On the Benefits of Active Data Collection in Operator Learning

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.