{"id":"e86fbdce-1368-49c5-8ab9-75926c83bff4","arxiv_id":"2607.23069","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Frozen neural operator blocks pretrained on a square transfer to arbitrary 2D domains via boundary-adapted Galerkin coordinates, enabling boundary-exact rollouts and short-time law discovery.","lead":"gLegONet lets a pretrained library of physics blocks (diffusion, transport) be reused on new 2D shapes without retraining, by rebuilding only a boundary-adapted coordinate system. It also recovers governing laws from short glimpses of data on unfamiliar domains, making physics reuse and discovery geometry-independent.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central reuse claim depends on an untested assumption that frozen ambient Fourier blocks remain accurate when evaluated at boundary-adapted coefficient states; the block's output may be contaminated by the chosen extension of the target field outside the domain.","rationale":"The reader's weakest-assumption diagnosis matches the most load-bearing vulnerability I can identify. The algebraic construction of N_Ω and the reduced dynamics in Eq. (5) are internally consistent, and the experiments provide broad, well-controlled evidence on smooth 2D domains. But the transfer premise—that a block trained on Q-Fourier coefficients remains a faithful Galerkin representation of the same mechanism when evaluated at boundary-adapted states—is tested only end-to-end, not isolated. My proposed test would directly compute the discrepancy between Eq. (4) and the true target-domain response, thereby separating the block-transfer assumption from approximation and integration errors. The paper does not prove this premise, and the deferred supplementary stability argument is not included in the submitted text, so the concern is not resolved by the current manuscript. Since this is an addressable empirical/analytical gap rather than a demonstrated contradiction, the existing CONDITIONAL verdict is appropriate; no verdict change is needed.","tokens_in":19036,"tokens_out":4458,"duration_ms":48886,"concrete_test":"For each benchmark domain and each frozen block, sample a set of admissible coefficient states z (e.g., random z within the training amplitude). Compute q_learned = N_Ω^T M_Ω F_i^θ(a_bc+N_Ω z) via Eq. (4). Independently compute q_exact = N_Ω^T Φ_Ω^T W_Ω L_i^Ω(Φ_Q(a_bc+N_Ω z)|_Ω), where L_i^Ω is evaluated exactly or with a high-order finite-element/spectral discretization on Ω only. If the average relative error ||q_learned - q_exact|| / ||q_exact|| is comparable to the block's held-out ambient error (Extended Data Figs. 1–2), the transfer assumption holds. If it is substantially larger—or if it grows as the target domain shrinks or shifts within Q—then Eq. (4) is not a faithful Galerkin realization and the central reuse claim is unsupported. This test isolates the transfer-in-state-space assumption from boundary sampling, rank, and time-integration errors.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that changing the domain changes only the algebraic realization layer, not the learned operator blocks. This rests on Eq. (4): q_i^Ω(z,t) = N_Ω^T M_Ω F_i^θ(a_bc(t)+N_Ω z), which is asserted to be the Galerkin projection of the target-domain mechanism. The load-bearing premise is that the frozen block F_i^θ, trained to approximate P_Q L_i^Q(R_Q a) on the ambient square Q, still approximates the same mechanism when evaluated at the restricted, boundary-adapted state a = a_bc+N_Ω z. This is not guaranteed: F_i^θ is a map on Q-Fourier coefficients, and for nonlinear or differential mechanisms its output at a point depends on the entire ambient field, including the part outside Ω. The boundary null-space construction selects one specific extension of the target field, but there is no argument that this extension matches the natural extension implicit in the target PDE or that F_i^θ is insensitive to the extension. Consequently, q_i^Ω may not equal N_Ω^T M_Ω applied to the true target-domain mechanism, even if F_i^θ is a very accurate ambient block. The manuscript's own Discussion lists 'accuracy of the pretrained blocks' as an error source but does not isolate this extension-dependence; the stability argument deferred to Supplementary Information is not present in the submitted text. This is the weakest point because the entire geometry-independence claim collapses if Eq. (4) is not a genuine Galerkin projection of the target operator.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes Geometry-aware LegONet (gLegONet), a method for reusing pretrained, frozen neural \"mechanism blocks\" (diffusion, transport, reaction) across arbitrary embedded domains. Physical mechanisms are pretrained once on an ambient square Q using Fourier coefficients. For a target domain Ω, sampled boundary conditions define an affine admissible set a(t)=a_bc(t)+N_Ω z(t), where N_Ω is a mass-orthonormal null-space basis. Frozen blocks are evaluated at such admissible states and their responses are Galerkin-projected onto the boundary-adapted coordinates, giving the reduced dynamics of Eq. (5). The paper claims that changing the domain changes only the algebraic realization layer (C, a_bc, N_Ω, M_Ω), not the learned operator blocks. This is tested on manufactured-solution benchmarks with Dirichlet, Neumann, Robin, mixed and clamped conditions; on Allen–Cahn, vector Burgers, Navier–Stokes cylinder wake and Swift–Hohenberg; and on sparse law identification from short-time data. Comparisons are made against PINN, FNO, UNO, finite-element/finite-volume references, and finite-difference baselines.","tokens_in":19450,"tokens_out":5218,"duration_ms":58664,"significance":"If the central transfer property holds, the paper offers a genuinely useful separation: learned physical mechanisms are amortized once in a geometry-free ambient space, while boundary conditions and geometry are imposed deterministically through an algebraic interface. The experiments are substantial and use independent references (manufactured exact solutions, FEM, finite volume), report boundary residuals and rank/sampling ablations, and provide code/data links. The forward tests are not circular: they use held-out domains, boundary conditions and reference solvers. The inverse discovery tests also appear to be genuine extrapolation, with identification on a short window and rollout 80 times longer. However, the paper's central claim rests on an unproved assumption about the behavior of frozen blocks on boundary-adapted coefficient states; the current evidence, while suggestive, does not isolate that assumption from other error sources. The manuscript also defers the key stability/error argument to a Supplementary Information not included in the submitted text.","major_comments":[{"comment":"The central identity q_i^Ω(z,t)=N_Ω^T M_Ω F_i^θ(a_bc(t)+N_Ω z) is not, by itself, a Galerkin projection of the target-domain mechanism L_i^Ω. F_i^θ is trained to approximate P_Q L_i^Q(R_Q a), where P_Q is the L^2(Q) Fourier projection over the whole ambient square. For any differential operator, the values of L_i^Q(R_Q a) on Q\\Ω enter F_i^θ(a). The boundary null-space construction selects a specific extension of the target field to Q\\Ω, but no argument is given that the frozen block is accurate on that extension, or that the outside-Ω contribution vanishes after multiplication by N_Ω^T M_Ω. The Fourier coefficients are global, so this cancellation is not automatic. If F_i^θ is inaccurate at the boundary-adapted states, then q_i^Ω is not the target mechanism response and Eq. (5) advances a different reduced ODE. This is load-bearing for the paper's main claim. I recommend adding a direct","section":"Methods, 'Galerkin realization and rollout of frozen blocks', Eq. (4)"},{"comment":"The manuscript states that 'A standard stability argument decomposes the resulting error into boundary-adapted approximation, mechanism-response mismatch and time-discretization terms; see Supplementary Information.' The Supplementary Information is not provided with the submitted manuscript, so this claim cannot be checked. More importantly, the Discussion lists 'accuracy of the pretrained blocks' as an error source but does not identify the extension-dependence highlighted above. Since the entire geometry-independence claim collapses if Eq. (4) is not a faithful Galerkin realization of the target operator, the stability argument or an explicit assumption with numerical validation must appear in the main text. Without it, the 'geometry changes only the algebraic layer' claim is asserted rather than demonstrated.","section":"Methods, Algorithm 1 and Discussion"},{"comment":"The inverse discovery claim depends on the accuracy of temporal derivatives ẑ_n estimated from a short trajectory (T_id=0.012) after solving the per-observation least-squares problem (6). The sensitivity of ẑ_n to sensor noise, sensor placement and the conditioning of Φ_obs is not analyzed. The numerical ablations show degradation with noise, but a theoretical or systematic conditioning study would support the claim that the method 'turns sparse observations into predictive laws'. This is secondary to the main geometry-transfer claim, but it is a stated contribution and should be addressed.","section":"Methods, Eq. (7) and Algorithm 2"}],"minor_comments":[{"comment":"Notation: M_Ω is used both for the mass matrix and, in the same paragraph, 'whereas N_Ω below denotes the boundary-adapted coordinate matrix'. The sentence is understandable but could be clearer. Also, the affine lift a_bc(t) is not unique; the text should state explicitly that the reduced state z reparametrizes the same admissible set and that the final evolution is independent of the chosen lift.","section":"Methods, 'Boundary-adapted coordinates on target domains'"},{"comment":"The FNO and UNO baselines are trained on nearby manufactured families (Extended Data Table 2) rather than on the target geometry/family. This is a reasonable baseline choice for a zero-shot transfer claim, but the figure captions should state this clearly in the main text; otherwise a reader may misinterpret the comparison as FNO/UNO being trained on the same target.","section":"Fig. 2, Extended Data Table 2"},{"comment":"The text reports e_coef=6.89e-5 for the peanut zero-noise case but the table entry includes a standard deviation. Please make the reporting consistent: give the mean±std in the text or state that the quoted value is a single-seed illustrative run.","section":"Results, 'Sparse physical-law discovery on unseen domains'"},{"comment":"Reference [21] (LegONet) is central to the proposed framework but is listed as 'manuscript under review'. If it is not publicly available, the paper should include sufficient details of the block training procedure and generator-form notation in the main text or an appendix so that the present work is self-contained.","section":"References"},{"comment":"There is a missing space in 'Correspondence and requests for materialsshould be addressed'. Also, the phrase 'boundary-guaranteed assembly' in the Abstract is stronger than what is demonstrated, since the boundary condition is enforced only at sampled boundary points to numerical rank tolerance; the paper later reports dense residual checks, so please qualify the abstract wording accordingly.","section":"Additional information"}],"recommendation":"major_revision","confidential_remarks":"The core idea is appealing and the experimental effort is substantial, but the central transfer property is currently an assumption rather than a demonstrated fact. The authors should either prove the required state-space transfer property under explicit conditions or add a focused numerical study that isolates the extension-dependence error. The missing Supplementary Information stability argument should be made available. This is a major revision, not a rejection, because the method is plausible and the experiments are extensive; the key missing piece is a load-bearing validation of Eq. (4)."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take on gLegONet. The genuinely new piece is the affine admissible manifold: for a target domain, you sample boundary constraints, take the nullspace of C, mass-orthonormalize against M_Ω, and represent every admissible state as a_bc + N_Ω z. Then the frozen ambient blocks are evaluated at those states and projected back, giving Eq. (5). Changing geometry changes only C, a_bc, N_Ω, M_Ω; the neural blocks stay fixed. That's a clean separation of physics and geometry, and the derivation is straightforward and correct.\n\nThe empirical work is the paper's strong suit. Five manufactured benchmarks with Dirichlet, Neumann, Robin, mixed, and transport boundaries; Allen-Cahn on a disk against FEM; vector Burgers against finite volume; Navier-Stokes around a cylinder; Swift-Hohenberg with a composed fourth-order operator; and inverse identification where coefficients are fit on T_id=0.012 and rolled to T=1.0. The baselines (PINN, FNO, UNO) are trained per target, so the comparison is fair. Code and data are public. The inverse tests are not fitting the answer; the short-window fit and long rollout is a real predictive claim.\n\nThe soft spot is the transfer guarantee, and it's exactly where the stress-test note lands. Eq. (4) assumes the frozen ambient block F_i^θ, trained to match the ambient mechanism on the square, is still faithful when evaluated at a restricted boundary-adapted state. For differential operators that's plausible because the operator is local, but for nonlinear or nonlocal mechanisms the output can depend on the part of the ambient field outside Ω, and the nullspace construction picks one extension without arguing it's the right one. The paper defers a stability argument to the Supplementary Information, which isn't in the submitted text. I'd want to see that proof before trusting the claim beyond the tested regimes. Also, the parent LegONet paper is unpublished, so the novelty attribution rests partly on an unreviewed manuscript. And 'arbitrary domains' overstates it: every test is a smooth 2D domain.\n\nThat said, the experiments are broad enough that this is not a fatal objection. The reader's conditional verdict is about right. I'd send it to peer review; the right referees will push for the SI proof and maybe a 3D or non-smooth case, and they should take the transfer assumption seriously. Worth citing if you work on geometry-aware neural operators.","headline":"A useful geometry-transfer layer for reusable neural PDE blocks, with broad empirical support; the transfer guarantee is tested rather than proved, and the parent LegONet paper remains unpublished.","tokens_in":19870,"tokens_out":3676,"would_cite":true,"duration_ms":39041,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65M70","65M60","68T07","35K55"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper shows that a frozen library of neural mechanism blocks, trained once on a regular domain, can be reused on unseen geometries through a boundary-adapted coordinate change, so that geometry and boundary conditions only alter an alge","keywords":["geometry-aware operator learning","boundary-adapted coordinates","neural mechanism blocks","PDE learning","physical-law discovery","Galerkin projection","transfer without retraining","modular operator learning"],"falsifier":"Evaluate the Galerkin projection residual N_Ωᵀ M_Ω [F_θ(a_bc + N_Ω z) − F_i(a_bc + N_Ω z)] over admissible states on a domain whose boundary features lie near or beyond the ambient Fourier cutoff K. The central claim predicts this residual stays near the ambient training error; if it rises to the order of the mechanism itself on such a domain, the frozen-block transfer premise fails.","tokens_in":18913,"feed_emoji":"🧩","tokens_out":6207,"duration_ms":57105,"temperature":0.7,"pith_summary":"The paper introduces gLegONet, which claims that a library of neural operator blocks—trained once on a regular square domain—can be reused unchanged on arbitrary new geometries and boundary conditions, with only an algebraic coordinate layer changing. Boundary constraints are built into the state coordinates by computing the null space of sampled boundary constraints and mass-orthonormalizing against the domain's L2 metric, so the reduced dynamics evolve in boundary-admissible coordinates and satisfy the boundary to round-off rather than via penalty terms. The same frozen blocks are shown to support both forward simulation and sparse-law discovery on unseen domains, without retraining or target-domain trajectories. If true, this turns arbitrary-domain PDE learning from per-geometry retraining into modular assembly over reusable physical mechanisms, and makes physical-law discovery geometry-independent.","feed_headline":"One frozen physics library solves PDEs on any new domain","feed_subtitle":"Physics blocks are trained once on a square, then reused on new shapes via boundary-adapted coordinates—no retraining.","key_machinery":"The boundary-adapted coordinate matrix N_Ω is the load-bearing object: it is built from the null space of the sampled boundary constraint matrix C (via SVD), then mass-normalized so that N_Ωᵀ M_Ω N_Ω = I. This matrix converts each frozen ambient block into a reduced Galerkin response q_i^Ω = N_Ωᵀ M_Ω F_i^θ(a_bc + N_Ω z), making the boundary condition an algebraic property of the coordinates. The same machinery supplies the feature space for law discovery: candidate mechanisms are realized through the same reduced response and regressed in the boundary-adapted coordinate z.","core_discovery":"The central claim is that the entanglement between learned physics and geometry in neural PDE solvers can be cut by a deterministic coordinate change. A pretrained ambient library B_Q^θ = {F_i^θ} on the ambient square is frozen; for a target domain Ω, boundary samples define an affine admissible manifold a(t) = a_bc(t) + N_Ω z(t), where N_Ω is a mass-orthonormal basis of the boundary-homogeneous directions (N_Ωᵀ M_Ω N_Ω = I). Each frozen block is realized on Ω through its reduced Galerkin response q_i^Ω(z,t) = N_Ωᵀ M_Ω F_i^θ(a_bc + N_Ω z), and the dynamics are ẏ = Σ_i c_i q_i^Ω − N_Ωᵀ M_Ω ȧ_bc. Thus geometry and boundary conditions change only the algebraic layer (C, a_bc, N_Ω, M_Ω), while t","pith_inferences":["If the transfer premise holds, the method implies a new unit of reusable scientific software: mechanism blocks pretrained on a reference domain and deployed on arbitrary geometries by solving a single linear algebra problem (null-space and mass-normalization), making geometry handling in learned PDE solvers as routine as mesh generation in classical codes.","The boundary-coordinate construction is not tied to the Fourier basis; any approximation-complete trial family (finite-element, wavelet, polynomial) could serve as the ambient baseplate. A testable extension is to substitute such a basis and check whether the transfer properties persist.","The observed sensitivity to the gap between the target domain and the ambient boundary (the paper's Fig. 4d) suggests a practical deployment guideline: keep target domains well inside the ambient square, and choose the ambient cutoff K to resolve the smallest boundary feature; handling tightly fitting domains would likely require a different ambient basis or adaptive refinement."],"forward_implications":["A single pretrained mechanism library can be reused for forward simulation on new domains, boundary types, and mixed constraints, with boundary residuals near machine precision instead of penalty losses.","Higher-order operators, such as the clamped biharmonic Swift–Hohenberg term, can be composed from lower-order frozen blocks inside the boundary-adapted space, so no high-order block needs to be trained.","The boundary-adapted coordinates provide a stable feature space for sparse law identification, recovering governing coefficients from short-time observations on unseen domains where finite-difference derivatives are unstable.","Because boundary enforcement is algebraic, remaining rollout error is explicitly attributable to ambient resolution, retained rank, block accuracy, and time integration—not boundary leakage—which redirects where improvements should focus."],"fun_headline_variants":["Frozen physics blocks solve PDEs on any domain shape","Pretrain once, reuse on arbitrary domains: LegONet","Boundary coordinates let one physics library fit all geometries","No retraining: same LEGO blocks handle PDEs on new shapes"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"A mechanism block trained to approximate a differential operator on unconstrained ambient coefficients remains accurate when evaluated at the restricted, boundary-adapted states used on a new domain; if this state-space transfer fails, the reduced response is not the true Galerkin projection of the target mechanism.","fun_headline_variants_meta":{"raw":{"variants":["Frozen physics blocks solve PDEs on any domain shape","Pretrain once, reuse on arbitrary domains: LegONet","Boundary coordinates let one physics library fit all geometries","No retraining: same LEGO blocks handle PDEs on new shapes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000546,"raw_usage":{"total_tokens":2461,"prompt_tokens":771,"completion_tokens":1690,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":515,"completion_tokens_details":{"reasoning_tokens":1629}},"tokens_in":515,"tokens_out":1690,"duration_ms":11534,"temperature":1.0,"reasoning_tokens":1629,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T03:41:24.593474+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the Galerkin projection residual N_Ωᵀ M_Ω [F_θ(a_bc + N_Ω z) − F_i(a_bc + N_Ω z)] over admissible states on a domain whose boundary features lie near or beyond the ambient Fourier cutoff K. The central claim predicts this residual stays near the ambient training error; if it rises to the order of the mechanism itself on such a domain, the frozen-block transfer premise fails.","supporting_citations":[],"review_version":1}