从点到边：面向物理敏感型偏微分方程学习的边条件谱算子
From Points to Edges: Edge-Conditioned Spectral Operators for Physics-Sensitive PDE Learning
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中文总结 AI 辅助
该研究针对现有谱算子无法响应PDE局部结构变化的问题，提出边条件谱算子（ESO），结合PVMM与PAR方法，在9个PDE基准测试中达到最先进性能，降低了物理敏感区域的解误差。
中文摘要 AI 辅助
神经算子已成为求解偏微分方程（PDE）的核心工具，其中谱算子可实现空间位置间高效的全局混合。然而，许多PDE包含对底层物理行为至关重要的物理敏感局部结构，例如在达西流中，局部材料界面通常表现为渗透率场的急剧变化，且会对解产生强烈影响。现有谱算子主要基于中心点表示调整模态混合，无法充分响应此类局部结构变化。我们提出边条件谱算子（ESO），这是一种新型谱算子框架，通过局部边方向变化调制全局谱混合。ESO引入成对变化模态混合器（PVMM）将局部边信息注入谱模态选择，在保留谱神经算子全局近似能力的同时，使学习到的核能适配物理敏感局部结构。此外，我们提出任务自适应物理感知重加权（PAR），该方法会强调由任务特定物理量识别的物理重要区域。在9个PDE基准测试中，ESO始终达到最先进性能；可视化和区域分析进一步表明，ESO可降低系数跃变、高梯度流结构及其他物理敏感区域附近的解误差。代码可在指定URL获取。
英文摘要
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.
