用神经算子扩展汤普森采样，高效优化未知算子的函数空间问题。
Thompson Sampling in Function Spaces via Neural Operators
- 用神经算子做近似采样，跳过复杂不确定性量化
- 在偏微分方程任务中实现更优样本效率与性能
- 适合高成本物理模拟或仿真优化场景
我们提出一种将汤普森采样扩展到函数空间优化的新方法，目标是优化一个已知泛函形式但依赖于未知算子输出的函数。假设对算子的查询（如运行高保真仿真或物理实验）代价高昂，而对算子输出的功能评估则廉价。算法采用先采样后优化的策略，利用神经算子代理模型。该方法通过将训练好的神经算子视为高斯过程后验的近似样本，避免了显式的不确定性量化。我们推导了后悔界，并建立了神经算子与高斯过程在无限维空间中的理论联系。在涉及物理系统偏微分方程的函数优化任务上，实验对比表明，该方法相较于其他贝叶斯优化基线，在样本效率和性能上均有显著提升。
原文摘要 · Abstract (English)
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.
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