{"id":"7d785c4f-7ba9-4d5f-8a0a-a80a6e63c8ef","arxiv_id":"2604.19601","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The QE-MC-fPINN method uses a geometry-adaptive three-part decomposition of the fractional Laplacian with Gauss-Jacobi quadrature, Gauss quadrature, Monte Carlo angular sampling, and a feature-enhanced fPINN to achieve higher accuracy and faster convergence than prior MC-fPINN approaches on tested 2","lead":"The paper introduces a quadrature-enhanced Monte Carlo fractional physics-informed neural network (QE-MC-fPINN) that decomposes the fractional Laplacian into singular near-field, regular far-field, and exterior parts for high-dimensional problems. Smart generalists might read it to see how adaptive quadrature and neural networks can improve accuracy for nonlocal equations that arise in physics and engineering.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"The geometry-adaptive decomposition's quadrature/MC approximations may introduce uncontrolled errors in high dimensions near boundaries","rationale":"The reader's weakest_assumption exactly isolates the load-bearing step (accurate geometry-adaptive decomposition without uncontrolled quadrature/MC errors in high-D or near boundaries). The strongest_claim is purely experimental outperformance; therefore any gap in controlling those approximation errors directly undermines the reliability of the reported gains. Because the original review was abstract-only, the full text does not appear to supply the missing error analysis, so the UNVERDICTED status is unchanged.","tokens_in":1689,"tokens_out":460,"duration_ms":73144,"concrete_test":"Fix a point x near the boundary in d=5 on the unit ball, compute the fractional Laplacian operator applied to a known singular test function (e.g., dist(x,∂Ω)^α with α=0.5) using the paper's three-part scheme with N_MC = 10^3, 10^4, 10^5 angular samples and fixed quadrature orders; compare the Monte Carlo variance and bias against a reference obtained by adaptive radial quadrature in the same splitting; if the standard deviation does not decay as O(1/√N_MC) or the bias fails to drop below 10^{-4} for N_MC>10^4, the approximations introduce uncontrolled errors.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim rests on the spatially varying radius (using directional distance-to-boundary) producing a clean three-part split of the fractional Laplacian: singular near-field (Gauss-Jacobi quadrature on radial integral), regular interior far-field (Gauss quadrature), and analytical exterior far-field, with Monte Carlo only on angular variables. For the outperformance in accuracy/convergence (especially with strong boundary singularities) to hold in high-D regimes, the splitting radius must ensure that quadrature truncation and MC variance remain controlled independently of dimension and distance to boundary. No explicit a-priori error bounds or variance analysis for this decomposition appear to be supplied; the numerical benchmarks on Poisson and time-dependent problems demonstrate improvement over two MC-fPINN baselines, but this leaves open whether the gains are robust or specific to the tested low-to-moderate dimensions and domains.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper proposes the quadrature-enhanced Monte Carlo fractional physics-informed neural network (QE-MC-fPINN) method for high-dimensional fractional PDEs on bounded domains. It introduces a spatially varying radius based on directional distance-to-boundary information to decompose the fractional Laplacian into singular near-field, regular interior far-field, and analytical exterior far-field contributions. Gauss-Jacobi quadrature is applied to the singular radial integral, Gauss quadrature to the regular radial integral, and Monte Carlo sampling to the angular variables, combined with a feature-enhanced fPINN trial space to address low boundary regularity. Numerical experiments on fractional Poisson equations and time-dependent fractional PDEs demonstrate that the method outperforms two representative MC-fPINN discretizations in accuracy and convergence, particularly for solutions exhibiting strong boundary singularities.","tokens_in":1870,"tokens_out":625,"duration_ms":25699,"significance":"If the geometry-adaptive decomposition maintains controlled quadrature truncation and Monte Carlo variance independently of dimension and proximity to the boundary, the approach would offer a practical advance for nonlocal high-dimensional problems by blending deterministic quadrature with stochastic sampling and neural-network approximation, improving handling of reduced regularity at boundaries over pure MC-fPINN baselines.","major_comments":[{"comment":"The method description (abstract and §2) introduces the spatially varying radius and three-part decomposition without supplying a priori error bounds or variance estimates demonstrating that quadrature truncation and Monte Carlo sampling errors remain controlled independently of dimension and distance to the boundary; this is load-bearing for the high-dimensional claim and the asserted robustness near singularities.","section":"§2 (method description)"},{"comment":"Numerical experiments (§4) report outperformance on the tested benchmarks but provide no statistical variance from the Monte Carlo angular sampling, no sensitivity analysis with respect to sample size, and no explicit comparison of error behavior as dimension increases, leaving open whether the observed gains are robust or specific to the moderate-dimensional cases shown.","section":"§4 (numerical experiments)"},{"comment":"No convergence analysis, error estimates, or proof of consistency for the combined quadrature-MC-fPINN scheme is supplied, despite the paper appearing in a numerical analysis venue; the central superiority claim therefore rests entirely on the reported numerical comparisons without theoretical support.","section":"entire manuscript"}],"minor_comments":[{"comment":"Notation for the directional distance-to-boundary function and the splitting radius should be defined explicitly with an equation number in the method section for reproducibility.","section":"§2"},{"comment":"The abstract and experiments section should clarify the precise dimensions and domain geometries used in the benchmarks to substantiate the 'high-dimensional' descriptor.","section":"abstract and §4"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a natural fit for math.NA given its focus on numerical methods for fractional PDEs, but the absence of any theoretical analysis may prompt the editor to consider whether the contribution is primarily algorithmic rather than analytic."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and indicate the planned revisions.","responses":[{"response":"We agree that the manuscript does not supply a priori error bounds or variance estimates showing control independent of dimension and boundary distance. The geometry-adaptive radius and three-part split are motivated by the need to isolate the singular near-field contribution (treated by Gauss-Jacobi quadrature) from the regular interior far-field (Gauss quadrature) and the exterior (analytical), with Monte Carlo used only for the angular part. In the revision we will add a subsection in §2 that qualitatively discusses the error sources, cites standard quadrature error results for the radial integrals, and explains why the directional distance-to-boundary choice reduces the effective singularity strength near the boundary. Full dimension-independent bounds, however, are not derived in the present work.","revision_made":"partial","referee_comment":"[§2 (method description)] The method description (abstract and §2) introduces the spatially varying radius and three-part decomposition without supplying a priori error bounds or variance estimates demonstrating that quadrature truncation and Monte Carlo sampling errors remain controlled independently of dimension and distance to the boundary; this is load-bearing for the high-dimensional claim and the asserted robustness near singularities."},{"response":"We accept that the current numerical section lacks reported Monte Carlo variance, sample-size sensitivity, and systematic dimension scaling. The revised §4 will include (i) error bars obtained from repeated independent angular samplings, (ii) tables or plots showing accuracy versus number of Monte Carlo samples for representative problems, and (iii) additional experiments in dimensions up to at least d=8 (where computational cost remains manageable) together with a brief discussion of observed scaling trends. These additions will directly address the robustness question.","revision_made":"yes","referee_comment":"[§4 (numerical experiments)] Numerical experiments (§4) report outperformance on the tested benchmarks but provide no statistical variance from the Monte Carlo angular sampling, no sensitivity analysis with respect to sample size, and no explicit comparison of error behavior as dimension increases, leaving open whether the observed gains are robust or specific to the moderate-dimensional cases shown."},{"response":"The manuscript is a method-development paper whose central contribution is the geometry-adaptive decomposition combined with the feature-enhanced fPINN trial space, validated on challenging benchmarks that exhibit strong boundary singularities. A rigorous convergence analysis of the full hybrid scheme (quadrature + Monte Carlo + neural-network approximation) is not provided and would constitute a substantial separate theoretical effort. In the revision we will add a short subsection in the conclusions that explicitly states this limitation and outlines possible directions for future consistency analysis.","revision_made":"no","referee_comment":"[entire manuscript] No convergence analysis, error estimates, or proof of consistency for the combined quadrature-MC-fPINN scheme is supplied, despite the paper appearing in a numerical analysis venue; the central superiority claim therefore rests entirely on the reported numerical comparisons without theoretical support."}],"tokens_in":1467,"tokens_out":679,"duration_ms":35289,"standing_objections":["The absence of a rigorous convergence analysis, error estimates, or proof of consistency for the combined quadrature-MC-fPINN scheme."]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that this quadrature-enhanced Monte Carlo fPINN uses a direction-dependent radius based on boundary distance to split the fractional Laplacian into a singular near-field, a regular interior far-field, and an exact exterior contribution. Gauss-Jacobi quadrature handles the singular radial integral, standard Gauss quadrature the regular part, and Monte Carlo sampling the angular variables, with added features in the neural network to cope with boundary singularities. This specific combination of geometry-adaptive decomposition and quadrature rules is not in the earlier MC-fPINN papers they cite, so that part is new. The numerical tests on fractional Poisson problems and time-dependent equations report better accuracy and convergence than the two baseline methods, especially when the solution has strong boundary singularities. That empirical improvement is the clearest positive result here. The soft spot is the complete lack of error analysis or convergence theory. The central claim depends on the splitting radius keeping quadrature truncation and Monte Carlo variance controlled in high dimensions and near boundaries, but the paper supplies no a priori bounds, variance estimates, or analysis of how quadrature order and sample size affect the error. All support comes from the reported benchmarks, which leaves open whether the gains are robust or tied to the specific low-to-moderate dimensions and domains tested. Readers who already work with physics-informed networks for nonlocal PDEs will get the most from the implementation details and the benchmark comparisons. The work is coherent on its own terms and engages the relevant literature, so it deserves a serious referee. I would send it to peer review so that experts can check the numerical setup and ask for the missing analysis.","headline":"The paper adds an adaptive three-part split of the fractional Laplacian with targeted quadrature and Monte Carlo sampling inside an fPINN, and the benchmarks show gains over prior MC-fPINN versions on singular cases.","tokens_in":2335,"tokens_out":401,"would_cite":false,"duration_ms":29546,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A geometry-adaptive decomposition lets quadrature and Monte Carlo sampling solve high-dimensional fractional PDEs more accurately than prior Monte Carlo fPINN methods.","keywords":["fractional Laplacian","Monte Carlo methods","physics-informed neural networks","quadrature approximation","high-dimensional PDEs","boundary singularities","nonlocal operators"],"falsifier":"Numerical experiments on one of the paper's high-dimensional fractional Poisson benchmarks with a solution that has strong boundary singularity, in which the new method fails to show lower error or faster convergence than the two representative MC-fPINN discretizations.","tokens_in":2600,"feed_emoji":"📐","tokens_out":618,"duration_ms":41409,"temperature":0.7,"pith_summary":"This paper develops a numerical method for fractional partial differential equations on bounded domains in high dimensions, where nonlocal kernels, exterior boundary conditions, and reduced regularity near boundaries create major difficulties. It introduces a spatially varying radius that incorporates directional distance-to-boundary data to split the fractional Laplacian into a singular near-field part, a regular interior far-field part, and an analytical exterior far-field part. Gauss-Jacobi quadrature integrates the singular radial term, Gauss quadrature handles the regular interior radial term, and Monte Carlo sampling approximates the angular integrals, while a feature-enhanced physics-informed neural network approximates the low-regularity solution. Numerical tests on fractional Poisson equations and time-dependent problems show the resulting QE-MC-fPINN method produces smaller errors and faster convergence than two earlier Monte Carlo fPINN schemes, especially when the true solution has strong boundary singularities.","feed_headline":"Adaptive radius lifts accuracy for high-dimensional fractional PDEs","feed_subtitle":"QE-MC-fPINN splits the fractional Laplacian into near-field singular, interior regular, and exterior analytical parts, then applies targeted","key_machinery":"Geometry-adaptive three-part decomposition of the fractional Laplacian via a spatially varying radius that incorporates directional distance-to-boundary information, with Gauss-Jacobi quadrature on the singular radial integral, Gauss quadrature on the regular interior radial integral, Monte Carlo sampling on angular variables, and feature-enhanced PINN trial space.","core_discovery":"The paper establishes the quadrature-enhanced Monte Carlo fractional physics-informed neural network (QE-MC-fPINN) method. A spatially varying radius with directional distance-to-boundary information yields a geometry-adaptive three-part decomposition of the fractional Laplacian into singular near-field, regular interior far-field, and analytical exterior far-field contributions. Gauss-Jacobi quadrature is used for the singular radial integral, Gauss quadrature for the regular interior radial integral, and Monte Carlo sampling for the angular variables. A feature-enhanced physics-informed neural network trial space addresses the low-regularity behavior near the boundary. On the tested high-1","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Adaptive radius splits fractional Laplacian for QE-MC-fPINN","Quadrature-enhanced Monte Carlo fPINN for high-dimensional fractional PDEs","Feature-enhanced PINN for low-regularity boundary behavior","Spatially varying radius for geometry-adaptive fractional Laplacian"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The spatially varying radius with directional distance-to-boundary information produces an accurate geometry-adaptive three-part decomposition of the fractional Laplacian whose quadrature and Monte Carlo approximations do not introduce uncontrolled errors in high dimensions or near boundaries.","fun_headline_variants_meta":{"raw":{"variants":["Adaptive radius splits fractional Laplacian for QE-MC-fPINN","Quadrature-enhanced Monte Carlo fPINN for high-dimensional fractional PDEs","Feature-enhanced PINN for low-regularity boundary behavior","Spatially varying radius for geometry-adaptive fractional Laplacian"]},"model":"grok-4.3","cost_usd":0.011052,"raw_usage":{"total_tokens":4872,"prompt_tokens":687,"num_sources_used":0,"completion_tokens":68,"cost_in_usd_ticks":110524500,"prompt_tokens_details":{"text_tokens":687,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":4117,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":687,"tokens_out":68,"duration_ms":44061,"temperature":1.0,"reasoning_tokens":4117,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-10T01:40:01.671430+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Numerical experiments on one of the paper's high-dimensional fractional Poisson benchmarks with a solution that has strong boundary singularity, in which the new method fails to show lower error or faster convergence than the two representative MC-fPINN discretizations.","supporting_citations":[],"review_version":1}