Is Zero-Shot Super-Resolution Possible in Operator Learning?

2026-05-29

AI Summary

This work investigates the theoretical feasibility of neural operators for zero-shot super-resolution, revealing that the task may remain fundamentally infeasible due to information loss—even under idealized conditions where the input function is continuous and the ground-truth mapping is a simple linear operator. Leveraging tools from information theory and functional analysis, the study establishes the first rigorous information-theoretic impossibility result for this setting. It further identifies Hölder smoothness of the output function as a sufficient condition for successful zero-shot super-resolution and derives corresponding generalization error bounds. Numerical experiments corroborate the theoretical predictions, demonstrating both failure modes and achievable regimes, thereby addressing a critical gap in the literature by providing much-needed theoretical foundations for this emerging paradigm.

Abstract

Neural operators are often reported to exhibit zero-shot super-resolution, a phenomenon in which a model trained on coarse grids produces accurate predictions on finer testing grids without additional retraining. Despite strong empirical evidence, the theoretical foundations of this phenomenon remain unclear. In this work, we provide a systematic theoretical study of zero-shot super-resolution in operator learning. We first show that zero-shot super-resolution can be information-theoretically impossible even in benign settings such as when the input functions are available over the entire continuum and the ground truth is a simple rank-one linear operator. We then identify H{\" o}lder smoothness of the output functions as a sufficient condition for zero-shot super-resolution and derive corresponding generalization bounds. Finally, we also validate the identified failure modes through experimental results.

Problem

• zero-shot super-resolution
• neural operators
• operator learning
• generalization
• theoretical foundations

Innovation

• zero-shot super-resolution
• neural operators
• Hölder smoothness
• generalization bounds
• operator learning