Operator Learning of Lipschitz Operators: An Information-Theoretic Perspective

TL;DR

This paper uses information theory to establish fundamental lower bounds on the size of neural operators needed to approximate Lipschitz operators, revealing exponential complexity requirements regardless of architecture or activation functions.

Contribution

It provides the first information-theoretic lower bounds on the parametric complexity of neural operators approximating Lipschitz operators, highlighting fundamental limitations.

Findings

Neural operator architectures require exponentially large size for a given approximation accuracy.

Lower bounds on metric entropy are established for uniform and probabilistic approximation settings.

Results are architecture-agnostic, applying to all neural operators approximating Lipschitz operators.

Abstract

Operator learning based on neural operators has emerged as a promising paradigm for the data-driven approximation of operators, mapping between infinite-dimensional Banach spaces. Despite significant empirical progress, our theoretical understanding regarding the efficiency of these approximations remains incomplete. This work addresses the parametric complexity of neural operator approximations for the general class of Lipschitz continuous operators. Motivated by recent findings on the limitations of specific architectures, termed curse of parametric complexity, we here adopt an information-theoretic perspective. Our main contribution establishes lower bounds on the metric entropy of Lipschitz operators in two approximation settings; uniform approximation over a compact set of input functions, and approximation in expectation, with input functions drawn from a probability measure. It is shown that these entropy bounds imply that, regardless of the activation function used, neural operator architectures attaining an approximation accuracy $\epsilon$ must have a size that is exponentially large in $\epsilon^{-1}$. The size of architectures is here measured by counting the number of encoded bits necessary to store the given model in computational memory. The results of this work elucidate fundamental trade-offs and limitations in operator learning.