Thompson Sampling in Function Spaces via Neural Operators

Rafael Oliveira (CSIRO's Data61) · Edwin Bonilla (CSIRO's Data61) · Xuesong Wang (CSIRO's Data61, Australia) · Kian Ming Chai (DSO National Laboratories)

algorithmic strategybayesian optimization function spacesfunctional optimizationgaussian process posteriorinfinite-dimensional settingsneural operator surrogatesoperator outputpartial differential equations performance gains physical systems regret bounds sample efficiencysample-then-optimizethompson sampling uncertainty quantification

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.