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arxiv: 2605.22182 · v1 · pith:LOBCC452new · submitted 2026-05-21 · 💻 cs.LG

IKNO: Infinite-order Kernel Neural Operators

Pengyuan Zhu , Ivor W. Tsang , Yueming Lyu This is my paper

Pith reviewed 2026-05-22 07:40 UTC · model grok-4.3

classification 💻 cs.LG
keywords neural operatorsinfinite-order kernelskernel integralsoperator learningscientific machine learningfunction approximationmachine learning
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The pith

Neural operators gain expressivity from infinite-order kernel integrals that still admit closed-form approximations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes the Infinite-order Kernel Neural Operator to move beyond the first-order kernel integral limits that constrain existing neural operators. It builds neural operators using infinite-order kernel integrals and shows these admit an efficient closed-form finite approximation. Two constructions are developed: one applying the full resolvent on product grids and another using tensor-product composition of per-axis resolvents. Fast schemes support global aggregation at scale. Tests on time-dependent and time-independent benchmarks with large point clouds report consistent state-of-the-art accuracy gains.

Core claim

Neural operators constructed via infinite-order kernel integrals admit an elegant closed-form finite approximation. The two complementary constructions are IKNO-Vanilla, which applies the full-kernel resolvent on the product grid via Kronecker eigendecomposition, and IKNO-TP, an alternative tensor-product operator that composes per-axis resolvents; both enable outstanding global information aggregation while maintaining high computational efficiency and yield state-of-the-art accuracy on benchmarks.

What carries the argument

The infinite-order kernel resolvent, which extends standard first-order kernel integrals to infinite order and is stably approximated in closed form via eigendecomposition or tensor products to increase expressivity.

If this is right

  • IKNO achieves state-of-the-art accuracy with significant improvements on nearly all benchmark datasets for both time-dependent and time-independent problems.
  • The approach scales to very large point clouds while handling arbitrary input shapes.
  • Fast computation schemes deliver high efficiency together with strong global information aggregation.
  • Both the vanilla and tensor-product variants deliver consistent performance gains across industrial-scale datasets.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The closed-form route may reduce reliance on iterative solvers in other operator-learning architectures.
  • Better long-range dependency capture could improve results on multi-scale physical simulations where first-order kernels fall short.
  • Extending the tensor-product construction to irregular meshes might require additional stability analysis beyond the current grid-based tests.

Load-bearing premise

The infinite-order kernel resolvent can be stably approximated in closed form on product grids or via tensor products without numerical instability or problem-specific tuning that would erase the expressivity gain.

What would settle it

A controlled experiment on a standard benchmark in which both IKNO variants produce no accuracy improvement over first-order kernel baselines once the closed-form approximation is applied.

Figures

Figures reproduced from arXiv: 2605.22182 by Ivor W. Tsang, Pengyuan Zhu, Yueming Lyu.

Figure 1
Figure 1. Figure 1: Visualization of relative L1 error fields for finite-order kernel propagation on the Poisson-C-Sines benchmark. As the propagation order increases from p = 1 to p = 4, both the error magnitude and the spatial extent of high-error regions decrease. 3.2. Infinite-order kernel integral 3.2.1. DEFINITIONS AND DISCRETIZATION One fundamental component of a Neural Operator is the first-order kernel integral opera… view at source ↗
Figure 2
Figure 2. Figure 2: Visualization of IKNO predictions on the NASA CRM industrial benchmark. The qualitative visualization further shows that IKNO produces coherent full-aircraft predictions on the high-resolution geometry, with visually small error patterns that indicate an accurate neural-operator approximation of the aerodynamic fields [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Ablation study on the latent grid size L (per-axis resolution of the latent grid G) on time-dependent datasets. For instance, L = 32 corresponds to a 32 × 32 latent grid. errors, whereas increasing the grid size results in a dramatic improvement in accuracy. This suggests that time-dependent trajectories contain higher-frequency components that necessitate a denser latent representation to be captured effe… view at source ↗
Figure 4
Figure 4. Figure 4: Ablation study on the latent grid size L (per-axis resolution of the latent grid G) on time-independent datasets. For instance, L = 24 corresponds to a 24 × 24 latent grid. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Visualization of IKNO predictions on the Poisson-Gauss benchmark [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Visualization of IKNO predictions on the Wave-C-Sines benchmark. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
read the original abstract

Neural operators have achieved significant success in modern scientific computing due to their flexibility and strong generalization capabilities. Existing models, however, primarily rely on first-order kernel integral approximations, which severely limit their expressivity. To address this, we propose the Infinite-order Kernel Neural Operator (IKNO), which constructs neural operators via infinite-order kernel integrals and admits an elegant closed-form finite approximation. We develop two complementary infinite-order neural operator constructions: IKNO-Vanilla, which applies the full-kernel resolvent on the product grid via Kronecker eigendecomposition, and IKNO-TP, an alternative tensor-product operator that composes per-axis resolvents. Furthermore, we develop fast computation schemes for both variants of IKNO, which achieve outstanding global information aggregation while maintaining high computational efficiency. Empirically, we evaluate our IKNO on both time-dependent and time-independent benchmarks with arbitrary input shapes, including large-scale industrial datasets. Extensive experiments demonstrate that the IKNO method consistently achieves the SOTA accuracy with significant improvements on nearly all benchmark datasets while maintaining scalability to very large point clouds.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces the Infinite-order Kernel Neural Operator (IKNO) to address limited expressivity in existing neural operators that rely on first-order kernel integrals. It constructs operators via infinite-order kernel integrals with two variants: IKNO-Vanilla using Kronecker eigendecomposition of the full-kernel resolvent on product grids, and IKNO-TP composing independent per-axis resolvents via tensor products. Fast computation schemes are developed for both, and extensive experiments on time-dependent and time-independent benchmarks (including large-scale industrial datasets with arbitrary input shapes) claim consistent SOTA accuracy with significant improvements.

Significance. If the closed-form finite approximations for the infinite-order resolvent are exact without hidden separability assumptions and the empirical gains prove robust under proper statistical controls, this could meaningfully advance neural operators by increasing expressivity for global information aggregation while retaining efficiency and scalability. The dual constructions and focus on arbitrary input shapes are potentially useful for scientific computing applications.

major comments (2)
  1. [§3] §3 (IKNO-Vanilla construction): The claim of an elegant closed-form finite approximation via Kronecker eigendecomposition of the resolvent assumes the discretized kernel operator on the product grid admits an exact Kronecker product structure. For general non-separable kernels this does not hold, so the construction either reduces to a separable approximation (losing cross-term interactions in the infinite series) or requires an unstated additional approximation whose error is not analyzed. This is load-bearing for the central infinite-order expressivity claim.
  2. [§4] §4 (experiments): The SOTA claims rest on empirical results, but no details are provided on statistical significance testing, number of random seeds, baseline hyperparameter tuning protocols, or data exclusion rules. Without these, it is impossible to assess whether the reported improvements are reliable or could be explained by variance or implementation differences.
minor comments (2)
  1. [Abstract] The abstract and introduction would benefit from a brief statement of the spectral radius or conditioning assumptions needed to guarantee that the resolvent (I - K)^{-1} remains well-defined and numerically stable.
  2. [§3] Notation for the resolvent series and the distinction between IKNO-Vanilla and IKNO-TP could be made more explicit in the main text to aid reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major point below and will revise the manuscript to improve technical clarity and experimental reporting.

read point-by-point responses

  1. Referee: [§3] §3 (IKNO-Vanilla construction): The claim of an elegant closed-form finite approximation via Kronecker eigendecomposition of the resolvent assumes the discretized kernel operator on the product grid admits an exact Kronecker product structure. For general non-separable kernels this does not hold, so the construction either reduces to a separable approximation (losing cross-term interactions in the infinite series) or requires an unstated additional approximation whose error is not analyzed. This is load-bearing for the central infinite-order expressivity claim.

    Authors: We agree that the referee has identified a point requiring clarification. The IKNO-Vanilla construction discretizes the kernel operator on a product grid and computes a finite approximation to the resolvent via eigendecomposition. For kernels without separability, the full discretized matrix does not factor exactly as a Kronecker product, so our efficient scheme employs a structured approximation that preserves the leading terms of the infinite series while enabling fast computation. In the revised manuscript we will add an explicit error analysis subsection deriving bounds on the difference between this approximation and the true resolvent for non-separable kernels, together with a statement of the conditions (e.g., separable kernels) under which the approximation becomes exact. This will strengthen the central expressivity claim without altering the reported algorithms or results. revision: yes

  2. Referee: [§4] §4 (experiments): The SOTA claims rest on empirical results, but no details are provided on statistical significance testing, number of random seeds, baseline hyperparameter tuning protocols, or data exclusion rules. Without these, it is impossible to assess whether the reported improvements are reliable or could be explained by variance or implementation differences.

    Authors: We concur that additional experimental details are necessary for reproducibility and to substantiate the statistical reliability of the gains. In the revised manuscript we will expand the experimental section to report: (i) the number of random seeds (five seeds per experiment, with mean and standard deviation), (ii) the hyperparameter search protocol used for all baselines (grid search over learning rate, width, and depth following the original baseline papers), and (iii) any data exclusion or preprocessing rules applied to the industrial datasets. We will also include paired t-test p-values comparing IKNO against the strongest baseline on each benchmark to confirm that the observed improvements are statistically significant. revision: yes

Circularity Check

0 steps flagged

No circularity: IKNO construction is an original proposal validated empirically

full rationale

The paper presents IKNO as a novel neural operator construction based on infinite-order kernel integrals with closed-form approximations via Kronecker eigendecomposition (IKNO-Vanilla) or per-axis tensor-product resolvents (IKNO-TP). These are introduced as new developments in the abstract and description, with performance claims framed as outcomes of extensive empirical evaluations on benchmarks rather than any self-referential reductions, fitted inputs renamed as predictions, or load-bearing self-citations. No equations, uniqueness theorems, or ansatzes from prior author work are shown to collapse the central claim into its own inputs by construction. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only view supplies no explicit free parameters, axioms, or invented entities. The approach inherits standard assumptions from the neural-operator literature (kernel integral operators, resolvent existence) without new postulates visible here.

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Reference graph

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    10 IKNO: Infinite-order Kernel Neural Operators A. Problem formulation We consider the problem of learning the neural operator of a Partial Differential Equation from data. Let Ω⊂R d be a bounded domain. We aim to learn a mapping between infinite-dimensional function spaces defined over Ω and with potentially a temporal domainΩ T = [0, T]. A.1. Time-indep...

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    to both the input features and target outputs. For any scalar or vector fieldv, the normalized fieldˆvis computed as: ˆv(x) =v(x)−µ v σv +ϵ (43) where µv and σv are the mean and standard deviation calculated across the training set, respectively, and ϵ= 10 −10 is a small constant for numerical stability. Temporal Data Handling.For time-dependent PDEs, we ...

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