We investigate active data collection strategies for operator learning when the target operator is linear and the input functions are drawn from a mean-zero stochastic process with continuous covariance kernels. With an active data collection strategy, we establish an error convergence rate in terms of the decay rate of the eigenvalues of the covariance kernel. Thus, with sufficiently rapid eigenvalue decay of the covariance kernels, arbitrarily fast error convergence rates can be achieved. This contrasts with the passive (i.i.d.) data collection strategies, where the convergence rate is never faster than $\sim n^{-1}$. In fact, for our setting, we establish a \emph{non-vanishing} lower bound for any passive data collection strategy, regardless of the eigenvalues decay rate of the covariance kernel. Overall, our results show the benefit of active over passive data collection strategies in operator learning.

翻译：本文研究了当目标算子为线性且输入函数服从具有连续协方差核的零均值随机过程时，算子学习中的主动数据收集策略。通过采用主动数据收集策略，我们建立了以协方差核特征值衰减率为基准的误差收敛速率。因此，当协方差核的特征值衰减足够快时，可以实现任意快的误差收敛速率。这与被动（独立同分布）数据收集策略形成鲜明对比，后者的收敛速率永远不会快于 $\sim n^{-1}$。事实上，在我们的设定下，我们证明了对于任何被动数据收集策略，无论协方差核的特征值衰减率如何，都存在一个非零下界。总体而言，我们的结果展示了在算子学习中主动数据收集策略相对于被动策略的优势。