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Neural Operator Processes for Probabilistic Operator Learning under Partial Observations

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Neural operators learn mappings between function spaces, but are typically developed with dense input-output training fields and fully observed inputs at inference. Many scientific problems require instead predicting solution fields from sparse, irregular, or partial observations under uncertainty.

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arXiv Machine Learning
Aug 20

Multi-stage neural operator learning with application for convolutions

The paper introduces two multi-stage neural operator learning frameworks—Deep Collocation Neural Operator (DCNO) and Deep Galerkin Neural Operator (DGNO)—for efficiently computing convolution integrals. DCNO is a supervised method that iteratively refines operator approximations by learning residuals from data pairs, while DGNO is an unsupervised approach that uses the weak form of a PDE residual when the operator can be represented by a PDE. Both frameworks build basis operators across multiple training stages, yielding markedly higher accuracy than one-shot learning and achieving near machine‑precision results for convolution problems, with significant efficiency gains for repeated queries or parametric variations.

By Zhiping Mao, Zhenye Wen, Yong Zhang, Xiaofei Zhao