Neural Operator Processes for Probabilistic Operator Learning under Partial Observations

2026-06-22

AI Summary

This work addresses the challenge of reliable prediction and uncertainty quantification in function space mapping under sparse and irregular observations. The authors propose a unified encoder–decoder architecture that integrates neural processes with neural operators, introducing for the first time the conditional mechanism of neural processes into the neural operator framework to enable uncertainty-aware operator learning sensitive to local geometry. Two conditioning strategies—convolutional pooling summaries and query-aligned attention—are employed alongside stochastic latent variables and local geometric pathways, and the model is trained on both function regression and partial differential equation (PDE) tasks. Experiments demonstrate that the method achieves efficient learning under sparse conditions across multiple PDE benchmarks, matching the performance of dense-grid methods in specific settings and highlighting the critical role of preserving local contextual geometry, particularly in non-periodic domains.

Abstract

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. We introduce Neural Operator Processes (NOPs), a framework that unifies neural-process conditioning with neural-operator decoding to predict full output fields from limited context. NOPs condition on sparse joint input-output observations and support deterministic and probabilistic prediction within a shared encoder-decoder architecture. We study two conditioning strategies, convolutional pooled summaries and query-aligned attention, and analyze how their interaction with latent stochastic variables depends on PDE geometry. Across function regression and three PDE benchmarks, we find that sparse conditional operator learning is viable and can match dense-grid behavior in several regimes, that preserving local context-query geometry is essential in non-periodic settings but less so in spectrally smooth periodic regimes, and that uncertainty-aware operator learning succeeds when latent conditioning complements rather than overwrites the local geometric pathway. These results provide a basis for probabilistic operator learning under partial observations and help bridge operator learning and probabilistic meta-learning in function space.

Problem

• neural operators
• partial observations
• probabilistic learning
• sparse data
• function space

Innovation

• Neural Operator Processes
• probabilistic operator learning
• partial observations
• neural processes
• function space meta-learning