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REVIEW 4 major objections 3 minor 48 references

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.

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2026-08-01 09:59 UTC pith:U4XUR25A

load-bearing objection A promising forward formulation undermined by an invalid inverse benchmark and overstated accuracy claims. the 4 major comments →

arxiv 2607.20378 v1 pith:U4XUR25A submitted 2026-07-22 cs.LG cs.NAmath.NA

PG-KINN: A Physics-Informed Petrov-Galerkin Kolmogorov-Arnold Network for Solving Forward and Inverse PDEs

Amirhossein Sadr , Nima Soltani , Vahideh Moghtadaiee , Aida Pakniyat , Dara Rahmati , Saeid Gorgin This is my paper
classification cs.LG cs.NAmath.NA MSC 65N3065N2168T07
keywords physics-informed neural networksKolmogorov-Arnold NetworksPetrov-Galerkin methodweak-form PDE solversinverse problemsparameter identificationGauss-Legendre quadraturecomputational mechanics
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

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.

Core claim

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

What carries the argument

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

Load-bearing premise

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.

What would settle it

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.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • 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.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • 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.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 3 minor

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.

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 (4)
  1. [§5.3.1, Eq. (33)] 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.
  2. [Abstract / Tables 2 and 5] 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.
  3. [§5, Tables 1–5] 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.
  4. [§4.2, Proposition 1] 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.
minor comments (3)
  1. [§4.4, Eq. (25)] 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.
  2. [Table 2] 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.
  3. [General] 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.

Circularity Check

0 steps flagged

No significant circularity: the weak-form derivation is direct and the inverse-loss proof is conditional on an explicit identifiability assumption; the main defects are benchmark inconsistency and omitted uniqueness proof, not circularity.

full rationale

The derivation chain is self-contained. Section 4.1 obtains Eq. (16) from Eq. (14) by the divergence theorem; no target result is assumed. Section 3.4/4.2's collapse-freeness follows from the affine residual Eq. (19): R_k is linear in kappa_h plus a data constant ell_k, so kappa_h = 0 gives L = (1/M) sum ell_k^2 > 0. Proposition 1 explicitly states its identifiability condition ('if the test space is rich enough that {v_k} determines kappa...'), so uniqueness is a conditional statement, not an imported conclusion. The benchmark tables are empirical comparisons; no fitted parameter is renamed as a prediction, and no load-bearing self-citation appears. Two non-circular caveats are worth flagging. (i) In Sec. 5.3.1, Eq. (33) states T = cos(15 pi x y) and f = cos(15 pi x y)[(15 pi x)^2 + (15 pi y)^2] k(x,y), but direct differentiation gives -div(k grad T) = [(15 pi x)^2 + (15 pi y)^2] k T - grad k dot grad T, so the stated f omits the -grad k dot grad T term. For nonconstant k, the manufactured T is not a solution of the stated forward problem, so Proposition 1's 'exact data generated by the true coefficient' premise fails for that experiment. This is a correctness defect, not a circular derivation. (ii) Proposition 1's uniqueness claim depends on an unproved identifiability condition on the test space. Both concerns affect validation strength, but neither makes a prediction equivalent to its input by construction.

Axiom & Free-Parameter Ledger

5 free parameters · 4 axioms · 0 invented entities

The central claim depends on several hand-chosen hyperparameters and on assumptions about quadrature exactness and inverse identifiability. The inverse benchmark's manufactured data is not consistent with the PDE, so the 'true' coefficient is not actually a solution of the forward model.

free parameters (5)
  • KAN grid size G = 10–15 (forward), 100 (inverse)
    Chosen per benchmark; controls the resolution of the trial field and directly affects accuracy.
  • Spline order r = 2–3
    Order of B-spline basis in the KAN, chosen per problem.
  • Test degree p_max = 2–4
    Degree of Legendre bubble functions, controls the number of test functions per cell.
  • Gauss–Legendre points n_g = 4–6
    Number of quadrature points per dimension; used in all weak-form integrals.
  • Dirichlet penalty weight λ_u = not reported
    Weight applied to the boundary loss in Eq. (9); value is not stated.
axioms (4)
  • standard math Kolmogorov–Arnold representation theorem
    Basis for representing functions with KANs, invoked in Section 3.1.
  • domain assumption Piecewise-polynomial integrands for quadrature exactness
    Section 4.4 assumes κ∇u_h·∇v_k is nearly polynomial on each cell for Gauss–Legendre exactness, but KANs use atanh activations, so the integrands are not polynomial.
  • domain assumption Identifiability of κ from the test space
    Proposition 1 proves uniqueness only if the test space is rich enough for the affine residual system to have a unique solution; this is assumed, not established.
  • domain assumption Measured data u and f are exact and consistent with the forward model
    The inverse formulation treats u and f as known; in the conductivity benchmark (Eq. 33) this consistency fails for spatially varying k.

pith-pipeline@v1.3.0-alltime-deepseek · 16145 in / 14336 out tokens · 109184 ms · 2026-08-01T09:59:19.890733+00:00 · methodology

0 comments
read the original abstract

Physics-informed learning of partial differential equations (PDEs) has been dominated by multilayer perceptrons (MLPs), whose spectral bias and dense parameterization limit both accuracy and interpretability. Kolmogorov Arnold Networks (KANs) mitigate these limitations because their learnable spline activations are structurally aligned with the piecewise-polynomial bases of classical discretizations. However, the way a PDE is cast into a loss functional is as decisive as the choice of approximator: strong-form residual minimization requires high-order derivatives and heavily weighted losses, the energy (Bubnov-Galerkin) form is restricted to self-adjoint operators and, as we show, collapses to a trivial solution for parameter-identification problems, and boundary integral forms require a known fundamental solution. We propose PG-KINN, a physics-informed KAN built on a Petrov-Galerkin formulation in which the trial space is a KAN and the test space is an independent, compactly supported, piecewise-polynomial space evaluated with Gauss-Legendre quadrature. Integration by parts lowers the differentiation order while retaining applicability to general non-self-adjoint, nonlinear, and inverse problems; the localized test functions turn the global residual into a set of element-wise weak residuals with favorable conditioning. On a suite of benchmarks spanning crack singularities, stress concentration, Neo-Hookean hyperelasticity, inverse parameter identification in heterogeneous media, and complex geometries, PG-KINN consistently outperforms legacy MLP baselines and state-of-the-art KAN-based strong/energy/inverse formulations (PIKAN). These results position the Petrov-Galerkin coupling of KAN trial spaces and polynomial test spaces as a robust and accurate route for AI-based computational mechanics.

Figures

Figures reproduced from arXiv: 2607.20378 by Aida Pakniyat, Amirhossein Sadr, Dara Rahmati, Nima Soltani, Saeid Gorgin, Vahideh Moghtadaiee.

Figure 1
Figure 1. Figure 1: The design space of physics-informed formulations. Strong, energy, and inverse forms each occupy a corner with a structural limitation. PG-KINN combines a KAN trial space with a Petrov–Galerkin test space to reach the favorable region: low derivative order, generality across operators, and robustness for inverse problems. • Localized, well-conditioned residuals. Compactly supported polynomial test function… view at source ↗
Figure 2
Figure 2. Figure 2: Petrov–Galerkin versus Bubnov–Galerkin. PG-KINN keeps the trial space (KAN) and the test space (compactly supported piecewise-polynomials) independent, so the weak form applies to general—including non-self-adjoint and nonlinear—operators, unlike the energy form in which the test space is the variation of the trial space. • In the Bubnov–Galerkin (energy) method, V is taken as the variation δu of the trial… view at source ↗
Figure 3
Figure 3. Figure 3: The PG-KINN pipeline. A KAN provides the trial field uh; an independent, compactly supported piecewise-polynomial test space {vk} is defined on a background partition; each pairing is integrated by Gauss–Legendre quadrature to form a weak residual Rk; the aggregated residual loss (plus a light boundary term) is minimized by Adam/L-BFGS, updating the KAN parameters θ. Algorithm 1 PG-KINN (Forward) 1: Input:… view at source ↗

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