发表机构
Scientific Computing Institute; University of Utah; Kahlert School of Computing（科学计算研究所; 犹他大学; 卡勒特计算学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对算子学习在复杂PDE上的困难，提出基于单位分解的HiRefPOU分层混合专家架构，实现局部化学习，提升精度与可解释性。
AI 中文摘要
诸如DeepONet和FNO等算子学习方法在处理具有尖锐界面、非均匀系数和局部多尺度结构的偏微分方程族时常常遇到困难。我们提出了一种用于局部算子学习的单位分解（POU）混合专家框架，其中几何感知的门控网络生成平滑的空间分区，以融合局部专家网络的贡献。我们的主要贡献是HiRefPOU，一种用于DeepONet的残差式分层POU架构，通过嵌套的父子分区组织局部表示，同时保持全局连续性。我们还表明，相同的POU原理可以融入傅里叶神经算子中，在不修改底层谱层的情况下引入空间自适应性。在非均匀达西和反应扩散基准测试中，HiRefPOU实现的误差显著低于全局DeepONet和静态POU-MoE基线，而更广泛的算子学习实验表明，局部化的益处取决于PDE结构和所选的神经算子主干。学习到的分区是可解释的，并与界面和快速解变化区域对齐。这些结果表明，显式的几何局部化可以提高神经算子学习的准确性和可解释性。
英文摘要
Operator learning methods such as DeepONets and FNOs often struggle with PDE families featuring sharp interfaces, heterogeneous coefficients, and localized multiscale structures. We introduce a partition-of-unity (POU) mixture-of-experts framework for localized operator learning, in which geometry-aware gating networks produce smooth spatial partitions which blend the contributions of local expert networks. Our main contribution is HiRefPOU, a residual-style hierarchical POU architecture for DeepONets that organizes localized representations through nested parent-child partitions while preserving global continuity. We also show that the same POU principle can be incorporated into Fourier Neural Operators to introduce spatial adaptivity without modifying the underlying spectral layers. On heterogeneous Darcy and reaction-diffusion benchmarks, HiRefPOU achieves substantially lower error than global DeepONet and static POU-MoE baselines, while the broader operator-learning experiments show that the benefits of localization depend on the PDE structure and the chosen neural-operator backbone. The learned partitions are interpretable and align with interfaces and regions of rapid solution variation. These results show that explicit geometric localization can improve both accuracy and interpretability in neural operator learning.
