Edge-Conditioned Spectral Operators Enhance Physics-Sensitive PDE Learning

Neural operators have become a powerful tool for solving Partial Differential Equations (PDEs), with spectral operators being particularly efficient for global spatial mixing. However, many real-world PDEs involve critical physics-sensitive local structures, such as material interfaces in Darcy flow, which cause sharp changes in physical properties and significantly influence solutions. Existing spectral operators primarily adapt their mixing based on central point representations, making them less responsive to these crucial localized variations. To address this, researchers propose the Edge-Conditioned Spectral Operator (ESO), a new spectral operator framework that modulates global spectral mixing using local edge-wise variations. ESO integrates a Pairwise-Variation Modal Mixer (PVMM) to inject local edge information directly into the spectral mode selection process. This allows the learned kernel to adapt to physics-sensitive local structures while retaining the global approximation capabilities inherent to spectral neural operators. Furthermore, the framework introduces Physics-Aware Reweighting (PAR), a task-adaptive mechanism that emphasizes physically important regions identified by specific physical quantities. Comprehensive evaluations across nine PDE benchmarks demonstrate that ESO consistently achieves state-of-the-art performance. Visual and region-wise analyses confirm that ESO significantly reduces solution errors, particularly near coefficient jumps, high-gradient flow structures, and other physically sensitive areas, making it a more accurate and robust tool for scientific machine learning.