Neural Operator Processes for Probabilistic Operator Learning under Partial Observations
2026-06-22 • Machine Learning
Machine Learning
AI summaryⓘ
The authors introduce Neural Operator Processes (NOPs), a new method that helps predict full solution fields from very limited and irregular data, unlike traditional approaches needing complete information. They combine ideas from neural processes and neural operators to handle uncertainty and partial observations effectively. Their experiments show that this approach works well across different problems, especially when the method respects local data structure in complex scenarios. This work helps connect learning operators that predict functions with probabilistic methods that deal with uncertainty.
Neural operatorsFunction spacesPartial differential equations (PDEs)Neural processesEncoder-decoder architectureSparse observationsProbabilistic predictionAttention mechanismsUncertainty quantificationMeta-learning
Authors
Jose Miguel Lara-Rangel, Serge Guillas
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.