Thompson Sampling in Function Spaces via Neural Operators

TL;DR

This paper extends Thompson sampling to optimize over function spaces using neural operators, achieving improved sample efficiency and performance in PDE-based tasks by leveraging neural surrogates as GP approximations.

Contribution

It introduces a neural operator-based Thompson sampling method for functional optimization, with theoretical regret bounds and practical advantages over existing Bayesian optimization techniques.

Findings

Outperforms baseline methods in PDE-based functional optimization tasks.

Achieves better sample efficiency and accuracy in experiments.

Provides theoretical connections between neural operators and Gaussian processes in infinite dimensions.

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.