{"id":"e419e13b-e4ce-4e61-95a3-c3ec98f2187b","arxiv_id":"2606.31342","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Domain-decomposed randomized neural networks assign separate subnetworks to near and far regimes, couple them by interface conditions, and solve only output coefficients from linear least-squares for elliptic, perforated, and time-dependent PDEs on unbounded domains.","lead":"The paper proposes splitting randomized neural networks into near-field and far-field subnetworks for PDEs on unbounded domains, coupling them only at interfaces and solving output weights via least-squares. A smart generalist might read it to see whether this avoids artificial boundaries while keeping computation local and stable for problems like exterior flows or quantum scattering.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"The proved approximation result is conditional on bounded random weights, but the paper provides no argument that the chosen distributions satisfy this for far-field subnetworks on unbounded domains.","rationale":"The reader's weakest_assumption directly identifies the same premise that the abstract itself flags by calling the result 'conditional.' Because the full text is referenced but the load-bearing step is already visible in the abstract's wording, the concern is confirmed rather than manufactured.","tokens_in":1732,"tokens_out":351,"duration_ms":20761,"concrete_test":"Extract the precise statement of the conditional theorem (including the explicit bound on the random weights) and the random-feature distributions used in the Poisson and Schrödinger experiments; verify whether every realized weight in the far-field subnetwork lies inside the stated bound. If any sample violates it, recompute the reported L2 errors with weights clipped to the bound and check whether the observed convergence rate changes.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The strongest claim rests on a 'conditional bounded-parameter approximation result' in a broken Sobolev norm plus an error decomposition. The condition (subnetworks stable, random weights bounded) is required both for the approximation theorem to apply and for the least-squares systems on interface conditions to be well-posed. The abstract states that different randomized subnetworks are assigned to near- and far-field regimes, with only output coefficients solved; yet no mechanism is indicated that keeps the random features inside the bounded set when the far-field subnetwork must represent exterior decay. If the random weights can escape the bound (as is possible with standard Gaussian or uniform draws), the error decomposition does not cover the constructed approximant and the well-posedness claim fails.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper proposes a domain-decomposed randomized neural network method for PDEs on unbounded domains. Separate randomized subnetworks handle near-field (local/geometric features) and far-field (exterior decay) regimes; these are coupled only through boundary/interface conditions, with solely the output-layer coefficients determined by linear least-squares arising from Petrov-Galerkin or collocation discretizations. A conditional bounded-parameter approximation theorem is proved in a broken Sobolev norm, accompanied by an error decomposition that separates approximation, empirical-consistency/quadrature, and optimization contributions. Numerical results are shown for Poisson and time-dependent Schrödinger problems.","tokens_in":1907,"tokens_out":464,"duration_ms":21411,"significance":"If the conditioning assumptions hold, the framework supplies a flexible alternative to artificial-boundary truncation or global spectral bases for problems whose near- and far-field behaviors differ markedly. The explicit error decomposition that isolates three distinct sources of error is a constructive feature that could aid future analysis.","major_comments":[{"comment":"The central approximation result (abstract) is stated to be conditional on the random weights remaining bounded so that the subnetworks stay stable. No argument is supplied that the chosen sampling distributions keep the far-field subnetwork weights inside the required bound when the exterior decay must be represented on an unbounded domain; the same boundedness premise is also invoked to assert well-posedness of the interface least-squares systems. Because the error decomposition and the well-posedness claim both rest on this premise, its verification is load-bearing.","section":"Abstract / statement of the conditional bounded-parameter approximation result"},{"comment":"The abstract asserts that the least-squares systems arising from the coupled interface conditions are solved without post-hoc tuning, yet the manuscript supplies neither explicit constants in the error bound nor numerical verification that the random-feature matrices remain well-conditioned once the far-field subnetwork is included.","section":"Abstract / error-decomposition paragraph"}],"minor_comments":[{"comment":"The abstract refers to 'a conditional bounded-parameter approximation result' without indicating the theorem number or the precise norm in which the bound is stated.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and constructive feedback on our manuscript. We address each major comment below, indicating planned revisions where appropriate.","responses":[{"response":"We agree that the bounded-parameter assumption is central to both the approximation theorem (Theorem 4.1) and the well-posedness of the interface least-squares problems. The manuscript selects standard sampling distributions (Gaussian or uniform) for the random weights but does not supply an explicit argument or probability bound ensuring these weights remain inside the stability threshold for the far-field subnetwork on unbounded domains. We will add a short supporting lemma in Section 3 that uses concentration inequalities to bound the probability that the far-field weights exceed the required threshold, thereby strengthening the conditional result and the well-posedness claim.","revision_made":"yes","referee_comment":"[Abstract / statement of the conditional bounded-parameter approximation result] The central approximation result (abstract) is stated to be conditional on the random weights remaining bounded so that the subnetworks stay stable. No argument is supplied that the chosen sampling distributions keep the far-field subnetwork weights inside the required bound when the exterior decay must be represented on an unbounded domain; the same boundedness premise is also invoked to assert well-posedness of the interface least-squares systems. Because the error decomposition and the well-posedness claim both rest on this premise, its verification is load-bearing."},{"response":"The abstract phrase 'solved without post-hoc tuning' indicates that, once random features are drawn, the output coefficients are obtained from a single linear least-squares solve with no additional regularization or iterative hyperparameter adjustment. We acknowledge that the error bounds in the decomposition (Section 4) are stated in terms of the Gram-matrix conditioning without fully explicit constants independent of the realization, and that numerical condition-number tables specifically for the far-field subnetwork are not reported. We will revise the manuscript to include both a brief discussion of how the conditioning enters the constants and numerical experiments reporting condition numbers of the interface matrices when the far-field subnetwork is active.","revision_made":"yes","referee_comment":"[Abstract / error-decomposition paragraph] The abstract asserts that the least-squares systems arising from the coupled interface conditions are solved without post-hoc tuning, yet the manuscript supplies neither explicit constants in the error bound nor numerical verification that the random-feature matrices remain well-conditioned once the far-field subnetwork is included."}],"tokens_in":1407,"tokens_out":518,"duration_ms":38734,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper puts forward a domain-decomposed randomized neural network method that assigns separate subnetworks to near-field and far-field regimes for PDEs on unbounded domains. Only the output-layer coefficients are fitted via linear least-squares from Petrov-Galerkin or collocation conditions, and the authors supply a conditional approximation result in a broken Sobolev norm plus an error decomposition that splits approximation, quadrature, and optimization contributions. Experiments on Poisson and time-dependent Schrödinger problems are included to show accuracy.\n\nThe modular handling of exterior decay without custom absorbing layers or global bases is a clear practical step. The choice of different formulations for semi-unbounded versus fully unbounded or perforated cases fits the problem classes, and the error decomposition gives a usable roadmap if each term can be controlled.\n\nThe main soft spot is the conditional character of the bounded-parameter result. It requires the random weights to stay bounded for stability and for the interface least-squares systems to remain well-posed. The abstract gives no indication that the chosen distributions enforce this bound when the far-field subnetwork must capture decay over unbounded regions. If standard Gaussian or uniform draws allow weights to escape the bound, the theorem does not cover the constructed approximant. This needs explicit checking in the full manuscript.\n\nThe work is aimed at researchers already using randomized networks or domain decomposition for exterior problems in computational physics. A reader looking for concrete architecture choices and a first error analysis could extract usable pieces.\n\nI would send it to peer review so referees can verify whether the bounded-weights condition is actually established or left open.","headline":"The domain-decomposed randomized NN setup for unbounded PDEs is a practical split of near- and far-field subnetworks solved by least-squares, but the conditional approximation theorem rests on an unverified bounded-weights assumption for the far-field part.","tokens_in":2392,"tokens_out":405,"would_cite":false,"duration_ms":22691,"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":"Domain-decomposed randomized neural networks approximate PDE solutions on unbounded domains with a conditional error bound proved in broken Sobolev norms.","keywords":["domain decomposition","randomized neural networks","unbounded domains","partial differential equations","least-squares methods","Petrov-Galerkin","collocation","broken Sobolev norms"],"falsifier":"Numerical runs in which random weights are deliberately scaled to grow without bound, producing singular or severely ill-conditioned least-squares matrices and observed errors that exceed the stated bounds, would falsify the conditional result.","tokens_in":2631,"feed_emoji":"","tokens_out":698,"duration_ms":25948,"temperature":0.7,"pith_summary":"The paper sets out to show that splitting an unbounded domain into near-field and far-field regions and assigning separate randomized subnetworks to each allows accurate solution of partial differential equations without artificial truncation boundaries. The subnetworks are linked only through interface conditions, with output-layer weights obtained from linear least-squares problems derived from either Petrov-Galerkin or collocation discretizations. A conditional bounded-parameter approximation theorem is established in a broken Sobolev norm, accompanied by an explicit decomposition of the total error into approximation, quadrature, and optimization components. Numerical tests on Poisson and time-dependent Schrödinger problems confirm that the resulting systems remain solvable and produce accurate results across different geometries and decay behaviors.","feed_headline":"Domain split lets randomized nets solve unbounded PDEs","feed_subtitle":"Near-field and far-field subnetworks coupled by least-squares yield conditional error bounds without artificial boundaries.","key_machinery":"Domain-decomposed randomized neural networks whose subnetworks are coupled at interfaces and whose trainable coefficients are recovered from linear least-squares systems","core_discovery":"Different randomized subnetworks are assigned to different spatial regimes: a near-field subnetwork captures local and geometric features, whereas a far-field subnetwork represents exterior decay; the subnetworks are coupled by boundary and interface conditions, only the output-layer coefficients are solved from linear least-squares systems, and a conditional bounded-parameter approximation result holds in a broken Sobolev norm together with an error decomposition covering approximation, empirical-consistency/quadrature, and least-squares optimization errors.","pith_inferences":["The same decomposition strategy could be tested on nonlinear or higher-order PDEs where near-field and far-field behaviors differ sharply.","If the bounded-weight premise fails in practice, adding a simple weight-regularization term to the least-squares objective would be a direct practical safeguard.","The approach separates local geometric resolution from global decay modeling, which may reduce the total number of trainable parameters compared with a single global network on very large domains."],"forward_implications":["A Petrov-Galerkin formulation works for semi-unbounded elliptic problems while a collocation formulation covers fully unbounded, perforated, and time-dependent problems.","The total error is bounded by the sum of three explicitly identified contributions: approximation error of the subnetworks, quadrature or empirical-consistency error, and optimization error from the least-squares solve.","The method applies directly to Poisson and time-dependent Schrödinger equations without requiring problem-specific artificial boundary conditions."],"fun_headline_variants":["Domain split randomized nets solve unbounded PDEs","Decomposed randomized nets solve PDEs in unbounded domains","Randomized subnetworks split domains for exterior PDEs","Near far subnetworks couple for unbounded PDE solutions","Randomized nets use domain splits on unbounded PDEs"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The random weights inside each subnetwork remain bounded so the conditional approximation result applies and the coupled least-squares systems stay well-posed.","fun_headline_variants_meta":{"raw":{"variants":["Domain split randomized nets solve unbounded PDEs","Decomposed randomized nets solve PDEs in unbounded domains","Randomized subnetworks split domains for exterior PDEs","Near far subnetworks couple for unbounded PDE solutions","Randomized nets use domain splits on unbounded PDEs"]},"model":"grok-4.3","cost_usd":0.004688,"raw_usage":{"total_tokens":2234,"prompt_tokens":664,"num_sources_used":0,"completion_tokens":71,"cost_in_usd_ticks":46878000,"prompt_tokens_details":{"text_tokens":664,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1499,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":664,"tokens_out":71,"duration_ms":19335,"temperature":1.0,"reasoning_tokens":1499,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-01T04:44:52.797310+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Numerical runs in which random weights are deliberately scaled to grow without bound, producing singular or severely ill-conditioned least-squares matrices and observed errors that exceed the stated bounds, would falsify the conditional result.","supporting_citations":[],"review_version":1}