Self-explainable Operator Learning for Discovering Spatial Patterns in Functional Data
Authors: Mojgan Alishiri, Amirhossein Arzani
Organizations: 1Scientific Computing and Imaging Institute, University of Utah, Salt Lake City, UT, 84112, United States · 2Mechanical Engineering Department, University of Utah, Salt Lake City, UT, 84112, United States
Abstract
Operator learning has emerged as a powerful tool for modeling complex physical systems in functional spaces. However, their neural network-based architectures make them opaque models, obscuring the reasoning behind their predictions. In this work, we introduce a self-explainable operator learning framework that overcomes this challenge by reformulating operator learning as a linear combination of generalized functional linear models expressed through integral equations. Exploiting the additive decomposability of these integral equations, we divide the input domain into subdomains and compute localized integrals to evaluate the contribution of each region to the final prediction. This decomposition enables direct interpretability where the model explains both inputs and outputs by linking specific input regions to corresponding output patterns, thereby revealing which spatial features drive predictions. We demonstrate the framework on function-to-scalar and function-to-function mappings in fluid flow problems involving blood flow and unsteady aerodynamics. The results show that the operator most often prioritizes regions with strong feature gradients, providing physically meaningful insight into the model's decision-making process. Comparisons with established post-hoc explainability methods demonstrate qualitative agreement while highlighting the key advantage of the proposed approach: explainability is embedded directly within the operator structure itself and does not require an external tool. Therefore, our framework provides a mathematically transparent and physically interpretable approach to uncover relationships within data, fostering trust in machine learning for scientific applications by enabling more informed data-driven analysis of physical systems.
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.
Joshua Stiller, Santo M. A. R. Thies, Felix Czaja +1
Physical systems often exhibit heterogeneous mechanisms, where rapidly evolving dynamics coexist with persistent structures. Capturing such multiscale physical behavior remains challenging for existing neural operators, which typically rely on single dominant inductive bias and therefore couple distinct physical responses into a shared representation. We introduce the Unified Green's Function Framework across domains and propose the Factorized Neural Operators (FaNO), which decompose spectral representations into equivariant dynamic responses and invariant persistent responses, leading to better interpretability and generalization. Mechanistically, we show that the two operator branches spontaneously specialize into distinct physical roles that remain consistent across scales and domains: the equivariant branch captures rapidly varying transient dynamics, whereas the invariant branch extracts coherent persistent structures. This factorized mechanism of FaNO improves prediction accuracy, parameter efficiency and cross-scale generalization across physical systems and domains. In particular, it maintains consistent predictions under long-horizon autoregressive rollout, cross-resolution extrapolation and physical-regime shifts. These findings suggest that scalable physical modeling may benefit from moving beyond single-inductive-bias formulations toward factorized operator representations that better reflect the heterogeneous organization of physical systems, accelerating the reliable deployment of machine learning for scientific computing and discovery.
Learning mappings between infinite-dimensional function spaces, or operator learning, is essential for many machine learning applications. Although transformer-based operators are popular, they often rely on token-wise attention. These methods treat continuous fields as discrete tokens and usually ignore the global functional structure. We introduce \emph{Functional Attention}, which reinterprets attention as a functional correspondence between adaptive bases. Inspired by geometric functional maps, our method replaces softmax affinities with structured linear operators. This yields a compact, generalizable, resolution-invariant representation that explicitly captures global dependencies. Experiments demonstrate that \emph{Functional Attention} can match state-of-the-art performance in many operator learning tasks, including solving PDEs, 3D segmentation, and regression, while remaining robust to varying discretizations. Project page is available at https://github.com/xjffff/FUNCATTN.