# On the Benefits of Active Data Collection in Operator Learning

arXiv: https://arxiv.org/abs/2410.19725 · HTML: https://mlsanity.com/paper/2410.19725/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Oct 2024.

## Built on by (most cited first, 5 of 5)

- [Operator Learning: A Statistical Perspective](https://mlsanity.com/paper/2504.03503/) (2025)
- [Learning where to learn: Training data distribution optimization for scientific machine learning](https://mlsanity.com/paper/2505.21626/) (2025)
- [Learning Where to Simulate: Generative Active Sampling for Online PDE Surrogate Training](https://mlsanity.com/paper/2606.09949/) (2026)
- [Operator Learning for Schr"odinger Equation: Unitarity, Error Bounds, and Time Generalization](https://mlsanity.com/paper/2505.18288/) (2025)
- [Sequential operator learning under dependent data](https://mlsanity.com/paper/2608.24426/) (2026)

## Abstract (arXiv)

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 (~ n-1). In fact, for our setting, we show a 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.