From Points to Edges: Edge-Conditioned Spectral Operators for Physics-Sensitive PDE Learning

2026-08-07

AI Summary

Existing spectral neural operators struggle to effectively capture the local structure of physics-sensitive regions in partial differential equations (PDEs), such as abrupt material interfaces. To address this limitation, this work proposes the Edge-conditioned Spectral Operator (ESO), which innovatively incorporates edge-level local variation information to modulate global spectral mixing via a Pairwise-Variation Modal Mixer (PVMM). Furthermore, a Physics-Aware Reweighting (PAR) mechanism is introduced to adaptively enhance representations in critical regions. The proposed method achieves state-of-the-art performance across nine PDE benchmarks, significantly reducing solution errors in areas characterized by coefficient discontinuities and high-gradient flows.

Abstract

Neural operators have become a central tool for solving partial differential equations (PDEs), with spectral operators offering efficient global mixing across spatial locations. However, many PDEs contain physics-sensitive local structures that are critical to the underlying physical behavior. For example, in Darcy flow, local material interfaces are often reflected by sharp changes in the permeability field and can strongly influence the solution. Existing spectral operators primarily adapt modal mixing based on center-point representations, making them insufficiently responsive to such localized structural variations. We propose the Edge-Conditioned Spectral Operator (ESO), a novel spectral operator framework that modulates global spectral mixing using local edge-wise variations. By incorporating the Pairwise-Variation Modal Mixer (PVMM) to inject local edge information into spectral mode selection, ESO preserves the global approximation capability of spectral neural operators while enabling the learned kernel to adapt to physics-sensitive local structures. Furthermore, we introduce a task-adaptive Physics-Aware Reweighting (PAR) that emphasizes physically important regions, identified by taskspecific physical quantities. Across nine PDE benchmarks, ESO consistently achieves state-of-the-art performance. Visual and region-wise analyses further demonstrate that ESO reduces solution errors near coefficient jumps, high-gradient flow structures, and other physically sensitive regions. The code is available at https://github.com/Tanpig-X/ESO.

Problem

• spectral operators
• physics-sensitive structures
• partial differential equations
• local variations
• neural operators

Innovation

• Edge-Conditioned Spectral Operator
• Neural Operators
• Physics-Sensitive PDEs
• Spectral Mixing
• Local Edge Information