{"id":"52da3dce-1b10-4319-a8ec-4eb860518290","arxiv_id":"2608.06428","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"Replacing point-sensor branch inputs with fixed or learned continuous linear functionals yields compact, discretization-portable DeepONet coordinates that beat point-sensor baselines on several PDE benchmarks.","lead":"This paper builds neural operator surrogates whose branch network reads continuous linear functionals of the input function, rather than point values on a fixed grid, in fixed and learned variants. It reports that these functional coordinates transfer across resolutions and, on a periodic Navier-Stokes benchmark, make the adaptive Topological DeepONet the best DeepONet variant, though a full-field Fourier neural operator remains more accurate.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The portability claim rests on an untested quadrature assumption: grid-dependent functional evaluations are never compared with continuum dual pairings, so the same-functionals-across-grids claim is not established.","rationale":"I read the paper's central assertion as a practical claim about discretization-portable functional coordinates, not merely a theoretical existence result. The most fragile link in that chain is the numerical evaluation of continuous linear functionals on arbitrary native grids. The reader's weakest-assumption statement identifies exactly this quadrature gap, and I agree with that assessment. I considered alternative concerns: the 'genuinely non-normable' benchmark in Section 5.9 is actually posed on the normable space of finite signed measures, and the paper lacks code and several seed-level error bars. These are real weaknesses, but they are less load-bearing than the quadrature issue because the method's deployment value depends on portability even for normable input spaces. My proposed test would settle whether the portability claim is supported: compare the implemented grid-quadrature coordinates against high-resolution reference evaluations of the same continuum functionals, and measure the effect of the discrepancy on the reported Darcy errors. This does not change the reader's conditional verdict; it sharpens the condition under which the central claim should be accepted.","tokens_in":39613,"tokens_out":6332,"duration_ms":70164,"concrete_test":"Take the 100 Darcy test fields from Section 5.5 and compute reference functional coordinates ℓ_j^ref(a) = ∫_Ω a φ_j using high-resolution quadrature on the original 421×421 fields. Then, for the unseen 57×57 and 97×97 grids, compute the paper's coordinates ℓ_j^grid(a) = Σ_m w_m a(x_m) φ_j(x_m). Report the relative L2 difference between the two coordinate vectors, and feed the reference coordinates into the trained Fixed and Adaptive branch networks, with all other components fixed, to recompute the Section 5.5 errors. If the coordinate discrepancy is comparable to or larger than the reported 5.5–5.6% predictive errors, or if using reference coordinates changes those errors materially, the discretization-portability claim is not established without explicit quadrature-error control.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central practical claim is that the same continuous linear functionals in V' provide discretization-portable coordinates. For this to hold, the discrete evaluation ℓ_j(a) ≈ Σ_m w_m a(x_m) φ_j(x_m) used in Appendix B.3 (Eq. 83) must be a controlled approximation of the continuum functional ∫_Ω a φ_j, with error that is small or at least calibrated across native grids. The paper supplies no quadrature-error analysis, no minimum-resolution condition, and no sensitivity study. Theorem 4.1 is entirely finite-dimensional: it assumes a discrete space V^h and a fixed reconstruction map R_q, and never bounds the gap between ℓ_{j,h} and a continuum ℓ_j. The experimental evidence in Section 5.5 is also narrower than advertised: the unseen 57×57, 73×73, and 97×97 grids are generated by subsampling a common stored fine grid, so the quadrature error is correlated with the training-resolution family and does not test arbitrary native meshes. Moreover, Appendix B.5 orthonormalizes the dictionary against the discrete weighted Gram matrix on the training grid, and Appendix B.6 selects active atoms by training-set variance; if these preprocessing steps are rerun on each native grid, the coordinates are not literally evaluations of the same continuum functionals. Because discretization portability is the main advertised advantage over the fixed-grid DeepONet, this missing verification is load-bearing.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes Fixed and Adaptive Topological DeepONets, which replace the point-sample branch input of a standard DeepONet with continuous linear functionals drawn from the dual V' of a Hausdorff locally convex input space (V,{p_α}). The adaptive measurements are learned as linear combinations of a fixed admissible dual dictionary, and both variants are combined with a Two-Step coefficient-space output projection. The theoretical section derives a finite-dimensional error decomposition (Theorem 4.1) separating measurement-reconstruction, output-basis truncation, and neural-approximation errors, together with a Barron-rate refinement (Corollary 4.1). Experiments span the antiderivative operator with per-sample verification of the bound, heterogeneous Darcy flow including transfer to unseen grids, a controlled operator with a known task-relevant input subspace, fixed-time and time-evolving Navier-Stokes vorticity prediction, and a screened-Poisson operator with source-list (measure-valued) inputs. The Adaptive Topological DeepONet is reported as the most accurate DeepONet-based model on the fixed-time Navier-Stokes problem (mean relative L2 error 1.685% ± 0.017% with 128 functional coordinates), while a parameter-matched FNO achieves lower error (0.832% ± 0.172%) at higher training time, peak GPU memory, and seed variance.","tokens_in":39910,"tokens_out":18596,"duration_ms":172843,"significance":"The computational study is substantial and largely carefully controlled: matched parameter budgets across models, an explicitly framed FNO comparison that concedes lower accuracy on the uniform periodic grid, per-sample coverage metrics, a controlled operator that isolates the task-relevant functional subspace, and detailed appendices documenting dictionaries, quadrature, SVD construction, and training schedules. Theorem 4.1 is correct but elementary (a triangle-inequality decomposition in finite dimensions), and its numerical verification in Section 5.1 is a strength, although the reported verification numbers are internally inconsistent (Major Comment 3). The paper's genuine strengths are the honest and mostly controlled experiments rather than the theory. If the load-bearing issues below are resolved — the misidentified 'non-normable' benchmark, the unsupported discretization-portability verification, and the contradictory sharpness ratios — the manuscript would be a solid computational contribution to operator learning. In its current form, the headline claims about non-normable input spaces and discretization portability exceed the evidence provided.","major_comments":[{"comment":"The abstract's final claim — that the formulation works 'including for input spaces that are not normable' — is supported in the manuscript only by the screened-Poisson benchmark, which is described as posed on 'a genuinely non-normable locally convex input space.' As written this is incorrect: the space M(Ω) of finite signed measures on a bounded domain is a Banach space under the total variation norm. Section 5.9 specifies no topology on M(Ω), and the algorithm only evaluates the finite sums ℓ_j(μ) = Σ_i a_i φ_j(x_i) directly from the source list; it never computes the seminorms p_α or otherwise engages a non-normable topology. If the intended input topology is the weak-* topology σ(M(Ω), C(Ω)), which is indeed non-normable, the paper must say so explicitly, since the continuity of the functionals and of G depends on the chosen topology. The benchmark therefore demonstrates learning from measure-valued source lists, but it does not demonstrate the claimed extension to non-normable input spaces, and Section 6's statement that the input topology 'cannot be represented by a single norm' is unsupported.","section":"Section 5.9; Abstract; Section 6"},{"comment":"The discretization-portability claim, which is the central advertised advantage over fixed-grid DeepONets, is not established. The discrete evaluation ℓ_j(a) ≈ Σ_m w_m a(x_m) φ_j(x_m) in Eq. (83) is used with no quadrature-error analysis, no minimum-resolution condition, and no direct comparison against the continuum pairing ∫_Ω a φ_j. The 'unseen' test grids in Section 5.5 (57×57, 73×73, 97×97) are, as the paper itself discloses, generated by subsampling a common stored fine (421×421) field, so the quadrature error is correlated with the training-resolution family and the test does not probe arbitrary native meshes. Moreover, the dictionary orthonormalization in Appendix B.5 is computed with the training-grid weighted Gram matrix G = Φ^T W_q Φ, and the active-atom selection in Appendix B.6 uses training-set variance; if these preprocessing steps are rerun on a new grid, the coordinates are not evaluations of the same continuum functionals, whereas if they are frozen, the quadrature error on a new grid is uncontrolled. Finally, Theorem 4.1 is entirely finite-dimensional and never bounds the gap between the discrete ℓ_{j,h} and a continuum ℓ_j, so the theory also provides no support for the cross-grid claim. A direct measurement of the quadrature error, or a test on genuinely independent meshes, is needed before the portability claim can be accepted.","section":"Section 5.5; Eq. (83); Appendices B.3, B.5, B.6"},{"comment":"The reported numerical verification of Theorem 4.1 is internally inconsistent. Section 5.1 and Figure 4(d) state that at q = 32 the maximum samplewise ratio is max_i E_tot/E_thm = 0.1249, whereas Section 6 states that 'all 400 test samples satisfied the samplewise bound, with maximum sharpness ratios of 0.94 and 0.97 for the adaptive and fixed models, respectively.' These numbers cannot both describe the verification of the same bound. Please report a single consistent set of values, define E_emp and E_thm precisely, and specify over which q the sharpness ratio is maximized.","section":"Section 5.1 and Section 6"}],"minor_comments":[{"comment":"The text says the adaptive coordinates 'remain continuous linear functionals,' but the coordinates actually fed to the branch network are standardized: z̃(v) = diag(σ_z)^{-1}(M^T Λ_m(v) − μ_z), which is an affine map of v, not a linear one. Please either absorb the standardization into the network or state explicitly that the linearity claim refers to the pre-standardization measurement map.","section":"Section 3.2.5; Algorithm 1"},{"comment":"The paper states that code and data 'will be made publicly available upon publication,' but no artifact is available in the submitted version. Given the number of benchmarks, dictionaries, and hyperparameters, providing the code at revision time would materially aid verification of the central claims.","section":"Code and Reproducibility"},{"comment":"The symbol m denotes the number of sensors/functionals in Section 3.1, the number of base observations in Section 3.2, and the grid-point index in Eq. (83); harmonizing this notation would improve readability.","section":"Notation"},{"comment":"The quantities E_emp, E_thm, and the 'operator-level measurement error' E_op,meas plotted in Figure 4 are not consistently defined in the text: Section 5.1 defines E_thm = E_Lip + E_out + E_NN, while Appendix A.5 defines E_bound = E_meas + E_out + E_NN and never defines E_emp; please align the notation between the figure, the text, and the appendix.","section":"Figure 4 and Appendix A.5"}],"recommendation":"major_revision","confidential_remarks":"The Section 5.9 non-normability claim is the most likely target of post-publication criticism; it is easily repaired either by specifying the weak-* topology on M(Ω) or by deleting the claim. The Section 5.1/6 sharpness-ratio discrepancy suggests that the verification experiment should be re-run and reported as one set of numbers. Given the empirical nature of the contribution, I would ask the authors to provide code and data at acceptance time even though the current text promises only future release. The self-citations are contextual and appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What you should know: this is a legitimate computational realization of Ismailov's Topological DeepONet, and the core idea is coherent. Replacing point sensors with fixed or learned continuous linear functionals is a sensible extension, and the integration with Two-Step SVD training is clean. The controlled operator experiment is the strongest part—it cleanly isolates input representation from network capacity and shows that adaptive functionals recover the task-relevant subspace. The error bound is a correct but elementary triangle inequality; the Barron refinement is standard. The main benchmark results are plausible, and the paper is honest that FNO beats it on the fixed-grid periodic NS problem while giving a fair accuracy–cost comparison.\n\nThe soft spots are real but not fatal. The 'genuinely non-normable' claim is factually wrong: the space of finite signed measures is normable under total variation. This does not break the method, but it undercuts a headline contribution. More importantly, the discretization-portability claim rests on an unexamined quadrature assumption. Appendix B.3 approximates the continuum functional by grid-dependent quadrature with no error analysis, and the 'unseen' grids in the Darcy experiment are subsampled from a common fine grid, so they do not test truly arbitrary native meshes. The paper also omits the closest variable-input baselines (VIDON, BelNet) from the benchmarks, and several key tables lack seed error bars. Table 8's adaptive training time looks internally inconsistent with Appendix D.14: 12.39 h for adaptive versus 23.74 h for fixed, when adaptive is supposed to include fixed pretraining. No code or data is currently available, which makes these checks harder.\n\nThe central methodological idea does hold up, and the paper is worth engaging. The target audience is neural-operator researchers who care about flexible, interpretable input representations. It deserves a serious referee, but the authors need to fix the non-normable framing, add quadrature-error analysis or at least a robustness study on genuinely different grids, resolve the training-time inconsistency, and release code/data before the claims should be accepted at face value.","headline":"A solid functional-measurement idea with two overclaimed results: the non-normable benchmark is mislabeled and the discretization-portability claim lacks quadrature analysis.","tokens_in":40442,"tokens_out":3428,"would_cite":true,"duration_ms":36341,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68T07","46A03"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper replaces point-sensor DeepONet inputs with continuous linear functionals from the dual of a Hausdorff locally convex space, yielding compact, interpretable coordinates that transfer across discretizations, including for…","keywords":["Topological DeepONet","continuous linear functionals","Hausdorff locally convex spaces","functional measurements","discretization portability","neural operators","Navier-Stokes vorticity","Darcy flow"],"falsifier":"Recompute the 32 Darcy functional coordinates $\\ell_j(a)=\\int_\\Omega a(x)\\phi_j(x)\\,dx$ by the paper's grid-dependent quadrature on a coarse and strongly nonuniform unseen mesh, and compare them with the same functionals evaluated on the fine reference field; if the quadrature-induced error is comparable to the model's $5.5\\%$ prediction error, the claim of discretization portability is not settled. A simpler check is to report the quadrature error itself for the $57\\times57$ grid before training.","tokens_in":39369,"feed_emoji":"🧮","tokens_out":7347,"duration_ms":70955,"temperature":0.7,"pith_summary":"This paper tries to establish that DeepONets should encode an input function not by point samples on a fixed grid but by finitely many continuous linear functionals from the continuous dual of a Hausdorff locally convex space, and that this switch yields operator-learning coordinates that are compact, interpretable, and portable across discretizations. The authors build both a fixed version, with prescribed dictionary functionals such as inner products against Legendre or multiscale test functions, and an adaptive version, where the measurement functionals are learned as linear combinations of a dual dictionary and stabilized by a training-only decoder. They combine both with a Two-Step output-basis construction and derive a discrete error decomposition separating measurement, output-basis, and neural-approximation errors, with a Barron-rate refinement. Numerically, the adaptive model reaches a mean relative $L^2$ error of $1.685\\%$ with $128$ functional coordinates on the fixed-time Navier-Stokes problem, and functional models keep roughly $5.5\\%$-$5.6\\%$ error on unseen Darcy grids, supporting the claim of discretization portability. The paper is careful to state that a Fourier neural operator still achieves lower error on that benchmark while consuming the full $64\\times64$ field, roughly twice the training time, and about $10.7\\times$ the peak GPU memory.","feed_headline":"Functional coordinates make DeepONets discretization-portable","feed_subtitle":"Adaptive topological inputs hit 1.685% Navier-Stokes error with 128 dual coordinates and keep 5.5% Darcy error on unseen grids.","key_machinery":"The load-bearing object is the functional measurement map $\\Lambda_m(v)=(\\lambda_1(v),\\dots,\\lambda_m(v))^\\top$ with $\\lambda_j\\in V'$, the continuous dual of a Hausdorff locally convex space whose topology is generated by a point-separating family of seminorms $\\{p_\\alpha\\}$. In the adaptive variant the coordinates are further compressed by a trainable linear map $M\\in\\mathbb{R}^{m\\times q}$ initialized from a structured dictionary, so each learned coordinate $\\ell_k^M(v)=\\sum_j M_{jk}\\lambda_j(v)$ is still a continuous linear functional. This map is combined with a Two-Step construction: a weighted SVD of output snapshots produces a rank-stable, weighted-orthonormal output basis, and the branch network predicts reduced coefficients in that basis, with a training-only decoder and soft orthogonality and drift losses stabilizing the adaptive measurements. The discrete error bound separates the total error into $L_h\\varepsilon_{\\mathrm{rec}}(q)+\\varepsilon_{\\mathrm{out}}(r,q)+\\varepsilon_{NN}$, and the Barron refinement gives a network-width rate $N^{-1/2}$ for the neural term.","core_discovery":"On the paper's own terms, the central discovery is that replacing the point-evaluation encoding $\\mathcal{S}_m(v)=(v(x_1),\\dots,v(x_m))$ of a DeepONet branch with a functional measurement map $\\mathcal{L}_m(v)=(\\ell_1(v),\\dots,\\ell_q(v))$, where each $\\ell_j$ belongs to the continuous dual $V'$ of the input space $(V,\\{p_\\alpha\\})$, preserves the branch-trunk architecture while freeing the coordinates from any prescribed mesh and extending the framework to non-normable locally convex spaces. The fixed model uses prescribed global functionals; the adaptive model learns a linear map $M\\in\\mathbb{R}^{m\\times q}$ inside the span of an admissible dual dictionary, so each learned coordinate remains a continuous linear functional. Combined with a weighted-SVD output basis from the Two-Step procedure, the framework yields a total error controlled by measurement reconstruction, output-basis truncation, and neural approximation. The paper argues that the empirical evidence, including best DeepONet accuracy on fixed-time Navier-Stokes, resolution-independent Darcy errors on unseen grids, and a benchmark on a non-normable locally convex input space, supports the conclusion that the continuous dual coordinates themselves, rather than universal superiority over grid-adapted architectures such as FNO, are the contribution.","pith_inferences":["The paper's portability claim is only as strong as its quadrature: a testable extension is to derive quadrature-error bounds for the dual pairings on coarse or irregular grids and to adapt the weights to each native mesh.","Because the learned coordinates are confined to the span of a prescribed dictionary, the dual-continuity guarantee is structural; extending the dictionary during training, or using unrestricted learned test functions, would break the guarantee and would need separate justification.","The error decomposition suggests that measurement compression and output-basis truncation can be tuned independently; tracking the three error terms during training could guide adaptive dictionary selection in future operator-learning pipelines.","The same functional-coordinate idea could be combined with physics-informed losses or used as interpretable features for inverse problems and uncertainty quantification on experimental data that arrive on irregular grids."],"forward_implications":["The same $q$ functionals can be evaluated on any native grid, so a model trained on mixed resolutions can be applied to unseen meshes without interpolating inputs to a common grid.","At equal representation budgets, global functional coordinates outperform point sensors: on Darcy, the Fixed Topological model reduces global relative $L^2$ error from $10.39\\%$ to $5.88\\%$ at $q=32$.","Learning the measurement functionals yields gains at nearly unchanged inference cost, with the Adaptive model reaching $1.685\\%$ mean relative error on fixed-time Navier-Stokes and $76.0\\%$ of test samples below the $2\\%$ threshold.","The framework extends to non-normable locally convex input spaces and to distribution-valued inputs where measurements are evaluated directly from source lists without rasterization.","A matched Fourier neural operator still achieves $0.832\\%$ on the same benchmark, so the contribution is compact, portable, interpretable coordinates rather than universal accuracy superiority."],"supporting_citations":[{"why":"Defines the DeepONet branch-trunk architecture and point-sensor encoding that this work extends.","marker":"[1]"},{"why":"Supplies the Topological DeepONet approximation result on compact subsets of Hausdorff locally convex spaces that grounds the framework.","marker":"[2]"},{"why":"Provides the Two-Step coefficient-space training procedure used for the output basis and branch prediction.","marker":"[3]"},{"why":"Defines the FNO baseline and the Darcy and Navier-Stokes benchmark datasets and evaluation protocols.","marker":"[4]"},{"why":"Gives the probabilistic DeepONet error bounds that this paper contrasts with uniform approximation on locally convex spaces.","marker":"[30]"},{"why":"Supplies the Deep Sets baseline used in the distribution-valued screened Poisson comparison.","marker":"[31]"}],"fun_headline_variants":["DeepONets swap point samples for dual-space coordinates","Adaptive functional inputs hit 1.685% Navier-Stokes error","Mesh-free DeepONets via continuous dual functionals","Functional coordinates enable non-normable input spaces"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing practical premise is that evaluating the same continuous linear functionals by grid-dependent quadrature on arbitrary unseen meshes accurately approximates the true continuum dual pairings, and the paper provides no quadrature-error analysis, no minimum-resolution condition, and no sensitivity study for that transfer step.","fun_headline_variants_meta":{"raw":{"variants":["DeepONets swap point samples for dual-space coordinates","Adaptive functional inputs hit 1.685% Navier-Stokes error","Mesh-free DeepONets via continuous dual functionals","Functional coordinates enable non-normable input spaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000969,"raw_usage":{"total_tokens":4256,"prompt_tokens":1217,"completion_tokens":3039,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":833,"completion_tokens_details":{"reasoning_tokens":2972}},"tokens_in":833,"tokens_out":3039,"duration_ms":25727,"temperature":1.0,"reasoning_tokens":2972,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T04:26:00.950836+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the 32 Darcy functional coordinates $\\ell_j(a)=\\int_\\Omega a(x)\\phi_j(x)\\,dx$ by the paper's grid-dependent quadrature on a coarse and strongly nonuniform unseen mesh, and compare them with the same functionals evaluated on the fine reference field; if the quadrature-induced error is comparable to the model's $5.5\\%$ prediction error, the claim of discretization portability is not settled. A simpler check is to report the quadrature error itself for the $57\\times57$ grid before training.","supporting_citations":[{"cited_title":"Topological deeponets and a generalization of the chen-chen operator approximation theorem.arXiv preprint arXiv:2603.11972, 2026","cited_arxiv_id":null,"evidence_quote":"Supplies the Topological DeepONet approximation result on compact subsets of Hausdorff locally convex spaces that grounds the framework."},{"cited_title":"On the training and generalization of deep operator networks.SIAM Journal on Scientific Computing, 46(4):C273–C296, 2024","cited_arxiv_id":null,"evidence_quote":"Provides the Two-Step coefficient-space training procedure used for the output basis and branch prediction."},{"cited_title":"Fourier neural operator for parametric partial differential equations","cited_arxiv_id":null,"evidence_quote":"Defines the FNO baseline and the Darcy and Navier-Stokes benchmark datasets and evaluation protocols."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Deep Sets baseline used in the distribution-valued screened Poisson comparison."}],"review_version":1}