HiRefPOU：用分区混合专家网络改进 PDE 算子学习
Localized Operator Learning with Adaptive Partition-of-Unity Mixture-of-Expert Networks
这篇论文把混合专家思路搬进了 DeepONet 和 FNO，专门对付尖界面、异质系数这类难解的 PDE，误差比基线低不少，做科学计算的可以看看。
论文提出 HiRefPOU，一种基于 partition-of-unity（POU）的层次化混合专家架构，通过几何感知门控网络生成平滑空间分区来组合局部专家网络。该架构以残差嵌套的父子分区组织 DeepONet 的局部表示，同时保持全局连续性。同样的 POU 原理也可嵌入 Fourier Neural Operator，在不改动谱层的前提下引入空间自适应性。在异质 Darcy 和 reaction-diffusion 基准上，HiRefPOU 的误差显著低于全局 DeepONet 和静态 POU-MoE 基线。学到的分区可解释，且与解的界面和剧烈变化区域对齐。
Localized Operator Learning with Adaptive Partition-of-Unity Mixture-of-Expert Networks
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.