{"id":"3f8f75a2-54e0-4f1b-889f-94d73b4e5e28","arxiv_id":"2606.27140","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"fTNN is a deterministic tensor neural network subspace method for fractional PDEs that decomposes the fractional Laplacian via spatially dependent integration splits and uses boundary-singularity-aware trial functions to achieve higher accuracy than fPINN and Monte Carlo methods on tested cases.","lead":"The paper presents fTNN, a tensor neural network approach that uses a geometry-adapted split and quadrature rules to compute the fractional Laplacian for solving fractional Poisson and advection-diffusion equations. A smart generalist might read it to understand how neural networks can be specialized for non-local operators that appear in models of anomalous transport.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Accuracy of geometry-adapted split hinges on whether the spatially varying near-field radius renders the interior far-field sufficiently regular for Gauss quadrature at all collocation points.","rationale":"The reader's weakest assumption is precisely the load-bearing step; the numerical superiority claim rests on that integration being accurate to machine precision on the tested singular cases. A single quadrature-consistency check on a canonical singular problem would confirm or refute it without requiring the full training pipeline.","tokens_in":1761,"tokens_out":328,"duration_ms":14574,"concrete_test":"On the unit disk with f=1 and s=0.5, fix a uniform grid of 200 interior points; for each point compute the far-field integral both with the paper's radius rule and with a reference radius 2× larger using 64-point Gauss quadrature; if the relative difference exceeds 1e-8 at more than 5 % of points, the regularity assumption fails for the chosen radius schedule.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The framework decomposes the fractional Laplacian via a spatially dependent near-field radius into singular near-field (Gauss-Jacobi), regular interior far-field (Gauss), and analytic exterior contributions. For the claim of high accuracy on singular solutions to hold, this radius choice must guarantee that the far-field integrand is C^∞ or at least high-order smooth everywhere inside the domain; otherwise quadrature error can dominate the loss and pollute the NN solution, especially near boundaries where the radius shrinks. The abstract provides no a-priori bound or adaptive criterion ensuring this regularity uniformly.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript presents fTNN, a tensor neural network approach for solving fractional partial differential equations involving the fractional Laplacian on bounded domains. Key components include a geometry-adapted integration split with spatially dependent near-field radius decomposing the operator into near-field (Gauss-Jacobi quadrature), interior far-field (Gauss quadrature), and exterior analytical parts; boundary-singularity-aware trial functions with strategies for selecting singularity exponents; and for time-dependent problems, a spatiotemporally separable neural network combined with alternating subspace optimization. The authors claim that this framework achieves high accuracy on benchmarks, outperforming fPINN and Monte Carlo methods, especially for problems with strong boundary singularities and long-time simulations.","tokens_in":1895,"tokens_out":430,"duration_ms":22984,"significance":"If the numerical performance claims are substantiated, the work offers a deterministic alternative to stochastic methods for fractional PDEs, potentially improving accuracy and reproducibility in handling singular solutions. The explicit incorporation of boundary features and the separable structure for time-dependent cases are notable technical contributions that could influence subsequent developments in neural methods for nonlocal operators.","major_comments":[{"comment":"The central accuracy claim depends on the spatially dependent near-field radius ensuring that the interior far-field integrand is sufficiently regular (C^∞ or high-order) for accurate Gauss quadrature at every collocation point. No a-priori analysis, adaptive criterion, or numerical check confirming this regularity—particularly near the domain boundary where the radius decreases—is provided, raising the risk that quadrature errors dominate the loss and affect solution accuracy for singular problems.","section":"Abstract and integration framework description"},{"comment":"The asserted high accuracy and substantial improvements over fPINN and Monte Carlo baselines are presented without error bars on the reported errors, convergence tables (e.g., vs. number of quadrature points or network parameters), or detailed descriptions of the benchmark problems, exact solutions, and datasets used. This absence prevents independent assessment of the performance claims.","section":"Numerical experiments"}],"minor_comments":[],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the constructive feedback on the integration framework and numerical validation. We address each major comment below, indicating where revisions will be made to strengthen the manuscript.","responses":[{"response":"We agree that the manuscript lacks an explicit a-priori regularity analysis or adaptive criterion for the far-field integrand under the spatially dependent radius. While the split is constructed so that the far-field kernel remains integrable and the radius choice excludes the singularity, no formal proof or boundary-specific numerical verification of quadrature accuracy is included. In the revised manuscript we will add a short subsection with a brief theoretical argument based on the kernel decay and radius scaling, together with numerical checks (e.g., quadrature-error tables near the boundary) to confirm that the far-field contribution remains below the target tolerance.","revision_made":"yes","referee_comment":"[Abstract and integration framework description] The central accuracy claim depends on the spatially dependent near-field radius ensuring that the interior far-field integrand is sufficiently regular (C^∞ or high-order) for accurate Gauss quadrature at every collocation point. No a-priori analysis, adaptive criterion, or numerical check confirming this regularity—particularly near the domain boundary where the radius decreases—is provided, raising the risk that quadrature errors dominate the loss and affect solution accuracy for singular problems."},{"response":"We acknowledge that the current numerical section does not report error bars, systematic convergence tables with respect to quadrature points or network parameters, or sufficiently detailed benchmark descriptions. In the revision we will expand the experiments section to include these elements: error bars from repeated runs where stochastic elements are present, convergence tables versus quadrature resolution and network size, and complete specifications of the benchmark problems, exact solutions, and data-generation procedures.","revision_made":"yes","referee_comment":"[Numerical experiments] The asserted high accuracy and substantial improvements over fPINN and Monte Carlo baselines are presented without error bars on the reported errors, convergence tables (e.g., vs. number of quadrature points or network parameters), or detailed descriptions of the benchmark problems, exact solutions, and datasets used. This absence prevents independent assessment of the performance claims."}],"tokens_in":1475,"tokens_out":464,"duration_ms":15511,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main thing here is the geometry-adapted integration split: a spatially varying near-field radius that carves the fractional Laplacian into a singular near-field piece handled by Gauss-Jacobi, a regular interior far-field piece handled by Gauss quadrature, and an analytic exterior piece. They combine this with trial functions that bake in the leading boundary singularity exponent, chosen either from the operator alone or jointly with the source. For the time-dependent case they factor the residual into separate temporal and spatial integrals and train with an alternating subspace scheme. That package is new relative to the fPINN and Monte Carlo baselines they cite.\n\nThe deterministic character is a genuine plus; it removes sampling variance and should help reproducibility on long-time or high-dimensional runs. The automatic exponent selection and the separable space-time network are also practical engineering moves that address real pain points in fractional problems.\n\nThe soft spot is the numerical evidence. The abstract asserts clear gains on benchmarks with strong singularities, yet supplies no error bars, no convergence tables, and no description of how the test problems were set up. Without those, it is hard to judge whether the claimed accuracy holds or whether quadrature error from the far-field piece leaks in near the boundary where the radius shrinks. The stress-test concern about uniform regularity of the interior far-field integrand is therefore live until the full paper shows either an a-priori bound or systematic numerical checks.\n\nThis is for people already working on deterministic solvers for nonlocal PDEs who need something more stable than PINNs for singular or long-time cases. The architecture is coherent and the deterministic integration is a real distinction, so it deserves a serious referee even if the experiments will need tightening.","headline":"The paper's core is a deterministic quadrature split for the fractional Laplacian using a position-dependent near-field radius plus singularity-aware trial functions, but the abstract's accuracy claims rest on unshown numerical details.","tokens_in":2358,"tokens_out":421,"would_cite":false,"duration_ms":19678,"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 tensor neural network with geometry-adapted quadrature solves fractional PDEs accurately by handling singularities deterministically.","keywords":["fractional PDEs","tensor neural networks","fractional Laplacian","deterministic quadrature","boundary singularities","neural network methods","advection-diffusion"],"falsifier":"Numerical experiments on a benchmark fractional Poisson problem with strong boundary singularity showing that the solution error does not decrease below Monte Carlo baselines or fails to improve with refined quadrature.","tokens_in":2670,"feed_emoji":"","tokens_out":545,"duration_ms":25433,"temperature":0.7,"pith_summary":"The paper presents fTNN, a deterministic tensor neural network method for solving fractional partial differential equations on bounded domains, such as the fractional Poisson equation and time-dependent fractional advection-diffusion equation. It establishes a geometry-adapted integration split that divides the fractional Laplacian into a singular near-field, regular far-field, and analytical exterior parts, each integrated with tailored quadrature rules to create a fully deterministic framework. Boundary-singularity-aware trial functions are constructed to resolve low-regularity solutions, with strategies for selecting exponents based on singularity structure. For time-dependent problems, a spatiotemporally separable network is combined with alternating optimization. Numerical experiments demonstrate high accuracy and substantial improvements over fPINN and Monte Carlo methods, especially for strong boundary singularities and long-time simulations. A sympathetic reader would care if this provides a reliable way to compute solutions to these nonlocal equations that appear in many applications.","feed_headline":"Tensor network solves fractional PDEs with high accuracy","feed_subtitle":"Geometry-adapted split and singularity-aware functions improve on fPINN and Monte Carlo for singular boundaries and long simulations.","key_machinery":"Geometry-adapted integration split featuring a spatially dependent near-field radius that decomposes the fractional Laplacian into three integrable contributions, paired with boundary-singularity-aware trial functions.","core_discovery":"We develop the fTNN as a deterministic tensor neural network subspace method for the fractional Laplacian on bounded domains. The geometry-adapted integration split with spatially dependent near-field radius decomposes the operator into singular near-field, regular interior far-field, and analytical exterior far-field contributions. These are integrated using Gauss-Jacobi quadrature for singular radial integrals, Gauss quadrature for regular ones, and deterministic angular quadrature. Boundary-singularity-aware trial functions enriched with explicit boundary features, along with automatic selection of leading exponent and loss evaluation from singularity structure, are used to resolve low-re","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["fTNN tensor network for fractional PDEs","Geometry adapted split for fractional Laplacian","Boundary singularity aware functions in fTNN","Spatiotemporal separation for time dependent PDEs","Gauss quadrature integration in deterministic fTNN"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The geometry-adapted integration split with a spatially dependent near-field radius decomposes the fractional Laplacian such that the chosen quadrature rules integrate the contributions accurately without errors that dominate the solution accuracy.","fun_headline_variants_meta":{"raw":{"variants":["fTNN tensor network for fractional PDEs","Geometry adapted split for fractional Laplacian","Boundary singularity aware functions in fTNN","Spatiotemporal separation for time dependent PDEs","Gauss quadrature integration in deterministic fTNN"]},"model":"grok-4.3","cost_usd":0.00634,"raw_usage":{"total_tokens":3014,"prompt_tokens":741,"num_sources_used":0,"completion_tokens":64,"cost_in_usd_ticks":63399500,"prompt_tokens_details":{"text_tokens":741,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2209,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":741,"tokens_out":64,"duration_ms":18396,"temperature":1.0,"reasoning_tokens":2209,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T05:07:06.312103+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Numerical experiments on a benchmark fractional Poisson problem with strong boundary singularity showing that the solution error does not decrease below Monte Carlo baselines or fails to improve with refined quadrature.","supporting_citations":[],"review_version":1}