On the Benefits of Active Data Collection in Operator Learning

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 thepassive (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.

In this paper, we study how to efficiently teach machines to predict the behavior of complex systems, such as those described by physical equations, using only a small number of examples. Instead of passively learning from randomly chosen data, we propose an active learning approach where the model carefully selects the most informative inputs. We show, both theoretically and through experiments, that this strategy can significantly reduce the amount of data needed to learn accurately. Our findings suggest that, especially in scientific settings where data is expensive to obtain, actively choosing what to learn from can lead to faster and more efficient modeling.