Operator Learning of Lipschitz Operators: An Information-Theoretic Perspective
Samuel Lanthaler
TL;DR
The paper investigates the parametric complexity of neural operator approximations for the broad class of Lipschitz operators between infinite-dimensional spaces. By framing operator learning through an information-theoretic lens, it connects bit-encoding length to Kolmogorov metric entropy and derives two rigorous minimax lower bounds: in uniform and in-expectation settings, the number of bits needed to achieve $\epsilon$-accuracy grows at least exponentially in $\epsilon^{-1}$ (and $\epsilon^{-1/(\alpha+1)}$ under eigenvalue decay). It further shows that, for generic Lipschitz operators, no sequence of bit-encoded neural operators can beat a logarithmic rate of approximation, and demonstrates a concrete curse for Fourier neural operators under mild weight-growth assumptions. These results illuminate fundamental trade-offs and limitations in operator learning and point to relying on structural properties beyond Lipschitz regularity to obtain efficient approximation. The findings have implications for the design and analysis of operator-learning architectures in high-dimensional settings, including guidance on when bit-precision limits memory and when architectural choices may or may not circumvent the curse.
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 $ε$ must have a size that is exponentially large in $ε^{-1}$. 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.