{"id":"f0b8f2c4-bbbc-4859-a3e0-a5996a230a6b","arxiv_id":"2506.10973","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A practical recipe to convert common neural architectures into discretization-agnostic neural operators, validated by Navier-Stokes experiments showing cross-resolution generalization of FNO-style models.","lead":"This paper lays out a recipe for turning standard neural network layers (MLPs, CNNs, GNNs, transformers) into neural operators, which learn maps between function spaces rather than finite-dimensional vectors. It validates the recipe on Navier-Stokes, showing that resolution-agnostic models like FNO generalize across resolutions while U-Nets do not, and provides practical guidance for building such models.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The discretization-agnostic guarantee in Eq. (19) rests on Riemann-sum quadrature convergence, which the paper only guarantees for continuous integrands; for discontinuous or highly oscillatory inputs, the recipe may fail and this limitation is not flagged in the main text.","rationale":"The reader's weakest_assumption identifies the same load-bearing concern: quadrature convergence for irregular functions. I agree that this is the most load-bearing condition because the entire discretization-agnostic property—and hence the central recipe—depends on the quadrature sums converging to the intended integrals. The paper does restrict its formal analysis to continuous functions (Appendix A.4) and states convergence for continuous K (Appendix A.5), so the concern is a limitation of the stated scope rather than an internal inconsistency. However, the paper's framing as a general recipe for scientific problems makes the caveat worth surfacing, and the empirical section in Section 5.2 does not cover non-smooth inputs. The concrete test above would settle whether the recipe extends to discontinuous data. No change to the reader's CONDITIONAL verdict is needed; the same limitation is already reflected there.","tokens_in":34190,"tokens_out":16173,"duration_ms":181677,"concrete_test":"On the unit square, define a discontinuous input f(x)=1 if x_1>0.5 else 0, fix a Gaussian kernel K(x,y)=exp(-|x-y|^2/(2 sigma^2)), and compute the GNO output (19) at a fixed query point y using Algorithm 1 weights for two sequences of nested refinements: one with grid lines aligned to the interface at x_1=0.5 and one offset. Track the outputs as n grows. If the two sequences do not converge to the same limit (the exact integral), the discretization-agnostic claim fails for discontinuous inputs. Also record the error at n=128, 256, and 512 to quantify the practical convergence rate.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Section 3.8, Eq. (19)) is that replacing the GNN neighbor sum with a quadrature-weighted sum yields a discretization-agnostic operator. This property depends on the sums converging to the integrals in (18) for the integrands in (4), (8), (11), and (13). Appendix A.5 explicitly guarantees convergence only for continuous functions K, and Appendix A.4 defines F as subspaces of continuous functions. However, the paper's abstract and applications target PDEs and scientific data, which frequently involve discontinuous fields (shocks, interfaces, piecewise constant coefficients). For such inputs, the composition K(x,y_j,f(x),f(y_j)) is discontinuous. While bounded piecewise-continuous functions remain Riemann integrable, the Delaunay-based quadrature weights in Algorithm 1 and the convergence rate are not analyzed; for non-Riemann-integrable or highly oscillatory inputs, the empirical sum may not have a unique refinement limit, so the operator would not be discretization-agnostic. The paper does not state this caveat in the main text or offer guidance for non-smooth data. Its own Section 6 acknowledges the need to quantify and control discretization error in practice, but the recipe in Section 3.8 is presented without this qualification. Thus the regularity of the input concatenation is the least secure condition on which the central claim rests.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a set of design principles for neural operators (discretization-agnostic behavior, fixed parameter count, and universal approximation) and derives a recipe for converting standard neural-network layers into operator layers. The recipe replaces index-based sums by quadrature-weighted sums over subsets of the underlying domain, parametrizes kernels by learnable functions, and fixes receptive fields independently of resolution. These ideas are applied to MLPs, CNNs, GNNs, transformers, and encoder-decoder architectures, recovering and unifying existing models such as GNO, FNO, DeepONet, and transformer neural operators. Experiments on a Navier-Stokes benchmark compare U-Net, ViT, FNO, OFormer, and composite operators across resolutions, and the appendices provide numerical integration details, an error decomposition for zero-shot resolution transfer, and experimental details.","tokens_in":34433,"tokens_out":4377,"duration_ms":55881,"significance":"The paper's main value is a unifying and practically actionable framework: it shows that many existing neural-operator architectures arise from one conversion recipe, gives clean, self-contained derivations in Section 3, and ships code. The paper also makes the useful point that naive discretizations of CNNs/GNNs have resolution-dependent receptive fields, and its experiments illustrate the practical importance of quadrature weights and fixed receptive fields. However, the central claims are stated more strongly than the evidence supports: the discretization-agnostic guarantee is established only under regularity conditions that are not flagged in the main text, and the empirical section rests on one dataset without reported seed variance. These issues are fixable with explicit caveats and additional experiments, so the paper is, in my view, a solid candidate after revision.","major_comments":[{"comment":"The central claim that Eq. (19) yields a discretization-agnostic operator is established only under regularity conditions that are not stated where the claim is made. Appendix A.5 guarantees Riemann-sum convergence for continuous integrands K, and Appendix A.4 defines F as subspaces of continuous functions; but the paper's motivation includes PDE applications with shocks, interfaces, and piecewise-constant coefficients, for which the composition K(x,y_j,f(x),f(y_j)) can be discontinuous. For such inputs the Delaunay-based quadrature weights of Algorithm 1 are not analyzed, and the refinement limit of the empirical sum may depend on the discretization. Section 6 acknowledges the need to quantify and control discretization error in practice, but Eq. (19) and the surrounding summary are presented without this qualification. Please add an explicit regularity assumption, or an analysis for piecewise-continuous data, to the statement of the recipe and flag the limitation in the main text.","section":"Section 3.8, Eq. (19); Appendix A.5"},{"comment":"The empirical section is the main support for the practical claims that quadrature weights and fixed receptive fields improve cross-resolution generalization, but it reports results on a single Navier-Stokes dataset and does not state the number of seeds or show error bars. The text claims in Section 5.2 that the experiments 'empirically verify three main principles,' which is too strong without variance information, particularly for conclusions about mixed-resolution training and kernel interpolation at unseen resolutions. Please report multiple seeds with standard deviations, or temper the claim to 'illustrate' rather than 'verify.'","section":"Section 5.2, Figure 5; Appendix C"}],"minor_comments":[{"comment":"The contribution statement says the design principles are 'necessary for well-posed operator learning,' but Section 2 motivates them as desirable properties and does not prove necessity. This wording overstates the formal content; either prove the necessity claim or soften it to 'central to' well-posed operator learning.","section":"Section 1, Contribution 1"},{"comment":"There is a typo: '(spherial) FNOs' should be '(spherical) FNOs.'","section":"Section 3.8, Figure 2 caption"},{"comment":"The notation in Eq. (11) uses f(x) and f(y_j) inside the integral, but the surrounding text switches between 'f(x)' and 'f(xi)' when discussing discretizations; this is understandable but could be clarified by using a dummy variable consistently, e.g., f(x) for the continuous integrand and f(x_i) for the discretized sum.","section":"Section 3.4, Eq. (11)"},{"comment":"The Monte-Carlo quadrature derivation is correct but the notation for the quadrature weight underbraces is visually confusing; a short sentence defining Delta_i = (n p(x_i))^{-1} would improve readability.","section":"Appendix A.5, Eq. (31)"}],"recommendation":"major_revision","confidential_remarks":"The paper is more of a synthesis and practical guide than a source of new architectures or new theory, but it does offer a clean unifying recipe and a useful empirical demonstration. For a journal venue, the main risk is overclaiming: the discretization-agnostic guarantee needs a regularity caveat, and the experimental section needs either error bars or a more modest wording. No concerns about citation practice beyond the heavy self-citation pattern, which is natural given the authors' prior work. The paper fits the scope of a machine-learning journal, though the intended audience may also include computational science practitioners."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a solid, useful synthesis rather than a new result. If you want one place that tells you how to turn a GNN/CNN/transformer layer into a neural operator, with the quadrature-weight fix and the fixed-receptive-field argument spelled out, this is it. The math in Section 3 is clean and self-contained; the empirical section is honest but thin.\n\nWhat's genuinely new: the recipe framing itself. The individual pieces—integral operators for MLPs, convolutional operators, GNO, attention with quadrature—are all in prior work, and the paper says so. The contribution is organizing them around two principles: replace sums over neighbors by integrals over domain subsets, and use quadrature weights so the discretized sum converges to the integral. That distillation is well done. The appendix error analysis (triangle inequality splitting approximation/optimization/discretization) is a nice, useful explanation of zero-shot super-resolution. Code is released, which helps practitioners.\n\nWhere it's soft: the experiments in 5.2 are one PDE dataset, no error bars, no multiple seeds. That's fine for illustrating a recipe but not for strong claims. The paper's language about the principles being 'necessary' is a bit stronger than what is actually shown; they are necessary for the specific notion of discretization convergence they define, not for all well-posed operator learning. The quadrature-convergence guarantee is stated for continuous functions; for shock-type or highly oscillatory inputs the Delaunay weights are not analyzed, and the paper doesn't flag this in the main text. I don't think that breaks the recipe—it's a tutorial, not a theorem paper—but a one-paragraph caveat in Section 3.8 would be fair.\n\nNet: the paper deserves a serious referee. It is a candidate for a good journal/venue survey or tutorial track. The empirical section should be strengthened or clearly labeled as illustrative. I'd be happy to see this in the literature, and I'd cite it for the recipe. Not a must-read for theory people, but for practitioners entering operator learning it's genuinely useful. Recommendation: send to review.","headline":"A useful and honest synthesis of neural-operator recipes from standard NN layers; not a new result, but the cleanest practical guide I've seen, with thin-but-illustrative experiments.","tokens_in":35059,"tokens_out":2264,"would_cite":true,"duration_ms":26349,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68T07","65D32"],"pacs":[],"model":"deepseek-v4-flash","headline":"Many neural-network layers become resolution-agnostic when their discrete sums are replaced by integrals with quadrature weights.","keywords":["neural operators","operator learning","discretization-agnostic","quadrature weights","graph neural operators","Fourier neural operator","function spaces","partial differential equations"],"falsifier":"On an irregular point cloud, take a discontinuous integrand such as a step function and compare the Delaunay-triangulation quadrature estimate against the true integral as the number of points grows; if the quadrature error does not tend to zero, the Riemann-sum assumption behind Eq. (19) is violated. In the operator setting, train a GNO-style model on a PDE with shock or discontinuous coefficient data at two resolutions and evaluate zero-shot at intermediate resolutions: the claim predicts relative error should decrease with resolution, so a systematic divergence or plateau would falsify it.","tokens_in":33982,"feed_emoji":"🧮","tokens_out":7324,"duration_ms":55429,"temperature":0.7,"pith_summary":"Many well-known neural-network layers—fully-connected, convolutional, graph, and transformer attention—can be converted into neural operators, mappings between function spaces, by a simple recipe: replace the discrete sum over neighboring points with an integral over a fixed patch of the domain, and approximate that integral by a sum with quadrature weights. The paper argues this replacement is what makes a model agnostic to the discretization, so the same trained model can be evaluated at arbitrary resolutions with errors that shrink as the mesh is refined. This matters for scientific machine learning because PDE solvers produce data on adaptive meshes, and operators that generalize across resolutions can exploit multi-resolution training data and do zero-shot super-resolution. The claim is supported by experiments on the Navier-Stokes equations, where a Fourier neural operator trained at one resolution keeps its accuracy at other resolutions while a U-Net and a vision transformer do not.","feed_headline":"Quadrature weights turn neural-network layers into neural operators","feed_subtitle":"The recipe replaces neighbor sums with domain integrals plus quadrature weights, enabling zero-shot super-resolution.","key_machinery":"The carrying identity is Eq. (19), the graph neural operator (GNO): $g(y_j)=\\sum_{i:\\,x_i\\in D(y_j)} K(x_i,y_j,f(x_i),f(y_j))\\,\\Delta_i$, which approximates the integral operator $\\int_{D(y_j)} K(x,y_j,f(x),f(y_j))\\,\\mathrm{d}x$. The quadrature weights $\\Delta_i$ (e.g., cell volumes from a Delaunay triangulation of the point cloud, or grid spacings on regular grids) make the discrete sum converge to the integral as the discretization is refined; fixed domain subsets $D(y_j)$ (radius balls, or the whole domain for global operators) keep the receptive field independent of resolution. Additional machinery includes pointwise operators for activations and liftings, spectral convolution via the Fourier transform for global convolutions, and inner-product encoders with parametric decoders for encoder-decoder layers. The recipe's role is to convert each finite-dimensional layer into a discretization-convergent layer, so the whole architecture inherits resolution-agnostic behavior.","core_discovery":"The paper's central claim is that most neural network architectures can be viewed as graph neural networks, and that turning them into neural operators requires one structural change: replace the sum over neighbors, $g_j=\\sum_{i\\in\\mathrm{Neighb}_j}K_{ij}(f_i,f_j)$, with an integral over a fixed subset $D(y_j)$ of the spatial domain, $g(y_j)=\\int_{x\\in D(y_j)} K(x,y_j,f(x),f(y_j))\\,\\mathrm{d}x$, and then discretize that integral with quadrature weights $\\Delta_i$, giving Eq.~(19). Because the neighborhood $D(y_j)$ is defined in the domain rather than by index proximity, the receptive field stays fixed as the resolution changes, and because the sum carries quadrature weights, the discrete model converges to the integral operator as discretizations are refined. The paper shows this recipe subsumes MLPs turned into integral transforms, convolutional operators with learned kernels, Fourier spectral convolutions, graph neural operators, resolution-agnostic self-attention, and encoder-decoder operators with finite-dimensional latents. Empirically, on the Navier-Stokes benchmark, models built this way generalize across resolutions, whereas standard U-Nets, vision transformers, and multi-resolution-augmented neural networks do not.","pith_inferences":["If the recipe is as general as claimed, porting mature architectures—segmentation U-Nets, vision transformers, and graph networks—into operator form with quadrature weights should reproduce their empirical strengths in scientific applications, something the paper tests only for a few cases.","The quadrature-weight analysis implies that on regular uniform grids, where weights cancel in self-attention, standard transformers are already discretization-consistent; the practical gap should appear mainly on irregular or adaptive meshes.","A testable extension is to apply the recipe to point-cloud data with strongly non-uniform density and compare quadrature-weighted GNOs against unweighted message passing; the paper predicts the weighted version will generalize across samplings while the unweighted one will not.","One could audit existing neural operators by checking whether their discrete layers converge to the claimed integral operator under refinement; any layer whose sums lack quadrature weights is a candidate for the same collapse-to-pointwise failure described for CNNs."],"forward_implications":["Any architecture expressible as a message-passing graph neural network can be converted into a neural operator by using domain-based neighborhoods and quadrature-weighted aggregation, making it applicable to arbitrary point clouds.","A neural operator trained at one resolution can be evaluated at higher and lower resolutions with only discretization error, enabling zero-shot super-resolution without retraining.","Multi-resolution training of ordinary neural networks helps at the resolutions seen during training but does not confer reliable generalization to unseen resolutions.","Interpolating inputs and outputs to a fixed resolution defines a naive neural operator but discards the high-resolution information in the input, limiting accuracy.","Losses, normalization statistics, and inner products used for training should be computed with quadrature weights so that the training objective itself is resolution-agnostic."],"supporting_citations":[{"why":"Defines the graph neural operator and integral kernel networks that generalize Eq. (11) and lead to Eq. (19).","marker":"Li et al. (2020b)"},{"why":"Introduces the Fourier neural operator whose spectral convolution is the global convolution recipe and which demonstrates cross-resolution generalization.","marker":"Li et al. (2020a)"},{"why":"Supplies the theoretical framework of discretization convergence, universal approximation, and encoder-decoder operator constructions on which the paper builds.","marker":"Kovachki et al. (2023)"},{"why":"Provides LocalConv and DiffConv kernels used in the composite FNO experiments and the analysis of continuous convolutional kernels.","marker":"Liu-Schiaffini et al. (2024)"},{"why":"Provides the Navier-Stokes data setting, FNO hyperparameters, and soft-gating skip-connections used in the empirical evaluations.","marker":"Kossaifi et al. (2023)"}],"fun_headline_variants":["From neighbor sums to domain integrals: a recipe for neural operators","Integral over neighbors: converting neural nets to operators","Quadrature weights make neural nets think continuous for operators","Turning discrete neural nets into continuous operators via quadrature"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The recipe assumes that the fields aggregated over each neighborhood are regular enough that Riemann-sum quadrature with the chosen point weights converges to the true integral, so for sharply discontinuous or highly irregular fields the quadrature weights from a Delaunay triangulation may not converge and the discretization-agnostic property could fail.","fun_headline_variants_meta":{"raw":{"variants":["From neighbor sums to domain integrals: a recipe for neural operators","Integral over neighbors: converting neural nets to operators","Quadrature weights make neural nets think continuous for operators","Turning discrete neural nets into continuous operators via quadrature"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000971,"raw_usage":{"total_tokens":4197,"prompt_tokens":1083,"completion_tokens":3114,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":699,"completion_tokens_details":{"reasoning_tokens":3048}},"tokens_in":699,"tokens_out":3114,"duration_ms":21064,"temperature":1.0,"reasoning_tokens":3048,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:12:58.552522+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On an irregular point cloud, take a discontinuous integrand such as a step function and compare the Delaunay-triangulation quadrature estimate against the true integral as the number of points grows; if the quadrature error does not tend to zero, the Riemann-sum assumption behind Eq. (19) is violated. In the operator setting, train a GNO-style model on a PDE with shock or discontinuous coefficient data at two resolutions and evaluate zero-shot at intermediate resolutions: the claim predicts relative error should decrease with resolution, so a systematic divergence or plateau would falsify it.","supporting_citations":[],"review_version":1}