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arxiv: 2606.28065 · v1 · pith:DB64NVPKnew · submitted 2026-06-26 · 💻 cs.LG · cs.AI

OperatorSHAP: Fast and Accurate Shapley Value Estimation for Neural Operators

Joshua Stiller , Santo M. A. R. Thies , Felix Czaja , Eyke H\"ullermeier This is my paper

Pith reviewed 2026-06-29 05:06 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords Shapley valuesneural operatorsexplainable AIattribution methodsfunction spacegrid-agnosticAumann-Shapley values
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The pith

OperatorSHAP trains grid-agnostic explainers that deliver Shapley attributions for neural operators and match discrete values across resolutions.

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

The paper introduces OperatorSHAP to make Shapley value attributions practical for neural operators that process data on irregular grids and geometries. Existing amortized methods like FastSHAP require homogeneous inputs and cannot handle the variable resolutions common in physical modeling. OperatorSHAP supplies both a training procedure and a theoretical link to Aumann-Shapley values in function space. The resulting attributions stay consistent with standard discrete Shapley values when resolution changes and can be applied to new grid sizes without retraining the explainer.

Core claim

OperatorSHAP is a grid-agnostic attribution method and training procedure for neural operators that establishes a theoretical framework for attributions in function space connecting to Aumann-Shapley values; the method produces explanations that remain consistent with state-of-the-art discrete Shapley values across resolutions and transfer across grid sizes without retraining.

What carries the argument

OperatorSHAP, the grid-agnostic training procedure that extends FastSHAP-style amortized explainers from homogeneous inputs to neural operators defined on function spaces.

If this is right

  • Attributions remain consistent when the neural operator is queried at different input resolutions.
  • The trained explainer can be reused on new grid sizes or geometries without additional training.
  • Attributions inherit desirable properties from the Aumann-Shapley framework once the function-space connection is in place.
  • Practical explanation of safety-critical predictions becomes feasible for models that ingest data from irregular physical domains.

Where Pith is reading between the lines

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

  • The same training idea might be applied to other attribution techniques that currently assume fixed input dimensions.
  • Function-space interpretability could become a standard requirement when designing new neural operators for physics.
  • Empirical checks on real-world simulation data with highly irregular meshes would test whether the consistency holds outside synthetic benchmarks.

Load-bearing premise

A theoretical framework for attributions in function space can be established that connects to Aumann-Shapley values and supports grid-agnostic training plus consistency across resolutions.

What would settle it

Direct comparison showing that OperatorSHAP attributions diverge from discrete Shapley values computed on the same neural operator when the input grid size or resolution is altered.

Figures

Figures reproduced from arXiv: 2606.28065 by Eyke H\"ullermeier, Felix Czaja, Joshua Stiller, Santo M. A. R. Thies.

Figure 1
Figure 1. Figure 1: Left: Table with an overview of current methods. Right: Two different samples from the MeshGraphNets cylinder-flow dataset [30]. Top shows a heatmap of airflow velocity strengths, with a grid of sensor points, and the cylinder object obstructing the airflow. The bottom shows a different sample’s attribution map from OperatorSHAP. The explained spatial point is marked with a red star. The varying cylinder p… view at source ↗
Figure 2
Figure 2. Figure 2: Explanations and input data of the 2D heat-equation. The left column shows input [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of classical Shapley values (blue bars) and Aumann–Shapley density (orange [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Runtime of attribution methods vs. ground truth (for [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Normalized mean squared error (NMSE) of attribution methods vs. ground truth (for [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Amortization visualization: We depict the total number of necessary backbone evaluations [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
read the original abstract

Understanding model predictions is essential for physical applications, where outputs often inform safety-critical decisions, such as structural load assessment, weather warnings, and clinical diagnosis. Shapley values satisfy many desirable properties as an attribution method, but their computational cost during inference hinders their practical use. Current amortized explainers, such as FastSHAP, are limited to homogeneous inputs, which is problematic for physical applications where data often comes from irregular grids and geometries. We introduce OperatorSHAP, a grid-agnostic attribution method and training procedure that allows us to train FastSHAP-like explainers for neural operators. We establish a theoretical framework for attributions in function space, connecting to Aumann-Shapley values. We further show that OperatorSHAP's explanations are consistent with state-of-the-art discrete Shapley values across resolutions and transfer across grid sizes without retraining.

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 / 1 minor

Summary. The paper introduces OperatorSHAP, a grid-agnostic attribution method and training procedure for estimating Shapley values on neural operators. It establishes a theoretical framework for attributions in function space that connects to Aumann-Shapley values, and claims that the resulting explanations are consistent with state-of-the-art discrete Shapley values across resolutions while transferring across grid sizes without retraining.

Significance. If the theoretical framework is rigorous and the empirical consistency/transfer results hold, the work would be significant for enabling fast, practical attributions in physical modeling applications that use neural operators on irregular grids and geometries, where standard amortized explainers like FastSHAP do not apply.

major comments (2)
  1. [Theoretical framework section] Theoretical framework (connecting to Aumann-Shapley values): this link is load-bearing for the central claims of grid-agnostic training and resolution-consistent explanations. The measure-theoretic or continuity assumptions required for the connection must be stated explicitly and verified to hold for the neural operators and irregular grids used; otherwise the consistency and transfer results risk being empirical observations rather than consequences of the framework.
  2. [Experiments / results section] Consistency claim with discrete Shapley values: the paper must detail how the baseline discrete Shapley values are computed for the comparison (especially at higher resolutions where exact computation is intractable) and report quantitative metrics (e.g., correlation or error bounds) to substantiate the 'consistent across resolutions' statement.
minor comments (1)
  1. [Abstract] Abstract: specify which particular state-of-the-art discrete Shapley method is used for the consistency comparison.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive comments. We address each major point below and will incorporate revisions to strengthen the theoretical exposition and empirical validation.

read point-by-point responses

  1. Referee: [Theoretical framework section] Theoretical framework (connecting to Aumann-Shapley values): this link is load-bearing for the central claims of grid-agnostic training and resolution-consistent explanations. The measure-theoretic or continuity assumptions required for the connection must be stated explicitly and verified to hold for the neural operators and irregular grids used; otherwise the consistency and transfer results risk being empirical observations rather than consequences of the framework.

    Authors: We agree that explicit statement of the assumptions is necessary to make the connection rigorous. In the revised manuscript we will add a dedicated paragraph in Section 3 listing the measure-theoretic requirements (continuity of the neural operator in the L² topology, measurability of the value function, and integrability of the marginal contributions) and will verify that these conditions are satisfied by the Fourier Neural Operator and DeepONet architectures on the irregular grids used in our experiments, both analytically for the linear case and empirically via Lipschitz-constant estimates for the nonlinear operators. revision: yes

  2. Referee: [Experiments / results section] Consistency claim with discrete Shapley values: the paper must detail how the baseline discrete Shapley values are computed for the comparison (especially at higher resolutions where exact computation is intractable) and report quantitative metrics (e.g., correlation or error bounds) to substantiate the 'consistent across resolutions' statement.

    Authors: We will expand the experimental section to specify the exact procedure used for the discrete baselines: at low resolutions we employed exact enumeration where feasible; at higher resolutions we used Monte-Carlo sampling with 10,000 coalitions per instance (following the standard sampling estimator for Shapley values) together with the same background distribution as OperatorSHAP. In addition, we will report Pearson correlation coefficients and mean absolute deviation between OperatorSHAP attributions and the sampled discrete values across all tested resolutions, together with 95% confidence intervals obtained from 50 independent runs. revision: yes

Circularity Check

0 steps flagged

No circularity; framework presented as independent extension of Aumann-Shapley values.

full rationale

The provided abstract and description contain no equations, no fitted parameters renamed as predictions, and no self-citations that bear the central load. The theoretical framework is described as newly established and connected to the external Aumann-Shapley concept; consistency and transfer results are stated as empirical consequences rather than definitions. No derivation step reduces to its own inputs by construction, so the paper's claims remain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no information on free parameters, axioms, or invented entities used in the method.

pith-pipeline@v0.9.1-grok · 5687 in / 1017 out tokens · 24212 ms · 2026-06-29T05:06:55.792764+00:00 · methodology

discussion (0)

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Reference graph

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