Localized Operator Learning with Adaptive Partition-of-Unity Mixture-of-Expert Networks

2026-10-03

AI Summary

This study addresses the limited accuracy of neural operators in solving partial differential equations characterized by sharp interfaces, heterogeneous coefficients, and multiscale structures. To this end, it proposes a localized operator learning framework based on a Partition of Unity (POU) Mixture-of-Experts. A geometry-aware gating network generates smooth spatial partitions to fuse local experts, while a novel HiRefPOU hierarchical residual architecture is introduced to achieve nested parent-child partitioning with global continuity. This design is further extended to Fourier Neural Operators to enhance spatial adaptivity. Evaluations on benchmarks such as Darcy flow demonstrate that the proposed method significantly outperforms global baselines. Moreover, the learned partitions exhibit strong interpretability, effectively improving both the accuracy and physical consistency of operator learning.

Abstract

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.

Problem

• Operator learning
• Partial differential equations
• Sharp interfaces
• Heterogeneous coefficients
• Multiscale structures

Innovation

• Partition-of-Unity
• Mixture-of-Experts
• Operator Learning
• Hierarchical Architecture
• Spatial Adaptivity