Data Complexity Estimates for Operator Learning

TL;DR

This paper investigates the data requirements for operator learning, establishing theoretical bounds and demonstrating that parametric efficiency can lead to data efficiency, especially with neural operators like FNO.

Contribution

It derives lower bounds on data complexity for general operators and shows that parametric efficiency implies data efficiency for certain neural operator classes.

Findings

Lower bounds on n-widths demonstrate exponential data requirements for general classes.

Efficient parametric approximation with FNO leads to corresponding data efficiency.

Data complexity is fundamentally linked to the parametric complexity of the operator class.

Abstract

Operator learning has emerged as a new paradigm for the data-driven approximation of nonlinear operators. Despite its empirical success, the theoretical underpinnings governing the conditions for efficient operator learning remain incomplete. The present work develops theory to study the data complexity of operator learning, complementing existing research on the parametric complexity. We investigate the fundamental question: How many input/output samples are needed in operator learning to achieve a desired accuracy $\epsilon$? This question is addressed from the point of view of $n$-widths, and this work makes two key contributions. The first contribution is to derive lower bounds on $n$-widths for general classes of Lipschitz and Fr\'echet differentiable operators. These bounds rigorously demonstrate a ``curse of data-complexity'', revealing that learning on such general classes requires a sample size exponential in the inverse of the desired accuracy $\epsilon$. The second contribution of this work is to show that ``parametric efficiency'' implies ``data efficiency''; using the Fourier neural operator (FNO) as a case study, we show rigorously that on a narrower class of operators, efficiently approximated by FNO in terms of the number of tunable parameters, efficient operator learning is attainable in data complexity as well. Specifically, we show that if only an algebraically increasing number of tunable parameters is needed to reach a desired approximation accuracy, then an algebraically bounded number of data samples is also sufficient to achieve the same accuracy.