's inverse errors and the claim of collapse-free recovery of the conductivity field from the PDE are therefore not supported by this experiment. This is the most load-bearing defect because the paper's central contribution includes a collapse-free inverse formulation; if the only inverse benchmark is inconsistent, the inverse claim rests on invalid data.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes PG-KINN, a Petrov-Galerkin weak-form PDE solver in which the trial space is a Kolmogorov-Arnold Network and the test space is a fixed, compactly supported piecewise-polynomial space. It derives the weak form for general second-order operators, gives forward and inverse algorithms, proves that the inverse residual loss does not have the trivial zero-coefficient minimizer that afflicts energy formulations, and reports numerical results on crack singularity, stress concentration, hyperelasticity, inverse identification, and complex geometry benchmarks. The weak-form derivation is standard and the collapse-freeness proposition is elementary, but the inverse benchmark in Eq. (33) uses manufactured data that do not satisfy the stated PDE for nonconstant coefficients, and several numerical claims are contradicted by the paper's own tables. As presented, the central claims are not sufficiently supported.","tokens_in":16752,"tokens_out":8579,"duration_ms":70895,"significance":"The collapse-freeness observation in Proposition 1 is a useful conceptual point for physics-informed inverse problems, and the combination of a KAN trial space with an independent polynomial test space is a plausible design. The forward benchmarks demonstrate reasonable accuracy in several settings. However, the only inverse experiment is invalid because the manufactured data are inconsistent with the PDE, the empirical claim of consistent outperformance is refuted by Table 2, and no error bars or repeated runs are reported. If the inverse experiment is corrected and the claims are calibrated, the paper could make a useful contribution. As it stands, the contribution is not established beyond the forward examples.","major_comments":[{"comment":"The manufactured inverse benchmark is internally inconsistent. For T=cos(15πxy), ΔT=-(15π)^2(x^2+y^2)T, so -∇·(k∇T)=-kΔT-∇k·∇T=(15π)^2(x^2+y^2)kT-∇k·∇T. The prescribed f=[(15πx)^2+(15πy)^2]T k omits the -∇k·∇T term. For the textured, strongly nonconstant k fields, T is not a solution of the stated forward problem; the residual at the true coefficient is -∇k·∇T ≠ 0, so Proposition 1's exact-data premise is not met. Consequently Table 4 and the claim that the PDE inverse loss recovers the conductivity field are unsupported. Fix by defining f=-∇·(k∇T) explicitly, including the ∇k term, and rerunning the experiment.","section":"§5.3.1, Eq. (33)"},{"comment":"The abstract and Section 1 claim that PG-KINN 'consistently outperforms' PIKAN. Table 2 shows PIKAN_BINN with displacement L2=2.3e-4 versus PG-KINN 3.5e-4, so PG-KINN is worse on that metric. Table 5 shows PG-KINN (flower) and PIKAN_BINN (flower) with identical L2=4.73e-3, yet Section 6 states PG-KINN 'does not match PIKAN_BINN on the flower problem.' The blanket superiority claim and the conclusion need to be corrected to match the reported numbers.","section":"Abstract / Tables 2 and 5"},{"comment":"All numerical results are single point estimates with no standard deviations, repeated seeds, or significance tests. Given that the paper's central empirical claim is 'consistent' outperformance, the absence of any variability measure is a load-bearing omission; a single run could be favorable or unfavorable by chance. At minimum, report multiple independent initializations and error bars for each benchmark.","section":"§5, Tables 1–5"},{"comment":"The uniqueness half of Proposition 1 is conditional on an identifiability assumption ('if the test space is rich enough') that is neither formulated precisely nor proved. The paper gives no conditions on the test basis, the measured field u, or the quadrature under which the weak residuals uniquely determine κ. Thus the collapse-freeness claim (zero coefficient is not a minimizer) is established, but the stronger statement that PG-KINN identifies the true coefficient is not. This should be stated as a limitation or proved in the revision.","section":"§4.2, Proposition 1"}],"minor_comments":[{"comment":"The claim that Gauss-Legendre quadrature integrates the weak residual 'to near machine precision' because the KAN trial field is piecewise-polynomial is not justified. Equation (2) includes a SiLU residual path and atanh nonlinearities, so the KAN output is not piecewise-polynomial on the fixed background grid; only the individual B-spline edge maps are piecewise-polynomial. This does not invalidate the method, but the stated mechanism for quadrature accuracy is overclaimed.","section":"§4.4, Eq. (25)"},{"comment":"PIKAN_BINN uses grid range [0,20] and a deeper architecture [2,5,5,5,2], while PG-KINN uses grid range [0,1]. The comparison is not fully controlled; the authors should state whether these choices affect the reported accuracy.","section":"Table 2"},{"comment":"No code or data availability statement is provided. For reproducibility, especially given the absence of error bars, the authors should release code and seeds. Minor textual issues include the typo 'Marchine Learning' in reference [41] and the use of 'PIKAN' to denote several distinct formulations.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The inverse benchmark defect in Eq. (33) is the most serious issue: it invalidates the only inverse identification experiment. The authors should be asked to correct f to include the ∇k·∇T term and re-run the experiment, and to soften the blanket outperformance claim to match Table 2. The forward results and the collapse-freeness argument are plausible, so I recommend major revision rather than rejection, but the revised version must contain a valid inverse benchmark and honest accuracy statements."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"If you read one section, read 5.3.1. The inverse conductivity benchmark is set up with a manufactured couple (T, f) that does not satisfy the PDE for spatially varying k. Direct differentiation: ΔT = -(15π)^2(x^2+y^2)T, so -∇·(k∇T) - f = -∇k·∇T, which is nonzero for any textured k. The paper then 'identifies' k from data that are not a solution of the forward problem, so Table 4's inverse errors and the collapse-freeness claims for inverse recovery rest on invalid input. The stress-test note is right; the reader's weakest_assumption is also right.\n\nWhat is genuinely new: pairing a KAN trial space with an independent piecewise-polynomial test space in a Petrov-Galerkin weak form, plus the simple observation that the residual-based inverse loss does not have the zero-modulus minimizer of energy methods. The collapse-freeness proof is a short linear-algebra argument and it is correct for the trivial coefficient; the uniqueness part is honestly conditional on identifiability. The forward machinery — Gauss-Legendre quadrature on a background partition, local test functions, first-order weak form — is standard but competently assembled. The paper also reports a limitation on fractal geometries and uses a fair set of baselines.\n\nThe soft spots are real. The abstract claims 'consistently outperforms' but Table 2 shows PIKAN_BINN has a lower displacement error (0.00023 vs 0.00035) on the plate with a hole. The inverse experiment is not just noisy; it is inconsistent. There are no error bars and no code, so it is hard to know whether the single-run numbers are meaningful. That said, the forward experiments (crack, hole, cantilever, Koch) do not appear to have the same structural defect; those results may hold up.\n\nThis paper is useful for someone tracking KAN-based PDE solvers, and the forward formulation seems worth building on. But as it stands, the central inverse claim is unsupported and the headline accuracy claim is overstated.\n\nFor peer review: I would send it to referees, expecting major revision. The flaws are fixable (re-run the inverse bench with a consistent f computed from the true k; add error bars; soften the 'consistently' language). Desk-rejecting would lose a potentially useful piece of incremental engineering. The authors seem capable of fixing it.","headline":"A promising forward formulation undermined by an invalid inverse benchmark and overstated accuracy claims.","tokens_in":17212,"tokens_out":3456,"would_cite":false,"duration_ms":27628,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65N30","65N21","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"PG-KINN claims a Petrov–Galerkin weak-form KAN that beats MLP and strong-form baselines on forward and inverse PDEs, with a provably collapse-free inverse loss.","keywords":["physics-informed neural networks","Kolmogorov-Arnold Networks","Petrov-Galerkin method","weak-form PDE solvers","inverse problems","parameter identification","Gauss-Legendre quadrature","computational mechanics"],"falsifier":"Take the three textured conductivity fields (Picasso, Caspar, Van Gogh) used in the inverse benchmark, compute −∇·(k∇T) numerically for T=cos(15πxy), and compare to the stated f = cos(15πxy)[(15πx)²+(15πy)²]k. If the residual is nonzero, the benchmark does not test the same forward problem; then recompute the inverse experiment with a consistent f, and check whether PG-KINN still recovers k at the reported error levels and maintains a nonzero loss for κ=0.","tokens_in":16293,"feed_emoji":"⚙️","tokens_out":5367,"duration_ms":42798,"temperature":0.7,"pith_summary":"The paper tries to show that the formulation of a PDE into a loss is as important as the network architecture, and that the right formulation can be built from a Kolmogorov–Arnold Network (KAN) trial space paired with an independent, compactly supported piecewise-polynomial test space in a Petrov–Galerkin weak form. It argues this combination lowers the derivative order, applies to general non-self-adjoint and nonlinear operators, and—crucially for inverse problems—produces a residual-based loss that cannot collapse to a trivial zero material coefficient. Across benchmarks on crack singularities, stress concentration, hyperelasticity, heterogeneous conductivity identification, and moderately complex geometries, the paper reports that PG-KINN consistently outperforms legacy MLP baselines and state-of-the-art KAN-based strong/energy/inverse formulations. The one admitted limitation is extreme non-rectangular geometries, where the axis-aligned spline grid becomes a bottleneck. The central proof establishes that the inverse loss is affine in the unknown coefficient, making the trivial zero field a non-minimizer, in contrast to energy-based inverse losses that drive the modulus to zero.","feed_headline":"Weak-form KAN beats MLP baselines; inverse loss can't collapse","feed_subtitle":"Independent polynomial test functions keep derivative order low and block the zero-modulus failure of energy-based inverse solvers.","key_machinery":"The key machinery is the Petrov–Galerkin weighted-residual form: a KAN trial space u_h(x;θ) and an independent test space spanned by piecewise polynomials v_k on a background partition, with all integrals mapped to the reference element and integrated by Gauss–Legendre quadrature. For an operator like −∇·(κ∇u)+b·∇u+cu=f, integration by parts transfers one derivative onto the test function, leaving only first derivatives of the KAN trial field. For inverse problems, the weak residual R_k(θκ) = ∫ κ_h ∇u·∇v_k dΩ − ∫Γt t̄ v_k dΓ − ∫ f v_k dΩ is affine in the unknown coefficient κ_h; this affine structure is what makes the inverse loss a genuine linear least-squares residual and blocks the zero-m","core_discovery":"The central discovery is the Petrov–Galerkin coupling itself: the trial space is a KAN with B-spline activations, and the test space is a fixed set of compactly supported piecewise-polynomial functions (hat and bubble functions) evaluated via Gauss–Legendre quadrature. Integration by parts lowers the differentiation order to first derivatives, the local test functions turn a single global residual into a well-conditioned vector of element-wise residuals, and for inverse problems the residual is affine in the unknown material field. The paper proves that this affine structure yields an inverse loss whose global minimizer is the true coefficient whenever the test space is rich enough to identi","pith_inferences":["The manufactured inverse benchmark in Eq. (33) may not actually satisfy the stated PDE for spatially varying k: the right-hand side f is chosen assuming constant k, so re-running the inverse experiment with a consistent f (computed numerically for each texture) would clarify whether the reported conductivity recovery is genuine.","The uniqueness claim in Proposition 1 depends on an identifiability condition—that the test space determines the coefficient from the residuals—which the paper states but does not verify; a natural extension is a quantitative identifiability bound or noise-stability analysis.","The admitted bottleneck on extreme geometries is specific to the axis-aligned spline grid; coupling PG-KINN with conformal or isoparametric mappings, or using adaptive h-refinement of the background partition, is a concrete next step that could extend the method to fractal boundaries.","Because the inverse loss is affine in the coefficient, the same collapse-free property should hold for any operator where the unknown appears linearly (e.g., Lamé parameters in elasticity), which could be tested directly on elastography problems."],"forward_implications":["Physics-informed solvers can be built for operators that lack an energy functional or a known fundamental solution, since the Petrov–Galerkin weak form only needs first derivatives and localized integration.","Inverse material identification becomes a well-posed least-squares problem: the loss is affine in the unknown field, and the trivial zero coefficient is provably not a minimizer as long as source or flux data are present.","Gauss–Legendre quadrature on piecewise-polynomial integrands reduces integration error by an order of magnitude compared with Monte-Carlo sampling used in energy-form baselines, improving accuracy on nonlinear hyperelasticity benchmarks.","The structural kinship between KAN splines and finite-element shape functions (the paper shows a linear KAN edge is exactly a linear FEM expansion) offers a path to transfer FEM analysis tools into neural PDE solvers.","The residual-based inverse loss could be applied to other parameter-identification problems in elasticity and beyond, wherever the unknown coefficient enters linearly in the weak form."],"fun_headline_variants":["Petrov-Galerkin KAN tames PDEs with piecewise-polynomial tests","KAN trial, polynomial test: PG-KINN solves forward/inverse PDEs","Weak-form KAN lowers derivative order, beats PIKAN on hard PDEs","PG-KINN: Petrov-Galerkin KAN with local polynomial tests for PDEs","Inverse PDEs solved by KAN with compact-support test functions"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The inverse conductivity experiment assumes that the manufactured data T=cos(15πxy) and f satisfy −∇·(k∇T)=f for a spatially varying, textured k, but this holds only for constant k; the uniqueness claim also assumes an unproved identifiability condition on the test space.","fun_headline_variants_meta":{"raw":{"variants":["Petrov-Galerkin KAN tames PDEs with piecewise-polynomial tests","KAN trial, polynomial test: PG-KINN solves forward/inverse PDEs","Weak-form KAN lowers derivative order, beats PIKAN on hard PDEs","PG-KINN: Petrov-Galerkin KAN with local polynomial tests for PDEs","Inverse PDEs solved by KAN with compact-support test functions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000268,"raw_usage":{"total_tokens":1510,"prompt_tokens":858,"completion_tokens":652,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":602,"completion_tokens_details":{"reasoning_tokens":544}},"tokens_in":602,"tokens_out":652,"duration_ms":5494,"temperature":1.0,"reasoning_tokens":544,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T09:59:19.890733+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the three textured conductivity fields (Picasso, Caspar, Van Gogh) used in the inverse benchmark, compute −∇·(k∇T) numerically for T=cos(15πxy), and compare to the stated f = cos(15πxy)[(15πx)²+(15πy)²]k. If the residual is nonzero, the benchmark does not test the same forward problem; then recompute the inverse experiment with a consistent f, and check whether PG-KINN still recovers k at the reported error levels and maintains a nonzero loss for κ=0.","supporting_citations":[],"review_version":1}