{"id":"41f6c341-ef1d-4493-990e-293d7b65ff8d","arxiv_id":"2607.06287","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":3,"one_line_summary":"An explicit budget allocation condition is derived for two-stage kernel-based operator learning, relating training set size, input observations, and output resolution, alongside a physics-informed online reconstruction extension.","lead":"This paper derives a budget allocation rule for kernel-based operator learning: it specifies how the number of training samples, input observation points, and output resolution must scale together to guarantee convergence. A smart generalist might read it to understand the theoretical limits of using kernel methods as fast surrogates for PDE solution operators.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"The budget allocation rule (10) requires α > np/2 (Assumption 23(3)), but experiments use Matérn-3/2 (α=2) with n_PCA=20–28, giving np≫2α=4. The experiments violate the theory's own assumptions.","rationale":"The reader correctly identified the CONDITIONAL verdict and the incompleteness of the PI extension. However, the reader's focus on Assumption 23(4) (existence of g) misses a more immediately testable concern: the experiments violate Assumption 23(3) (α > np/2), which is necessary for the L^∞ sampling inequality used in Theorem 24. The budget allocation rule (10) — the paper's central contribution — has (2α−np) in its denominator, which is negative in the experimental regime, making the rule vacuous. The experiments show convergence, but this convergence is not covered by the theory. The paper's claims of experimental validation are therefore not supported.\n\nThis concern is distinct from the reader's: even if g exists and has the required smoothness (Assumption 23(4)), the sampling inequality's dimension-dependent exponent requires α > np/2, which fails for any reasonable input dimension with Matérn-3/2 kernels. The theory is only applicable in very low-dimensional settings (np < 4 for α=2), which are of limited practical interest.\n\nThe theoretical derivation itself is sound given the assumptions — the error decomposition (7) is clean, the sampling inequalities are correctly applied, and the budget allocation rule follows logically. The issue is the gap between theory and experiments: the experiments don't test the theory in a regime where it applies. The paper does not acknowledge or discuss this limitation.\n\nThe reader's verdict of CONDITIONAL is appropriate, but the specific concern should be sharpened: the experimental validation doesn't merely lack error bars or code release — it tests a regime where the theory's assumptions are violated, so it doesn't validate the theory at all. The PI extension lacking convergence analysis (acknowledged by the authors) is a secondary issue.\n\nI recommend keeping the verdict at CONDITIONAL but noting that the experimental validation is not just limited but inapplicable. The theory is correct but its applicability is severely limited by the condition α > np/2, and this limitation is not acknowledged in the paper.","tokens_in":23821,"tokens_out":15338,"duration_ms":785535,"concrete_test":"Re-run the Darcy flow budget allocation experiment (Fig. 4) with n_PCA = 2 and Matérn-3/2 (α=2, σ=2, d=2), so that α > np/2 is satisfied (2 > 1). The predicted exponent is np(2σ−d)/[d(2α−np)] = 2·2/[2·2] = 1. Check whether the transition from stagnation to convergence occurs at κ=1 as predicted. If it does, the theory is validated in an applicable regime; if not, the budget allocation rule may not correctly capture the scaling even when its assumptions are met.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim is the budget allocation condition (10): log N / log m ≥ np(2σ−d)/[d(2α−np)]. This is derived from Theorem 24, where f₂ = (1/√λ) h_{U,B}^{α−np/2} ∥g∥_{H^α}. The exponent α−np/2 comes from applying the sampling inequality (Proposition 20, case 1) with q=∞ (L^∞ bound) on B⊂R^{np}, which requires α > np/2 for convergence — the standard Sobolev embedding H^α(B) ↪ C^0(B) when dim(B)=np. Assumption 23(3) explicitly states α > np/2.\n\nIn the Darcy flow experiments (Section 6.1), the bottom kernel K_b is Matérn-3/2 (α=2), and inputs are reduced via PCA to n_PCA=20 or 28 components. Whether np = n_PCA (if PCA coefficients serve as sampling functionals) or np = 4096 (if spatial grid points are used), the condition α > np/2 is massively violated: 2 > 10 (or 2 > 2048). The budget allocation exponent np(2σ−d)/[d(2α−np)] becomes negative, rendering condition (10) vacuous — it is automatically satisfied for any N, m > 1.\n\nThe experiments show convergence at the oracle rate m^{-1} and a transition from stagnation (κ=0.5) to convergence (κ≥1.0) in the budget allocation experiment (Fig. 4). However, this behavior is not covered by the theory, which does not apply in this regime. The paper claims to 'validate the theoretical findings' (abstract) and that 'theoretical predictions are validated on the Darcy flow benchmark' (conclusion), but the experiments test a regime where the theory's assumptions fail.\n\nThe reader's weakest_assumption focuses on Assumption 23(4) (existence of g). While that assumption is indeed unverified for infinite-dimensional M, the more immediately pressing concern is Assumption 23(3): the regularity condition α > np/2 is violated in all experiments, making the budget allocation rule inapplicable to the very benchmarks used to validate it.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"The paper studies kernel-based operator learning in a two-stage sampling framework (offline kernel regression + online kernel reconstruction). The main theoretical contribution is an explicit budget allocation condition (Eq. 10) relating the number of training pairs N, the number of input observations n, and the output resolution m, derived from a coupled error decomposition (Eq. 7) using sampling inequalities. A physics-informed extension augmenting the online reconstruction with a soft PDE collocation penalty is also introduced, with a representer theorem (Theorem 36). Numerical experiments on Darcy flow and Poisson equations are presented.","tokens_in":24216,"tokens_out":1554,"duration_ms":394756,"significance":"The budget allocation rule (Corollary 26, Eq. 10) is a useful, explicit contribution to the approximation-theoretic foundation of kernel-based operator learning, extending the framework of Batlle et al. (2024) by providing convergence rates directly in terms of N, n, and m. The asymmetric formulation removing the need for an input-space reconstruction operator A_le (Observations 1-3) is a clean simplification. The physics-informed representer theorem (Theorem 36) and its block linear system (Eq. 15) are well-derived and practically relevant. The error decomposition (7) and the optimal lambda choice (8) are clearly presented.","major_comments":[{"comment":"§4.1, Assumption 23(3) and §6.1: The budget allocation condition (10) requires alpha > n_p/2 (Assumption 23(3)), which is the standard Sobolev embedding condition for H^alpha(B) -> C^0(B) when dim(B) = n_p. In the Darcy flow experiments (§6.1), the bottom kernel is Matérn-3/2 (alpha=2) and inputs are reduced via PCA to n_PCA = 20 or 28 components. Whether n_p = n_PCA or n_p = 4096 (spatial grid points), the condition alpha > n_p/2 is massively violated (2 > 10 or 2 > 2048). This renders the exponent in condition (10) negative, making the condition vacuous. The paper claims to 'validate the theoretical findings' (abstract) and that 'theoretical predictions are validated on the Darcy flow benchmark' (§7), but the experiments operate in a regime where the theory's assumptions do not hold. The paper should either (a) explicitly acknowledge that the experiments are in a regime not covered by ","section":null},{"comment":"§4.3, Assumption 23(4): The existence of g in H_{K_b} satisfying g(S_X(u)) = S_Y(G(u)) for all u in M is load-bearing for all error bounds (Theorems 24, 28) and the budget allocation condition (10). Section 4.3 discusses this only for the finite-dimensional case M = B_R[0] cap U_{n_0}, where S_X is injective on U_{n_0}. However, the paper does not verify that g (which maps R^{n_p} to R^{m*ell}) has the required Sobolev regularity alpha > n_p/2 in H^alpha(B)^{m*ell}. Injectivity of S_X on M ensures g is well-defined as a function, but says nothing about its smoothness as an element of the RKHS. The paper should state clearly what regularity of G and M are needed to guarantee g in H^alpha(B)^{m*ell}, or at minimum flag this as an open assumption with a discussion of when it can be expected to hold.","section":null}],"minor_comments":[{"comment":"§6.1, Fig. 3: The reference slope is labeled '-1' but the caption states the theoretical rate is m^{-sigma/d} = m^{-1}. It would help to state the oracle rate formula explicitly in the figure caption for readers who skip the text.","section":null},{"comment":"§6.1: The regularization schedule used is lambda(m) = m^{-5/2}, but the theoretical optimum (Eq. 8) gives lambda* proportional to h_{Y,D}^{sigma - d/2} which, for quasi-uniform Y in d=2 with sigma=2, gives lambda* ~ m^{-(sigma-d/2)/d} = m^{-1/2}. The paper notes lambda(m) = m^{-5/2} is 'well within the flat plateau' (Fig. 5), but the discrepancy between the theoretical optimum and the practical schedule should be discussed more explicitly.","section":null},{"comment":"§5.1, Remark 37: The smoothness requirement sigma > 2*nu + d/2 is explained, but the Poisson experiment (§6.2) uses Matérn-9/2 (sigma=5.5) with nu=2, d=2, requiring sigma > 5. The choice sigma=5.5 barely satisfies this. This should be noted as a near-marginal case.","section":null},{"comment":"§2.4, Theorem 2: The condition (1) is a Lipschitz stability condition for the inverse sampling map. It would help to state this interpretation in the theorem statement itself, not only in the subsequent paragraph, to aid readability.","section":null},{"comment":"§3.3, Proposition 20: The proof references 'Theorem 20' but should reference 'Proposition 20'. Similarly, 'Theorem 14' should be 'Proposition 14' and 'Theorem 17' should be 'Proposition 17'.","section":null},{"comment":"§4.1, Corollary 26: The condition (10) requires 2*alpha - n_p > 0 for the exponent to be positive. This is equivalent to Assumption 23(3) but should be stated explicitly in the corollary for clarity.","section":null},{"comment":"§6.2, Fig. 6: The PI surrogate saturates at approximately 0.023 from m=121 onward. The caption attributes this to 'irreducible learning error of A_off in n_PCA=28 input dimensions.' A brief quantitative connection to the budget allocation analysis would strengthen this claim.","section":null},{"comment":"The abstract states 'Numerical experiments illustrate the theoretical findings.' Given the regime mismatch in §6.1, this phrasing should be softened or qualified.","section":null}],"recommendation":"major_revision","confidential_remarks":"The core theory (Sections 3-4) is sound and well-constructed. The main issue is the gap between theory and experiments: the Darcy flow experiments violate Assumption 23(3) by a large margin, yet the paper claims validation. This is fixable — the author needs to either reframe the experiments as testing the method beyond the theory's regime, or add experiments with n_p small enough to satisfy alpha > n_p/2 (e.g., n_PCA <= 3 with Matérn-3/2). The physics-informed extension (Section 5) is promising but lacks error analysis, which the author acknowledges. The representer theorem is correct and the computational cost discussion is adequate. I would recommend major revision primarily to address the theory-experiment gap and the g-existence regularity question."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for a careful and substantive reading of our manuscript. Both major comments identify genuine gaps between the theory and the experiments/assumptions that we will address in revision. We respond to each below.","responses":[{"response":"The referee is correct that Assumption 23(3) (α > n_p/2) is violated in the Darcy flow experiments. With α = 2 (Matérn-3/2) and n_p = n_PCA = 20 or 28, the condition 2 > n_p/2 fails, and the exponent (2α - n_p) in condition (10) becomes negative, rendering the budget allocation condition vacuous in this regime. We acknowledge this without reservation. We will revise the manuscript in the following ways: (1) We will add an explicit remark in Section 6.1 stating that the experiments do not satisfy Assumption 23(3) and that condition (10) is not directly applicable. (2) We will soften the claims in the abstract and Section 7 from 'validate the theoretical findings' to language that accurately reflects what is and is not validated. Specifically, the oracle convergence experiment (Fig. 3) validates the reconstruction rate m^{-(σ-τ)/d} = m^{-1}, which depends only on the output kernel smoothness σ and the spatial dimension d, not on α or n_p. This part of the theory is genuinely tested. The budget allocation experiment (Fig. 4) validates the qualitative prediction that N must grow superlinearly with m to avoid stagnation, but the specific threshold exponent in (10) is not tested because the assumptions underlying it are not met. We will state this distinction clearly. (3) We will add a discussion of what kernel smoothness would be required to satisfy α > n_p/2 in the experimental setting (e.g., α > 10 for n_PCA = 20, corresponding to Matérn kernels of sufficiently high order) and note that this is a practical limitation of high-dimensional kernel methods, not specific to our framework.","revision_made":"yes","referee_comment":"Major Comment 1: Assumption 23(3) requires α > n_p/2, but in the Darcy experiments (Matérn-3/2, α=2, n_PCA=20 or 28), this condition is massively violated, making condition (10) vacuous. The paper claims to 'validate theoretical findings' but experiments are outside the theory's regime."},{"response":"The referee is correct. Section 4.3 establishes that g is well-defined as a function (via injectivity of S_X on the finite-dimensional model class M), but we do not verify that g belongs to H^α(B)^{m·ℓ} with the required smoothness α > n_p/2. This is a genuine gap: injectivity ensures existence of g as a mapping, but says nothing about its Sobolev regularity. We will address this in revision as follows: (1) We will add an explicit remark after Assumption 23(4) flagging that the Sobolev regularity of g is an open assumption, distinct from its well-definedness. (2) We will add a discussion of settings where the regularity of g can be expected to hold. In the finite-dimensional case M = B_R[0] ∩ U_{n_0} with U_{n_0} finite-dimensional, if G: U → V is sufficiently smooth (e.g., G ∈ C^k(U, V) for k large enough) and S_X, S_Y are point evaluation operators, then g = S_Y ∘ G ∘ S_X^{-1} inherits smoothness from G and S_X^{-1}. Since S_X^{-1}: S_X(U_{n_0}) → U_{n_0} is a linear map between finite-dimensional spaces, g is as smooth as G composed with finite-dimensional linear maps, which for analytic or C^∞ operators G (as in the Darcy setting with smooth coefficients) yields g ∈ C^∞ and hence g ∈ H^α for any α. However, the quantitative requirement α > n_p/2 becomes increasingly stringent as n_p grows, and we will note this tension explicitly. (3) We will also note that in the mismatch framework of Remark 25, one can work with g ∈ H^β(B)^{m·ℓ} for β < α, at the cost of modified convergence rates, which partially mitigates the issue but does not eliminate the need for β > n_p/2.","revision_made":"yes","referee_comment":"Major Comment 2: The existence of g in H_{K_b} with the required Sobolev regularity α > n_p/2 is load-bearing but not verified. Section 4.3 only shows g is well-defined (injectivity of S_X on M) but does not address its smoothness as an element of the RKHS."}],"tokens_in":23620,"tokens_out":1598,"duration_ms":119025,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"The main thing to know: this paper derives an explicit budget allocation rule (Eq. 10) for two-stage kernel operator learning, relating training pairs N, input observations n, and output resolution m. The error decomposition into reconstruction and learning terms via the perturbed-data interpretation is a clean structural contribution, and the asymmetric formulation dropping A_le is a genuine simplification over Batlle et al. (2024). The representer theorem for the physics-informed extension (Theorem 36) is correct and practically motivated.","headline":"Solid budget allocation theory for kernel operator learning, but experiments violate the theory's own regularity assumptions","tokens_in":24757,"tokens_out":1929,"would_cite":false,"duration_ms":172112,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Budget allocation rule for kernel operator learning","keywords":[],"falsifier":"If one constructs an operator G and model class M where the discretized operator g either does not exist as an element of the assumed Sobolev RKHS or has substantially lower smoothness than α, the budget allocation condition would predict convergence that does not occur, and the error bounds of Theorems 24 and 28 would not hold.","tokens_in":23875,"feed_emoji":"📐","tokens_out":1337,"duration_ms":93828,"temperature":0.7,"pith_summary":"This paper proves that a two-stage kernel surrogate for operator learning converges to the true operator at an explicit rate, provided the number of training pairs N, the number of input observation points n, and the output resolution m are coupled by a precise scaling law. The surrogate works in two stages: an offline kernel regression learns a finite-dimensional map from discretized inputs to discretized outputs, and an online kernel reconstruction recovers the output function from those predicted values. The author's key insight is to treat the online stage as a reconstruction from perturbed data, which splits the total error into a reconstruction term (governed by output grid density) and a learning term (governed by training data density), coupled only through a single regularization parameter. Balancing that parameter yields the budget allocation condition: the total error converges to zero as m and N grow, with the oracle rate m^{-(σ-τ)/d}, whenever log N / log m ≥ np(2σ-d) / [d(2α-np)]. If N scales exactly at this threshold, both error terms decay at the same rate and the learning error does not pollute the reconstruction rate. The paper also introduces a physics-informed variant: at evaluation time, the online reconstruction is augmented with a soft penalty on PDE residuals at collocation points, implemented via a closed-form representer theorem. This requires no retraining for new inputs and no full PDE solve, and numerically reduces reconstruction error below the data-driven baseline.","feed_headline":"How many training samples does a kernel operator learner need?","feed_subtitle":"An explicit scaling law ties training set size to output resolution and input dimension, with a physics-informed boost at evaluation time.","key_machinery":"Two-stage kernel surrogate A = A_on ∘ A_off ∘ S_X; error decomposition into reconstruction error (Term I) and learning error (Term II) coupled via regularization parameter λ; sampling inequalities bounding Sobolev norms by fill-distance-dependent terms; physics-informed Tikhonov functional with PDE residual penalty at collocation points","core_discovery":"The central object is the budget allocation condition (Equation 10): log N / log m ≥ np(2σ−d) / [d(2α−np)]. This inequality is derived from a decomposition of the total surrogate error into a reconstruction error term scaling as m^{-(σ−τ)/d} and a learning error term scaling as N^{-(2α−np)/(2np)} · m^{(2σ−d)/(2d)}, with the two coupled through the regularization parameter λ. When N grows at the threshold rate, both terms match and the surrogate achieves the same convergence rate as reconstruction from exact data. The decomposition holds for both interpolation-based and regularized least-squares offline learning, with identical asymptotic rates after optimal parameter tuning.","pith_inferences":["The budget allocation condition implies a curse of dimensionality in the input observation dimension n: the required N grows as m raised to a power proportional to np, so increasing the input resolution n demands disproportionately more training data. This suggests that PCA compression of inputs (as used in the experiments) is not merely convenient but may be essential for tractability.","The physics-informed extension's convergence rate is left unanalyzed. If the PDE residual penalty effectively increases the smoothness of the reconstruction (as the numerical experiments suggest), one might expect an improved effective σ in the reconstruction error term, potentially relaxing the budget allocation condition.","The saturation of the PI surrogate at the learning error of A_off (observed in the Poisson experiments) is a direct manifestation of the budget allocation theory: once reconstruction is accurate enough, the fixed learning error dominates, and further increasing m or m_L yields no benefit without also increasing N."],"forward_implications":["The budget allocation rule gives practitioners a concrete formula for sizing training datasets relative to output resolution, rather than relying on heuristics or empirical scaling experiments.","The asymmetry of the framework (no reconstruction needed on the input side) opens a direct path to kernel-based inverse problem surrogates, where the roles of input and output spaces are swapped.","The physics-informed reconstruction can be layered onto any already-trained kernel operator surrogate at deployment time, improving output fidelity without retraining or invoking a PDE solver.","The identical asymptotic rates for interpolation and regularized least-squares suggest that the choice between the two offline methods should be driven by numerical conditioning and noise robustness rather than convergence rate."],"fun_headline_variants":["Scaling laws for kernel operator learning training budgets","Coupling training size and output resolution in operator learning","Physics-informed reconstruction without kernel retraining","Error bounds for two-stage kernel operator learning","Optimal budget allocation for kernel-based operator learning"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The entire error analysis depends on the existence of a function g living in a specific Sobolev-type reproducing kernel Hilbert space that exactly maps discretized inputs to discretized outputs. If the true operator G does not induce such a well-defined, sufficiently smooth finite-dimensional map, the convergence rates degrade or fail entirely.","fun_headline_variants_meta":{"raw":{"variants":["Scaling laws for kernel operator learning training budgets","Coupling training size and output resolution in operator learning","Physics-informed reconstruction without kernel retraining","Error bounds for two-stage kernel operator learning","Optimal budget allocation for kernel-based operator learning"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":990,"prompt_tokens":565,"completion_tokens":425,"prompt_tokens_details":null},"tokens_in":565,"tokens_out":425,"duration_ms":22574,"temperature":1.0,"reasoning_tokens":441,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-08T10:46:01.113811+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"If one constructs an operator G and model class M where the discretized operator g either does not exist as an element of the assumed Sobolev RKHS or has substantially lower smoothness than α, the budget allocation condition would predict convergence that does not occur, and the error bounds of Theorems 24 and 28 would not hold.","supporting_citations":[],"review_version":1}