🤖 machine learning

OperatorSHAP: Fast and Accurate Shapley Value Estimation for Neural Operators

The paper introduces OperatorSHAP, a grid-agnostic method that enables fast and accurate Shapley value estimation for neural operators by establishing a theoretical framework in function space and demonstrating consistent, transferable explanations across varying resolutions and grid sizes.

Original authors: Joshua Stiller, Santo M. A. R. Thies, Felix Czaja, Eyke Hüllermeier

Published 2026-06-29

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Joshua Stiller, Santo M. A. R. Thies, Felix Czaja, Eyke Hüllermeier

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Problem: The "Black Box" Weather Forecaster

Imagine you have a super-smart AI that predicts the weather or calculates how much weight a bridge can hold. This AI is like a Neural Operator. Unlike normal AI that looks at a fixed grid of pixels (like a photo), this AI is special: it can understand shapes and patterns whether they are drawn on a tiny napkin or a massive billboard. It doesn't care about the grid size; it just understands the flow of the data.

However, there is a catch. When this AI makes a prediction (e.g., "This bridge will hold 5 tons"), nobody knows why. Which part of the input mattered? Was it the wind speed? The shape of the bridge? The temperature?

In the world of AI, we use a tool called Shapley Values to answer this. Think of Shapley Values as a way to split a "prize" (the prediction) fairly among all the "players" (the input data points) based on how much each one contributed.

The Problem:

It's too slow: Calculating the fair split for every single data point takes forever. It's like trying to count every grain of sand on a beach to see which ones are most important.
It's too rigid: Existing fast tools (like FastSHAP) only work if the sand is arranged in a perfect, uniform grid. But in the real world (physics, engineering), data often comes from messy, irregular shapes (like a cylinder with sensors placed randomly). Current tools break when the grid isn't perfect.

The Solution: OperatorSHAP

The authors introduce OperatorSHAP. Think of this as a "universal translator" that can instantly explain the AI's decisions, no matter how messy or irregular the data grid is.

Here is how they did it, broken down into three simple concepts:

1. The "Infinite Player" Game (Aumann–Shapley)

Usually, Shapley values count discrete players (Player 1, Player 2, Player 3). But in physics, space is continuous. You don't have "Point 1" and "Point 2"; you have a smooth flow of air or heat.

The authors realized that instead of counting players, they should treat the input as a continuous fluid. They used a mathematical concept called Aumann–Shapley values, which is like measuring the "density" of importance across a river rather than counting individual fish. This allows them to handle any shape or grid size without getting confused.

2. The "Trained Detective" (Amortized Explainer)

Normally, to explain one prediction, you have to run a massive calculation. That's like hiring a detective to solve one specific crime, then hiring a new detective for the next crime.

OperatorSHAP trains a single "Detective AI" (an explainer model).

Training: They teach this detective by showing it thousands of examples and asking, "If we hide this part of the input, how much does the prediction change?"
The Magic: Once trained, this detective can look at any new input (even on a weird, irregular grid) and instantly spit out the explanation in a single second. It doesn't need to re-calculate from scratch every time.

3. Grid Agnosticism (The "Shape-Shifter")

Most explainers are like a pair of shoes that only fit size 10. If you have a size 8 or a size 12 foot (a different grid resolution), the shoe doesn't fit.

OperatorSHAP is like a pair of magic shoes that stretch to fit any foot. Because it was trained on the "fluid" concept of Aumann–Shapley values, it can take an input with 16 data points, 1,024 data points, or a messy mesh with 5,000 points, and give you a correct explanation without needing to be retrained.

What They Found (The Results)

The paper tested this on various physics problems, like heat flowing through metal, air moving around a cylinder, and water waves.

Speed: OperatorSHAP is orders of magnitude faster than the old methods. While other methods might take hours to explain a single complex simulation, OperatorSHAP does it in a fraction of a second.
Accuracy: Even though it's fast, it's just as accurate as the slow, perfect methods.
Flexibility: They trained it on one grid size, and it worked perfectly on much larger or smaller grids without any extra work. It even worked on the "cylinder flow" dataset where the sensors were placed on irregular, shifting grids.

The Bottom Line

The paper claims that OperatorSHAP is a new way to make complex physics-AI models understandable. It combines a deep mathematical theory (treating space as a continuous flow) with a practical training trick (teaching one AI to explain all grids).

This means that in fields like structural engineering or weather forecasting, where data is often messy and decisions are critical, we can now get fast, reliable, and grid-independent explanations for why an AI made a specific prediction.

Technical Summary: OperatorSHAP

Problem Statement

Neural operators are designed to learn mappings between function spaces, offering grid-agnostic flexibility essential for physical applications involving partial differential equations (PDEs). These models can accept inputs sampled on irregular grids or varying resolutions (e.g., 128 vs. 1,024 points) without retraining. However, interpreting these models remains a significant challenge.

In safety-critical physical domains (e.g., structural load assessment, weather warnings), understanding model predictions is vital. While Shapley values provide a theoretically sound attribution method satisfying desirable axioms, their computational cost during inference is prohibitive. Existing amortized explainers, such as FastSHAP, are limited to homogeneous inputs (fixed, regular grids). They cannot directly handle the heterogeneous grids and irregular geometries common in scientific data (e.g., unstructured meshes from sensor placements or adaptive refinement). Furthermore, defining amortized Shapley values on heterogeneous grids is non-trivial because attributions depend on both the total player count and the spatial distances between neighboring players.

Methodology

The authors introduce OperatorSHAP, a grid-agnostic attribution method and training procedure that extends amortized Shapley estimation to neural operators. The approach bridges the gap between discrete Shapley values and continuous function spaces through the following steps:

1. Theoretical Framework: Aumann–Shapley Values

The paper establishes a theoretical foundation by connecting neural operator games to the Aumann–Shapley value, an extension of Shapley values to games with infinite players (non-atomic measure spaces).

Smooth Operator Games: The authors define a "smooth operator game" where the input space is a Sobolev space $H^s$ (with $s > d/2$ ). By using smooth masks (bump functions) to partition the spatial domain, they ensure the operator is Fréchet differentiable.
pNA Class: They prove that these smooth operator games belong to the pNA class (games with finite-dimensional representations via non-atomic measures). This guarantees the existence of a unique Aumann–Shapley value, calculable via a diagonal formula involving the Fréchet derivative.
Connection to Integrated Gradients: The derived Aumann–Shapley value is shown to be equivalent to a function-valued Integrated Gradient (IG) formula, where the attribution is the integral of the directional derivative along a path from a baseline to the input.

2. Amortized Grid-Agnostic Explainer

To make these values computationally feasible, the authors propose training a neural operator explainer (e.g., an FNO) to approximate the Aumann–Shapley density.

Training Objective: The explainer $\Phi_\theta$ is trained using a FastSHAP-like loss. Instead of calculating Shapley values for each sample individually, the explainer outputs a continuous attribution field $\phi$ for a given input $u$ .
Discretization: During training, the continuous output field is integrated over specific coalition regions (partitions) to produce discrete values. The loss minimizes the weighted Mean Squared Error (MSE) between these integrated values and the true coalition values $v(S)$ .
Grid Agnosticism: Because the explainer is a neural operator, it learns a mapping from function spaces to function spaces. It can be trained on samples with varying grid resolutions and irregular meshes. Once trained, the same model can generate attributions for inputs on any grid resolution without retraining, effectively learning the underlying Aumann–Shapley density.

Key Contributions

Theoretical Foundations: The paper provides a rigorous mathematical framework for attributions in function spaces, demonstrating that smooth neural operator games admit well-defined Aumann–Shapley values. It connects these values to the diagonal formula and Integrated Gradients.
Amortized Grid-Agnostic Explainer: The authors propose a training procedure (Algorithm 1) that allows for the training of FastSHAP-like explainers on irregular and heterogeneous grids. This enables the generation of Shapley-like attributions for neural operators across arbitrary resolutions.
Empirical Validation: The method is validated on nine PDE datasets (including 1D/2D structured grids and unstructured MeshGraphNets benchmarks). The results show that OperatorSHAP produces attributions consistent with state-of-the-art discrete Shapley approximators (RegressionMSR) across resolutions, while offering significantly faster inference speeds.

Experimental Results

The authors evaluated OperatorSHAP against KernelSHAP, RegressionMSR, and Integrated Gradients on various PDEs (Burgers, Heat, Shallow Water, etc.) and MeshGraphNets datasets.

Attribution Quality: Across resolutions ranging from $n=16$ to $n=1024$ , OperatorSHAP achieved Pearson correlations and Normalized RMSE (NRMSE) comparable to or better than RegressionMSR. Notably, while classical methods required re-running at every resolution, OperatorSHAP used a single trained model for all resolutions.
Inference Speed: OperatorSHAP is orders of magnitude faster than baselines. It returns full attributions in a single forward pass with zero per-sample budget, whereas baselines require thousands of backbone evaluations per sample.
Transferability: The trained explainer successfully transferred across grid sizes without retraining, maintaining high fidelity to the reference attributions even when the input grid structure changed (e.g., from 64 to 1024 points).
Limitations: The authors note that while amortization saves inference time, the initial training cost is significant. Additionally, for very low player counts, classical methods with low budgets may outperform OperatorSHAP, but this gap widens significantly as resolution increases.

Significance

The paper claims that OperatorSHAP enables the practical application of Shapley value explanations in physical sciences, where data is inherently heterogeneous and grid-agnostic. By connecting the discrete Shapley value to the continuous Aumann–Shapley value, the authors provide a theoretical justification for training explainers on function spaces. This allows for the interpretation of neural operators on irregular meshes and varying resolutions—a capability previously unavailable to amortized explainers. The work suggests that for industrial applications requiring high-throughput, high-stakes decision support (e.g., real-time weather warnings or structural monitoring), amortized explainers like OperatorSHAP are a necessary step toward scalable and reliable AI interpretability in dynamical systems.