On the Benefits of Active Data Collection in Operator Learning

TL;DR

This paper demonstrates that active data collection strategies significantly outperform passive strategies in operator learning, achieving faster error convergence rates when input functions are drawn from stochastic processes with rapidly decaying eigenvalues.

Contribution

The paper introduces and analyzes active data collection methods for operator learning, showing they can attain arbitrarily fast convergence rates unlike passive strategies.

Findings

Active strategies outperform passive ones in convergence speed.

Error decay rate depends on the eigenvalue decay of the covariance kernel.

Passive strategies have a non-vanishing lower bound on error regardless of kernel decay.

Abstract

We study 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. We can achieve arbitrarily fast error convergence rates with sufficiently rapid eigenvalue decay of the covariance kernels. This contrasts with the passive (i.i.d.) data collection strategies, where the convergence rate is never faster than linear decay ($\sim n^{-1}$). In fact, for our setting, we show 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 data collection strategies in operator learning over their passive counterparts.