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.
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Cite this article
APA 7
Cooley, M., Sharma, R., Zhe, S., Kirby, R. M., & Shankar, V. (2026). Localized Operator Learning with Adaptive Partition-of-Unity Mixture-of-Expert Networks. https://omanscience.com/en/articles/localized-operator-learning-with-adaptive-partition-of-unity-mixture-of-expert-networks
MLA 9
Cooley, Madison, et al. "Localized Operator Learning with Adaptive Partition-of-Unity Mixture-of-Expert Networks." https://omanscience.com/en/articles/localized-operator-learning-with-adaptive-partition-of-unity-mixture-of-expert-networks.
Chicago (author–date)
Cooley, Madison, Ramansh Sharma, Shandian Zhe, Robert M. Kirby, and Varun Shankar. 2026. "Localized Operator Learning with Adaptive Partition-of-Unity Mixture-of-Expert Networks." https://omanscience.com/en/articles/localized-operator-learning-with-adaptive-partition-of-unity-mixture-of-expert-networks.
Harvard
Cooley, M., Sharma, R., Zhe, S., Kirby, R. M. and Shankar, V. (2026) 'Localized Operator Learning with Adaptive Partition-of-Unity Mixture-of-Expert Networks', Available at: https://omanscience.com/en/articles/localized-operator-learning-with-adaptive-partition-of-unity-mixture-of-expert-networks.
Vancouver
Cooley M, Sharma R, Zhe S, Kirby RM, Shankar V. Localized Operator Learning with Adaptive Partition-of-Unity Mixture-of-Expert Networks. https://omanscience.com/en/articles/localized-operator-learning-with-adaptive-partition-of-unity-mixture-of-expert-networks
IEEE
M. Cooley, R. Sharma, S. Zhe, R. M. Kirby, and V. Shankar, "Localized Operator Learning with Adaptive Partition-of-Unity Mixture-of-Expert Networks," https://omanscience.com/en/articles/localized-operator-learning-with-adaptive-partition-of-unity-mixture-of-expert-networks.