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Self-explainable Operator Learning for Discovering Spatial Patterns in Functional Data

This paper introduces a self-explainable operator learning framework that reformulates neural operator predictions as decomposable integral equations to directly link specific input spatial regions to output patterns, thereby providing mathematically transparent and physically interpretable insights into complex fluid dynamics systems without relying on external post-hoc explanation tools.

Original authors: Mojgan Alishiri, Amirhossein Arzani

Published 2026-07-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mojgan Alishiri, Amirhossein Arzani

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

Imagine you have a super-smart computer program that can predict how blood flows through a brain artery or how air moves over a wing. This program is incredibly accurate, but it's a "black box." You put data in, and it gives you an answer, but it won't tell you why it gave that answer. It's like asking a genius chef for a recipe, and they just say, "I mixed some stuff, and it tasted good," without telling you which ingredients were the secret.

Scientists need to know the "why" to trust the computer, especially when dealing with life-or-death situations like heart surgery or airplane design. This paper introduces a new way to build these computer programs so they come with their own "recipe book" built right in.

Here is the simple breakdown of what the authors did:

1. The Problem: The Opaque Black Box

Most modern AI models are like complex magic tricks. They use deep neural networks (layers of digital neurons) to find patterns. They are great at guessing the right answer, but because they are so complicated, no human can look inside and say, "Ah, the computer decided this because the blood flow was fast in this specific corner."

2. The Solution: The "Lego" Approach

The authors built a new type of AI that works differently. Instead of a tangled web of neurons, they built their model using Integral Equations.

Think of this like building a picture out of Lego bricks.

Old AI: A smooth, solid sculpture made of clay. You can't take it apart to see how it was made.
New AI: A sculpture made of distinct, separate Lego bricks. You can take the sculpture apart, look at each brick, and say, "This red brick represents the left side of the input, and this blue brick represents the right side."

Because their model is built on math that allows it to be broken down into a simple sum of parts, the computer can naturally explain itself. It doesn't need a separate tool to figure out what it did; the explanation is part of the math itself.

3. How It Works: The "Four-Quadrant" Test

To show how this works, the authors took a map of a fluid flow (like blood in an artery) and chopped it into four equal squares (like a pizza cut into quarters).

They asked the computer: "If I only give you the data from the top-left square, what part of the answer do you get? What if I only give you the bottom-right square?"

Because their model is built like a sum of Lego bricks, it can calculate the answer for just the top-left square, then just the bottom-right square, and so on.

The Result: They found that for predicting the "maximum stress" on the artery wall, the computer relied almost entirely on the bottom-right square.
The Insight: This tells scientists, "Hey, the computer isn't guessing randomly. It's correctly identifying that the most dangerous part of the blood flow happens in that specific corner."

4. The Trade-Off: Accuracy vs. Clarity

The paper admits a small catch. If you just want the most accurate number possible, the old "black box" models are slightly better. They are like a master chef who can cook a perfect meal but won't tell you the ingredients.

The new "self-explainable" model is like a student chef who is 90% as good at cooking but writes down every single step and ingredient.

The Paper's Claim: The new model is accurate enough to capture the main patterns (the "flavor" of the data) but gives you the "recipe" (the explanation) for free.
The Proof: They tested their new model against the old "black box" models and also against other tools that try to explain black boxes (like SHAP or LIME). Their new model agreed with those tools on where the important parts were, but it did it without needing those extra, complicated tools.

5. Real-World Tests

They tested this on two things:

Blood Flow: Predicting how much stress the blood puts on the walls of a brain aneurysm. They found the model correctly identified that the "fastest, most turbulent" parts of the flow (usually in a specific zone) were the ones driving the stress predictions.
Airplane Wings: Predicting how air moves over a flapping wing. Again, the model broke the wing down into zones and told them exactly which part of the wing's movement mattered most for the lift or drag.

The Bottom Line

This paper presents a new way to build AI for science. Instead of building a "black box" that guesses the answer and then trying to guess why it guessed that, they built a "glass box."

The computer is designed so that its answer is just a sum of contributions from different parts of the input. This means the computer can point to a specific spot on a map and say, "I made this prediction because of what happened here," without needing any extra software to explain itself. It trades a tiny bit of raw guessing power for a huge gain in trust and understanding.

Technical Summary: Self-explainable Operator Learning for Discovering Spatial Patterns in Functional Data

Problem Statement
Operator learning has emerged as a powerful tool for modeling complex physical systems governed by partial differential equations (PDEs) within functional spaces. However, state-of-the-art neural operator architectures, such as Fourier Neural Operators (FNO) and Deep Operator Networks (DeepONet), function as opaque "black-box" universal approximators. Their complex, non-convex training processes and massive parameter counts obscure the reasoning behind their predictions, hindering scientific trust and the discovery of underlying physical laws. While post-hoc explainability methods (e.g., SHAP, LIME, Grad-CAM) exist, they suffer from limitations: they are often model-agnostic external tools that can produce conflicting explanations, lack inherent confidence intervals, and fail to provide simultaneous interpretability at both the input and output levels. There is a critical need for a "by-design" framework that is inherently interpretable, mathematically transparent, and capable of linking specific spatial features in input functions to corresponding patterns in output functions without relying on external attribution tools.

Methodology
The authors propose a self-explainable operator learning framework based on Generalized Functional Linear Models (gFLM). The methodology diverges from standard neural operators by reformulating the learning problem as a linear combination of integral equations with learnable kernels, rather than a deep neural network.

Model Formulation:
- The mapping between input functions $f(\xi)$ and output functions (or scalars) $u(x)$ is expressed as a sum of integral equations:
  $u(x) = \sum_{n,m} w_{n,m} \iint \psi_n(x-\zeta, y-\eta) T_m f(\zeta, \eta) \, d\zeta d\eta$
- Here, $\psi_n$ represents a library of predefined analytical kernel functions, $T_m$ denotes lifting transformations (e.g., polynomial, exponential) to capture non-linearities in a linear feature space, and $w_{n,m}$ are coefficients learned via regularized linear regression.
- Unlike previous work that used L1 regularization for sparsity, this framework employs L2 regularization to ensure training stability and retain all candidate terms, which is essential for the subsequent decomposition analysis.
Self-Explainable Decomposition Strategy:
- The core innovation is a post-training decomposition strategy that exploits the additive nature of the integral operator.
- The input domain is partitioned into $k$ subdomains (specifically $k=4$ equal zones).
- The model computes localized integrals for each subdomain independently. Due to the linearity of the operator, the total output is exactly the sum of these localized contributions.
- To isolate the influence of specific regions, a mean-centering procedure is applied: the expected value of the prediction for each zone (across the dataset) is subtracted from the local prediction. This yields "zonal contributions" ( $\Phi_i$ ) that reveal how deviations in specific input regions drive deviations in the output.
Validation and Baselines:
- The framework is tested on two physics-based datasets: pulsatile blood flow in a brain aneurysm (predicting Wall Shear Stress) and unsteady aerodynamics of a pitching airfoil (predicting drag and lift coefficients).
- Performance is benchmarked against a Fourier Neural Operator (FNO) for accuracy and against three post-hoc methods (Kernel SHAP, Occlusion Sensitivity, Grad-CAM) applied to a trained U-Net for explainability consistency.

Key Contributions

Intrinsic Explainability: The framework provides a closed-form analytical surrogate where explainability is embedded directly into the model structure. It does not require external tools or post-hoc processing to interpret the model's logic.
Dual-Level Interpretability: Unlike most XAI methods that only highlight important input regions, this approach reveals the spatial patterns in the output generated by specific input regions. It explicitly links input subdomains to the formation of output features.
Mathematical Transparency: By utilizing a linear superposition of integral operators with a convex optimization solution, the model ensures that the total prediction can be decomposed exactly into additive, physically localizable contributions.
Quantitative Metrics: The authors introduce a method to quantify the significance of spatial regions using Pearson correlation coefficients between zonal contributions and the total prediction, facilitating the development of standardized XAI metrics.

Results

Accuracy vs. Interpretability Trade-off: As expected, the FNO baseline achieves significantly higher predictive accuracy (lower Mean Absolute Error) than the gFLM model, particularly in complex, high-frequency scenarios. The gFLM model exhibits errors roughly one order of magnitude higher than FNO in some cases.
Qualitative Fidelity: Despite lower numerical accuracy, the gFLM model successfully captures the dominant qualitative spatial patterns and temporal evolution of the physical phenomena.
Physical Insights:
- In the blood flow case, the model identified that Zone 3 (a region with strong velocity gradients) is the primary driver for predicting maximum Wall Shear Stress (WSS), while Zone 1 and Zone 2 dominate the prediction of spatially averaged WSS and divergence metrics, respectively. This aligns with known fluid dynamics principles where velocity gradients drive shear stress.
- In the airfoil case, the model successfully identified distinct spatial regions influencing drag and lift coefficients at different pitching frequencies.
Consistency with Post-hoc Methods: When compared to Kernel SHAP and Occlusion Sensitivity applied to a U-Net, the gFLM framework showed qualitative agreement in identifying the most influential regions (e.g., both methods identified Zone 3 as the dominant driver for WSS). While the ranking of secondary zones differed slightly due to the distinct mechanisms of the methods (perturbation vs. additive decomposition), the convergence on the primary driver validates the reliability of the proposed approach.

Significance and Claims
The paper claims that this framework addresses a critical gap in scientific machine learning by providing a mathematically transparent and physically interpretable approach to operator learning. Its significance lies in:

Fostering Trust: By making the decision-making process of the model simulatable and decomposable, it allows scientists to verify consistency with physical principles, moving ML from a mere predictive surrogate to a potential new scientific model.
Eliminating External Dependencies: It removes the need for unreliable, computationally expensive post-hoc tools, offering a self-contained explanation mechanism.
Revealing Hidden Relationships: It enables the discovery of which specific spatial features in functional data drive predictions, offering insights into the underlying physics (e.g., the relationship between velocity gradients and shear stress) without prior physical constraints.

The authors maintain a modest stance, acknowledging that while the model sacrifices some predictive accuracy compared to black-box neural operators, this trade-off is necessary and beneficial for scientific applications where understanding the "why" behind a prediction is as critical as the prediction itself.