{"id":"29c34109-7069-4d8a-b310-76b4cbe14d68","arxiv_id":"2507.22678","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A boundary-only holomorphic KAN framework solves Laplace, Helmholtz, and elasticity problems on simply and multiply connected 2D domains, outperforming standard PINNs in the tested cases.","lead":"The paper introduces a holomorphic Kolmogorov-Arnold network that solves 2D elliptic PDEs by training only on the boundary, and extends the approach to Helmholtz problems and to domains with holes. On the tested benchmarks it beats both classic physics-informed networks and the earlier holomorphic MLP in accuracy for similar or smaller network sizes.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The multiply-connected extension rests on an unsupported transfer of Axler's harmonic logarithmic-conjugation theorem to all Table 1 potentials; for the biharmonic row the statement is false, since r^2 log r on an annulus is not representable by the proposed Laurent-plus-log ansatz.","rationale":"The reader's weakest assumption identifies exactly the load-bearing risk: Section 2.4 extends a harmonic-function theorem to all four rows of Table 1 without proof, and the only multiply-connected experiment does not exercise logarithmic terms. My analysis strengthens this from 'unproved' to 'false for the biharmonic row,' because the annulus solution r^2 log r is biharmonic yet cannot lie in the span of the proposed Taylor/Laurent/log ansatz. That directly undercuts the abstract's claim of removing the simply-connected limitation for arbitrary 2D elliptic problems. It does not invalidate the paper's other contributions: the PIHKAN architecture, the initialization scheme, and the simply-connected Laplace and Helmholtz benchmarks are independent and appear sound. The paper is already CONDITIONAL, and the appropriate response remains CONDITIONAL: require either a correct proof or a restricted statement of the multiply-connected extension, plus at least one nonsymmetric or biharmonic multiply-connected test.","tokens_in":22316,"tokens_out":18006,"duration_ms":235745,"concrete_test":"Implement the biharmonic benchmark on the annulus 1<|z|<2 with exact solution u=r^2 log r, prescribing u and ∂u/∂n on both boundary circles. Use the Section 2.4 Laurent PIHKAN with z_s=0, following the architecture of Section 3.3, varying polynomial degree P=4,6,8 and several seeds. If the relative L2 error saturates at O(10^-2) or above while a control problem with representable solution (e.g., u=1+|z|^2) reaches O(10^-3) on the same geometry, then Eq. (15) fails to extend the biharmonic representation to multiply connected domains.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the sentence in Section 2.4: 'Since all representations in Table 1 stem from the holomorphic characterization of harmonic functions, we conclude that applying the transformation in Equation (15) to each complex potential is sufficient to extend these representations to multiply connected domains.' Axler's Theorem 1 is a statement about real harmonic functions; it does not automatically transfer to the biharmonic combination Re(\\bar z φ1 + φ2), to the Kolosov-Muskhelishvili formulas with derivatives, or to the Vekua operator. The biharmonic row is not merely unproved but incorrect as stated. On the annulus 1<|z|<2, u(r)=r^2 log r is biharmonic. With the proposed ansatz φ_n(z)=φ_{n,0}(z)+φ_{n,1}(1/z)+c_n log z (z_s=0), the Taylor and Laurent parts generate only radial modes r^2 times constants and constants, while the logarithmic part contributes only log r from φ_2 or the multivalued term Re(\\bar z c_1 log z)=r log r cosθ+r θ sinθ from φ_1. No combination, even with infinite Laurent series, produces the radial mode r^2 log r, so the exact solution lies outside the model class. The same unsupported transfer affects elasticity and Helmholtz; for Helmholtz, the Vekua integral in Eq. (3) also assumes star-shapedness around the origin and is not analyzed when the origin lies inside a hole. The only multiply-connected numerical test is the symmetric centered-hole elasticity benchmark, whose exact potentials are single-valued and need no logarithmic terms, so it cannot validate Eq. (15). The paper itself concedes in the conclusion that 'little is known on Vekua operators... whether the map is surjective,' which reinforces that this extension is asserted rather than established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes three extensions to physics-informed holomorphic neural networks (PIHNNs): a Kolmogorov-Arnold variant called PIHKAN based on monomial (polynomial) activation functions; a Vekua-operator treatment of the 2D Helmholtz equation; and a Laurent-series/logarithmic-conjugation ansatz, Eq. (15)-(16), intended to remove the restriction of holomorphic networks to simply-connected domains. Numerical tests are reported for the Poisson/Laplace problem on an L-shaped domain, the Helmholtz equation on a square, and linear elasticity on a square with a circular hole, comparing PIHKAN, PIHNN, classical PINN, and FEM. The central claim is that holomorphic networks can now solve Laplace, biharmonic, elasticity, and Helmholtz problems on arbitrary multiply-connected 2D domains with boundary-only training and high accuracy.","tokens_in":22641,"tokens_out":13078,"duration_ms":159402,"significance":"If the mathematical claims were fully supported, the paper would make a valuable contribution to physics-informed machine learning: boundary-only training, hard PDE satisfaction, sharply smaller parameter counts, and an alternative to domain decomposition for multiply-connected geometries. The numerical study is reported with concrete hyperparameters, parameter counts, and FEM reference solutions, and the PIHKAN results on the three benchmarks show consistently lower relative L2 errors than the compared PINN/domain-decomposition baselines; the ablation study in Table 3 and the RAD experiment in Table 4 are useful additions. However, the multiply-connected extension -- the paper's headline novelty -- is not established for the biharmonic and Helmholtz rows of Table 1, and the single multiply-connected test does not exercise the logarithmic-conjugation terms at all. The central claim is therefore stronger than the evidence and mathematics in the manuscript support.","major_comments":[{"comment":"The assertion that applying Eq. (15) to each complex potential is sufficient to extend all representations in Table 1 to multiply-connected domains is false for the biharmonic equation. On the annulus 1<|z|<2, u(r)=r^2 log r is biharmonic. With z_s=0, the ansatz (16) generates terms of the form Re(zbar z^n) (radial only for n=1, giving r^2), Re(zbar z^{-k}) for k>=1 (all carrying angular factors e^{-i(k+1)theta}), and logarithmic terms whose real parts are log r or r theta sin theta; no combination, even with infinite Laurent series, produces the radial mode r^2 log r. The exact solution therefore lies outside the model class. A correct extension needs additional multivalued potentials (for example z log z in phi_1) or a separate biharmonic logarithmic-conjugation theorem with proof; the sentence 'Since all representations in Table 1 stem from the holomorphic characterization of harmonic functions' does not supply this.","section":"Section 2.4, Eq. (16) and Table 1 (biharmonic row)"},{"comment":"For the Helmholtz equation, the claimed transfer to multiply-connected domains is unsupported. The Vekua operator in Eq. (3) integrates g(tx) along the segment from 0 to x; if the domain is not star-shaped with respect to the chosen origin, or if the origin lies inside a hole, the integral samples points outside Omega and the formula is not defined. The paper includes no multiply-connected Helmholtz test and no modification of the Vekua operator for such domains. Moreover, Eq. (2) requires the image of F* to equal the kernel of F, so the Helmholtz representation needs a concrete surjectivity statement; Section 4 itself notes that 'little is known' about whether the Vekua map is surjective. The authors should either state the precise hypotheses from [44] under which surjectivity holds or explicitly restrict the Helmholtz claim to simply-connected (e.g., star-shaped) domains.","section":"Section 2.4, Eq. (3) (Helmholtz row)"},{"comment":"The only multiply-connected numerical experiment, the centered-hole elasticity benchmark, does not validate the logarithmic-conjugation part of the proposed method. For this symmetric problem with a self-equilibrated traction system, the Kolosov-Muskhelishvili potentials are single-valued and no log(z-z_s) term is active; the Laurent ansatz is exercised only through the negative powers 1/(z-z_1). Table 6 therefore demonstrates that negative powers help here, but it gives no evidence that Eq. (15) transfers Axler's theorem to elasticity, let alone to the biharmonic and Helmholtz rows. A test in which log terms are genuinely required (for plate bending with a hole, or an elasticity problem with nonzero resultant force) would be needed to support the central claim.","section":"Section 3.3, Table 6"}],"minor_comments":[{"comment":"The caption of Table 5 says 'comparison between the different methods employed in Section 3.1', but the table reports the Helmholtz experiment of Section 3.2.","section":"Table 5 caption"},{"comment":"In the displayed Kolosov-Muskhelishvili formulas, the complex-conjugate bars appear to be missing: the standard displacement formula has kappa phi_1 - z \\overline{phi_1'} - \\overline{phi_2}, not kappa phi_1 - z phi_1' - phi_2. If the bars are present in the typeset version, the plain-text rendering is misleading; if they are absent, the formulas are incorrect.","section":"Table 1 and Section 3.3"},{"comment":"The phrase that z_s 'should be chosen as the most internal point of B_s' is ambiguous for non-circular holes; the operative mathematical requirement is that the Laurent expansion with center z_s converges on the annular region between the hole and the outer boundary, which depends on the distance to the boundary of the hole, not just on a point selected inside it.","section":"Section 2.4, Eq. (16)"},{"comment":"The paper gives no universal-approximation statement for PIHKANs. For the method to be a convergent solver as network size grows, the ansatz class must be dense in the relevant holomorphic function spaces; for polynomial activations this can presumably be obtained from polynomial/Runge approximation, but the assumption is not stated in the manuscript.","section":"Section 2.3"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is likely suitable for the journal once the multiply-connected claims are corrected or explicitly restricted. The numerical contributions are solid, but the false biharmonic statement in Section 2.4 and the unproved Helmholtz transfer are central to the advertised 'arbitrary 2D domains' contribution, so they should be resolved before acceptance. No issues with citation practices or novelty disclosure were apparent."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know up front. First, the PIHKAN architecture is real and the Helmholtz results are the first sensible holomorphic-network treatment I've seen; the benchmarks are clean and the accuracy gains over PIHNN/PINN are believable. Second, the headline claim about multiply-connected domains is too strong. The stress-test note is right: Section 2.4 takes Axler's logarithmic-conjugation theorem, which is a statement about harmonic functions, and applies it without proof to every row of Table 1. That doesn't work. On an annulus, u = r^2 log r is biharmonic, but the proposed ansatz phi_n = phi_{n,0} + phi_{n,1}(1/z) + c_n log z generates only radial modes like r^2, 1, log r, and angle-weighted terms like r log r e^{i theta}; it cannot produce r^2 log r. So the biharmonic representation is incomplete, not merely unproved. The same transfer is suspect for elasticity and Helmholtz, where the Vekua integral also assumes star-shapedness around the expansion center. The only multiply-connected test, a centered-hole Kirsch-like elasticity problem, has single-valued potentials with no log terms, so it doesn't exercise the new logarithmic terms at all.\n\nWhat earns credit: the monomial-basis KAN with variance-derived initialization is a sensible way to keep activations holomorphic and bounded; the weight-init analysis is straightforward and useful. The Vekua-operator route to Helmholtz is new and the square-domain test shows a real accuracy/parameter improvement over PIHNN and PINN. The paper is also honest, conceding in the conclusion that Vekua surjectivity is unknown. No code was shipped, which is a minor negative given the 'available on request' data statement.\n\nNet: the core PIHKAN and Helmholtz material is solid and worth building on. The multiply-connected section needs re-scoping or a real proof per problem; as written, 'removing the previous limitation' is false for at least the biharmonic equation. I'd send it to peer review with a request for major revision on Section 2.4 and a new numerical test that actually contains a log term, like a hole with nonzero resultant traction.","headline":"A genuinely useful PIHKAN and a clean Helmholtz demo, but the multiply-connected extension is oversold: the log-conjugation transfer to Table 1 is unproven and false for the biharmonic row.","tokens_in":23233,"tokens_out":6361,"would_cite":true,"duration_ms":69915,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that holomorphic neural networks, extended via Vekua operators and Laurent-series/logarithmic-conjugation terms, solve Laplace, biharmonic, elasticity, and Helmholtz problems on arbitrary 2D domains with boundary-only…","keywords":["physics-informed neural networks","Kolmogorov-Arnold networks","holomorphic neural networks","Laurent series","multiply-connected domains","Vekua operators","Helmholtz equation","linear elasticity"],"falsifier":"Run the Laurent-based PIHKAN on Helmholtz or biharmonic problems over an annular or multi-hole domain with non-constant boundary data and compare against a high-resolution FEM reference; if the relative $L^2(\\Omega)$ error does not converge toward the FEM solution as epochs, polynomial degree, and boundary points are increased, the claimed completeness of the log-term extension for non-Laplace problems is false.","tokens_in":22095,"feed_emoji":"📐","tokens_out":12514,"duration_ms":123794,"temperature":0.7,"pith_summary":"Physics-informed holomorphic networks solve an elliptic PDE by construction: the solution is written through holomorphic potentials, so the differential equation is satisfied identically and training only fits boundary data. This paper claims to remove three limits of that idea in one framework. A new Kolmogorov-Arnold variant, PIHKAN, uses monomial activations with variance-preserving initialization and reaches higher accuracy with fewer parameters than the earlier MLP-based PIHNN and vanilla PINNs. A Vekua operator extends the approach to the Helmholtz equation, and a Laurent-series/logarithmic-conjugation construction allows domains with holes without domain decomposition. If the claims hold, boundary-only neural surrogates become a practical option for a broad class of 2D elliptic problems, including acoustics and elasticity on perforated geometries.","feed_headline":"Holomorphic networks solve Laplace, Helmholtz, elasticity with holes","feed_subtitle":"A Laurent/log extension handles multiply connected domains while a KAN variant cuts error and parameter count.","key_machinery":"The load-bearing object is the holomorphic representation of PDE solutions: for Laplace, biharmonic, linear elasticity, and Helmholtz, the unknown field is built from holomorphic potentials by formulas such as those in the paper's Table 1, so the PDE is satisfied by construction and only boundary conditions need to be learned. The new architecture PIHKAN replaces exponential-activation MLPs with Kolmogorov-Arnold layers whose trainable activations are monomials $\\sum_{p=0}^{P} W_{p}z^{p}$, chosen because the monomials are $L^2$-orthogonal on the unit disc and on the unit circle, and it is stabilized by a variance-preserving weight initialization. The Helmholtz extension uses the Vekua operator $V_\\beta[g](x)=g(x)-\\int_0^1 \\frac{J_1(\\beta|x|\\sqrt{1-t})}{2\\sqrt{1-t}\\beta|x|}g(tx)\\,dt$, which maps harmonic functions to Helmholtz solutions. The multiply connected extension uses the logarithmic-conjugation and Laurent form $\\phi_n(z)=\\phi_{n,0}(z)+\\sum_s \\phi_{n,s}(1/(z-z_s))+\\sum_s c_{n,s}\\log(z-z_s)$, with trainable real coefficients $c_{n,s}$, so a single network plus small singular subnetworks covers each hole without domain decomposition.","core_discovery":"The central claim is that every 2D elliptic problem admitting a holomorphic potential representation — Laplace, biharmonic, linear elasticity, and now Helmholtz via the Vekua operator $V_\\beta$ — can be solved by a holomorphic neural network trained only on the boundary, and that multiply connected domains are no longer a barrier. The paper introduces PIHKAN, a KAN whose trainable activations are complex monomials $\\sum_{p=0}^{P} W_{p}z^{p}$, with an initialization that keeps variance near unity across layers, and augments each potential with terms $\\sum_{s} c_{n,s}\\log(z-z_s)$ and $\\sum_{s} \\phi_{n,s}(1/(z-z_s))$ to cover holes. In the reported tests PIHKAN reaches relative $L^2(\\Omega)$ errors of $1.76\\times10^{-2}$ on the L-shaped Laplace benchmark and $4.89\\times10^{-3}$ on a Helmholtz square with wave number $\\beta\\approx18$, versus $1.18\\times10^{-1}$ and $2.66\\times10^{-2}$ for vanilla PINNs; on the multiply connected elasticity benchmark the Laurent-based PIHKAN reduces the $\\sigma_{yy}$ error to $8.43\\times10^{-3}$, where domain-decomposition PIHNN reaches $4.70\\times10^{-2}$.","pith_inferences":["The paper validates the Laurent/log construction on only one symmetric elasticity example; if the log-completion conjecture transfers, the same one-network construction should handle asymmetric multi-hole Helmholtz and biharmonic problems, which is a direct next experiment the paper leaves open.","Because the PDE is hard-coded, inverse problems such as identifying hole location, shape, or boundary impedance from interior measurements would reduce to fitting a boundary/interior measurement loss; the paper does not explore this direction.","The monomial-basis KAN inherits polynomial approximation theory, so for smooth potentials error should decay rapidly as the polynomial degree $P$ increases; this implies $P$-refinement may be a cheaper accuracy path than adding layers, which the paper does not demonstrate.","The reported KAN-versus-MLP wall-clock gap is attributed to unoptimized evaluation of the polynomial activations; a closed-form or recurrence-based evaluation would likely close most of the gap, leaving PIHKAN both smaller and faster than PIHNN."],"forward_implications":["On all tested benchmarks the holomorphic methods match or beat FEM reference accuracy using only boundary training points, so interior collocation becomes unnecessary for these elliptic PDE classes.","PIHKAN reaches the reported accuracies with roughly half to one-third the parameters of the MLP-based PIHNN, implying similar savings should appear in any 2D problem with holomorphic potentials.","The Laurent/log construction replaces domain decomposition for multiply connected domains, so holes, cracks, and inclusions can be handled by one base network plus small subnetworks per hole.","The Vekua-operator route opens boundary-only training for wave problems, and the Helmholtz square test at $\\beta\\approx18$ shows acoustic benchmarks are within reach.","Boundary-based residual adaptive sampling cuts the Laplace error from $1.76\\times10^{-2}$ to $7.13\\times10^{-3}$ at negligible extra cost, identifying adaptive boundary collocation as a cheap accuracy lever."],"supporting_citations":[{"why":"Supplies the original physics-informed holomorphic neural network construction, its universal approximation analysis, and the multiply connected elasticity benchmark that this work extends.","marker":"[30]"},{"why":"Introduces the Kolmogorov-Arnold network architecture whose layered composition PIHKAN adapts to complex monomial activations.","marker":"[13]"},{"why":"Establishes the Vekua-type representation $V_\\beta$ that maps harmonic functions to Helmholtz solutions, providing the Helmholtz row of Table 1.","marker":"[44]"},{"why":"States the logarithmic-conjugation theorem for harmonic functions on multiply connected domains, which the paper's Laurent/log construction generalizes.","marker":"[64]"},{"why":"Defines the class of Vekua operators mapping harmonic functions to solutions of general elliptic equations, framing the Helmholtz extension.","marker":"[45]"},{"why":"Demonstrates the accuracy benefits of polynomial-activation KANs, motivating the monomial basis chosen for PIHKAN.","marker":"[24]"},{"why":"Provides the systematic MLP-versus-KAN comparison for differential equations that supports the claim of smaller KAN architectures with comparable or better accuracy.","marker":"[28]"},{"why":"Supplies the residual-based adaptive distribution resampling method that the paper adapts to boundary training points and shows to improve accuracy at negligible cost.","marker":"[72]"}],"fun_headline_variants":["Holomorphic KAN solves elliptic PDEs on holes and Helmholtz","Laurent terms make holomorphic nets handle multiply connected domains","PIHKAN: KAN architecture improves holomorphic net accuracy","Multiply connected elliptic problems yield to holomorphic networks"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method relies on the assumption that the logarithmic-conjugation correction proven for harmonic functions also completes the complex-potential representations for biharmonic, linear-elasticity, and Helmholtz equations on domains with holes, an assumption the paper verifies numerically only for one symmetric linear-elasticity case.","fun_headline_variants_meta":{"raw":{"variants":["Holomorphic KAN solves elliptic PDEs on holes and Helmholtz","Laurent terms make holomorphic nets handle multiply connected domains","PIHKAN: KAN architecture improves holomorphic net accuracy","Multiply connected elliptic problems yield to holomorphic networks"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000481,"raw_usage":{"total_tokens":2405,"prompt_tokens":996,"completion_tokens":1409,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":612,"completion_tokens_details":{"reasoning_tokens":1341}},"tokens_in":612,"tokens_out":1409,"duration_ms":13910,"temperature":1.0,"reasoning_tokens":1341,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T11:24:52.828635+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the Laurent-based PIHKAN on Helmholtz or biharmonic problems over an annular or multi-hole domain with non-constant boundary data and compare against a high-resolution FEM reference; if the relative $L^2(\\Omega)$ error does not converge toward the FEM solution as epochs, polynomial degree, and boundary points are increased, the claimed completeness of the log-term extension for non-Laplace problems is false.","supporting_citations":[{"cited_title":"Engsig-Karup, and Tito Andriollo","cited_arxiv_id":null,"evidence_quote":"Supplies the original physics-informed holomorphic neural network construction, its universal approximation analysis, and the multiply connected elasticity benchmark that this work extends."},{"cited_title":"Vekua theory for the helmholtz operator.Zeitschrift f¨ ur angewandte Mathematik und Physik, 62(5):779–807, 2011","cited_arxiv_id":null,"evidence_quote":"Establishes the Vekua-type representation $V_\\beta$ that maps harmonic functions to Helmholtz solutions, providing the Helmholtz row of Table 1."},{"cited_title":"Harmonic functions from a complex analysis viewpoint.The American mathematical monthly, 93(4):246–258, 1986","cited_arxiv_id":null,"evidence_quote":"States the logarithmic-conjugation theorem for harmonic functions on multiply connected domains, which the paper's Laurent/log construction generalizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the class of Vekua operators mapping harmonic functions to solutions of general elliptic equations, framing the Helmholtz extension."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the systematic MLP-versus-KAN comparison for differential equations that supports the claim of smaller KAN architectures with comparable or better accuracy."}],"review_version":1}