Self-explainable Operator Learning for Discovering Spatial Patterns in Functional Data
For scientists and engineers using operator learning in physical systems, this work provides a transparent and interpretable model that builds trust in data-driven analysis, though it is an incremental improvement over existing post-hoc methods.
The paper introduces a self-explainable operator learning framework that reformulates operator learning as a linear combination of generalized functional linear models, enabling direct interpretability by linking input regions to output patterns. Applied to fluid flow problems, the model prioritizes regions with strong feature gradients, providing physically meaningful insights without external tools.
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.