Thompson Sampling in Function Spaces via Neural Operators
Abstract
We propose an extension of Thompson sampling to optimization problems over function spaces where the objective is a known functional of an unknown operator's output. We assume that queries to the operator (such as running a high-fidelity simulator or physical experiment) are costly, while functional evaluations on the operator's output are inexpensive. Our algorithm employs a sample-then-optimize approach using neural operator surrogates. This strategy avoids explicit uncertainty quantification by treating trained neural operators as approximate samples from a Gaussian process (GP) posterior. We derive regret bounds and theoretical results connecting neural operators with GPs in infinite-dimensional settings. Experiments benchmark our method against other Bayesian optimization baselines on functional optimization tasks involving partial differential equations of physical systems, demonstrating better sample efficiency and significant performance gains.
Cite
@article{arxiv.2506.21894,
title = {Thompson Sampling in Function Spaces via Neural Operators},
author = {Rafael Oliveira and Xuesong Wang and Kian Ming A. Chai and Edwin V. Bonilla},
journal= {arXiv preprint arXiv:2506.21894},
year = {2026}
}
Comments
Final revision to appear at NeurIPS 2025 proceedings, expanded proof of Proposition 2, added Remark 2 on sublinear information gain, and revised discussion at the end of Appendix C.4