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OperatorSHAP: Fast and Accurate Shapley Value Estimation for Neural Operators

NeurIPS 2026 · Learning theory and algorithms › Prediction and decision losses · approach: Shapley value attribution · 0 citations (OpenAlex)

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Proposes OperatorSHAP, a grid-agnostic amortized Shapley explainer for neural operators, trained with a FastSHAP-style loss and tied theoretically to Aumann-Shapley values via pNA games.

Problem

Shapley-value attributions for neural operators are expensive at inference and existing amortized explainers (FastSHAP) only handle fixed-dimensional homogeneous inputs, not irregular or varying grids common in physical simulations.

Approach

Treat spatial points as infinite players (Aumann-Shapley), define smooth operator games that lie in pNA so the value exists, and train an FNO-based explainer that outputs a continuous attribution density, coarsened by integration over partition cells to match discrete Shapley values at any resolution.

Key results

• On Burgers 1D at n=1024, OperatorSHAP reaches Pearson 0.907 and NRMSE 0.472 vs. IG 0.664/0.853 (RegressionMSR 0.212/1.386 at that resolution); OperatorSHAP has higher faithfulness R^2 across most Burgers resolutions (e.g. 0.865-0.882 vs RegressionMSR 0.662-0.892).
• On Heat 2D at n=256, OperatorSHAP achieves R^2 1.000 and Pearson 0.993 with NRMSE 0.123, outperforming RegressionMSR (Pearson 0.711) and IG.
• OperatorSHAP is reported as orders of magnitude faster than all baselines at inference, returning full attributions in a single forward pass with the same model across resolutions.

Surprising findings

• On the MGN cylinder dataset, RegressionMSR and IG have higher R^2 (-0.409 and 0.702) than OperatorSHAP (0.134), and on Shallow Water 2D at n=1024 RegressionMSR beats OperatorSHAP in Pearson and NRMSE.
• Empirically the framework works with the non-smooth ReLU activation despite the theory requiring C^1 smoothness.

Limitations

• Theory restricted to smooth masks and C^1 operators; ReLU and non-smooth cases not covered theoretically.
• Baseline comparisons rely on RegressionMSR as reference without ground truth; NRMSE and Pearson only relative; no alternative baselines tested.
• Training cost of the amortized explainer is high; OperatorSHAP underperforms RegressionMSR on some settings (e.g., MGN cylinder, Shallow Water 1024).

Experimental setup

Datasets: Advection 1D Heat 1D Burgers 1D Allen-Cahn 1D Diffusion-sorption 1D Heat 2D Gray-Scott 2D Shallow water 2D Darcy 2D MeshGraphNets airfoil MeshGraphNets cylinder

Benchmarks: Burgers 1D Heat 2D Shallow Water 2D MGN Cylinder

Metrics: Pearson correlation NRMSE weighted R^2 faithfulness NMSE runtime

Baselines: KernelSHAP RegressionMSR Integrated Gradients exact Shapley

Builds on: FastSHAP Fourier neural operator Aumann-Shapley values integrated gradients KernelSHAP

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